$$ (k_{\varepsilon}(\tilde{\lambda}, s, t))_{\varepsilon} = \left( \lim_{\delta(\Delta(\varepsilon)) \to 0} a^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, s, t)) \right)_{\varepsilon} \quad (0.8.32) $$

exist in the function space $D(\mathbb{R}_x^n \times \mathbb{R}_y^n)$. Moreover, for any integers $m > 0$ and $N > 0$, there exists a positive constant $\gamma_{\varepsilon} = \gamma(m, N, T, \varepsilon)$ such that

$$ \left( \left\| k_{\varepsilon}(\tilde{\lambda}, t, s) - a_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s) \right\|_{m} \right)_{\varepsilon} \le \left( \left( \gamma_{\varepsilon}(\delta(\Delta(\varepsilon)))^{N+1} \right)_{\varepsilon} \right) |\tilde{\lambda}|^{-N} |t-s| \quad (0.8.33) $$

and

$$ \left( \left\| k_{\varepsilon}(\tilde{\lambda}, s, t) - a_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, s, t) \right\|_{m} \right)_{\varepsilon} \le \left( \left( \gamma_{\varepsilon}(\delta(\Delta(\varepsilon)))^{N+1} \right)_{\varepsilon} \right) |\tilde{\lambda}|^{-N} |t-s| \quad (0.8.34) $$

if $|t-s| < \delta(T)$. The constant $\gamma(m, N, T, \varepsilon)$ is independent of any particular choice of $\varepsilon, t, s, \Delta(\varepsilon)$ and $\tilde{\lambda}$ if $|\tilde{\lambda}|$ is bounded away from 0. The function $(k_{\varepsilon}(\tilde{\lambda}, t, s, x, y))_{\varepsilon}$ of is independent of $N$. For any $t \in [s - \delta(T), s + \delta(T)]$, we define

$$ \begin{gathered} (U_{\varepsilon}(\tilde{\lambda}, t, s)\varphi_{\varepsilon}(x))_{\varepsilon} = \\ \left(\frac{-\tilde{\lambda}}{2\pi(t-s)}\right)^{n/2} \left( \int_{\mathbb{R}^n} dy \varphi_{\varepsilon}(y) k_{\varepsilon}(\tilde{\lambda}, t, s, x, y) \exp[\tilde{\lambda} S_{\varepsilon}(t, s, x, y)] \right)_{\varepsilon} \end{gathered} \quad (0.8.35) $$

Then generalized function $(u_{\varepsilon}(t, x))_{\varepsilon} = (U_{\varepsilon}(\tilde{\lambda}, t, s)\varphi_{\varepsilon}(x))_{\varepsilon}$ satisfies the Colombeau-Schrödinger equation (0.8.22) and the initial condition $(u_{\varepsilon}(s, x))_{\varepsilon} = (\varphi_{\varepsilon}(x))_{\varepsilon}$.

(ii) Let $\forall \varepsilon \in (0, 1]$, $f_{\varepsilon}(x) \in \mathcal{L}_2(\mathbb{R}^n)$ then
$$ (k_{\varepsilon}(\tilde{\lambda}, s, t))_{\varepsilon} = \left( \lim_{\delta(\Delta(\varepsilon)) \to 0} a^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, s, t)) \right)_{\varepsilon} \quad (0.8.32) $$

exist in the function space $D(\mathbb{R}_x^n \times \mathbb{R}_y^n)$. Moreover, for any integers $m > 0$ and $N > 0$, there exists a positive constant $\gamma_{\varepsilon} = \gamma(m, N, T, \varepsilon)$ such that

$$ \left( \left\| k_{\varepsilon}(\tilde{\lambda}, t, s) - a_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s) \right\|_{m} \right)_{\varepsilon} \le \left( \left( \gamma_{\varepsilon}(\delta(\Delta(\varepsilon)))^{N+1} \right)_{\varepsilon} \right) |\tilde{\lambda}|^{-N} |t-s| \quad (0.8.33) $$

and

$$ \left( \left\| k_{\varepsilon}(\tilde{\lambda}, s, t) - a_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, s, t) \right\|_{m} \right)_{\varepsilon} \le \left( \left( \gamma_{\varepsilon}(\delta(\Delta(\varepsilon)))^{N+1} \right)_{\varepsilon} \right) |\tilde{\lambda}|^{-N} |t-s| \quad (0.8.34) $$

if $|t-s| < \delta(T)$. The constant $\gamma(m, N, T, \varepsilon)$ is independent of any particular choice of $\varepsilon, t, s, \Delta(\varepsilon)$ and $\tilde{\lambda}$ if $|\tilde{\lambda}|$ is bounded away from 0. The function $(k_{\varepsilon}(\tilde{\lambda}, t, s, x, y))_{\varepsilon}$ of is independent of $N$. For any $t \in [s - \delta(T), s + \delta(T)]$, we define

$$ \begin{gathered} (U_{\varepsilon}(\tilde{\lambda}, t, s)\varphi_{\varepsilon}(x))_{\varepsilon} = \\ \left(\frac{-\tilde{\lambda}}{2\pi(t-s)}\right)^{n/2} \left( \int_{\mathbb{R}^n} dy \varphi_{\varepsilon}(y) k_{\varepsilon}(\tilde{\lambda}, t, s, x, y) \exp[\tilde{\lambda} S_{\varepsilon}(t, s, x, y)] \right)_{\varepsilon} \end{gathered} \quad (0.8.35) $$

Then generalized function $(u_{\varepsilon}(t, x))_{\varepsilon} = (U_{\varepsilon}(\tilde{\lambda}, t, s)\varphi_{\varepsilon}(x))_{\varepsilon}$ satisfies the Colombeau-Schrödinger equation (0.8.22) and the initial condition $(u_{\varepsilon}(s, x))_{\varepsilon} = (\varphi_{\varepsilon}(x))_{\varepsilon}$.

(ii) Let $\forall \varepsilon \in (0, 1]$, $f_{\varepsilon}(x) \in \mathcal{L}_2(\mathbb{R}^n)$ then
$$ (U_{\varepsilon}(\tilde{\lambda}, t, s)f_{\varepsilon}(x))_{\varepsilon} = \left( \int_{\mathbb{R}^n} \left\{ \lim_{\delta(\Delta) \to 0} \mathbf{I}_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s, x, y) \right\} f_{\varepsilon}(y) dy \right)_{\varepsilon} = \\ \left( \lim_{\delta(\Delta) \to 0} E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s)f_{\varepsilon}(x) \right)_{\varepsilon} \qquad (0.8.36) $$

in the sense of convergence in $L_2(\mathbb{R}^n)$, i.e. one can interchange the order of operations $\lim_{\delta(\Delta) \to 0}$ and $\int dy$ if $f_{\varepsilon}(x) \in L_2(\mathbb{R}^n)$.

**Theorem.0.8.4.** If $|t-s| < \delta(T)$, then for any integers $N \ge 0$ and $m \ge 0$, we have

$$ \left( \| k_{\varepsilon}(\tilde{\lambda}, t, s) - a^{(N)}(\tilde{\lambda}, t, s) \|_{m} \right)_{\varepsilon} \le ((\gamma_{\varepsilon})_{\varepsilon}) |\tilde{\lambda}|^{-N} |t-s|^{N+1}, \\ \left( \| k_{\varepsilon}(\tilde{\lambda}, s, t) - a^{(N)}_{\varepsilon}(\tilde{\lambda}, s, t) \|_{m} \right)_{\varepsilon} \le ((\gamma_{\varepsilon})_{\varepsilon}) |\tilde{\lambda}|^{-N} |t-s|^{N+1} \qquad (0.8.37) $$

with some constant $\gamma_{\varepsilon} = \gamma(m, N, T, \varepsilon)$. Here $(a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s))_{\varepsilon}$ is the generalized amplitude function of $(E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s))_{\varepsilon}$ of Eq. (0.8.23).

## I.1. Colombeau generalized functions.

We use [1],[2],[3],[4],[5] as standard references for the foundations and various applications of standard Colombeau theory. We briefly recall the basic Colombeau construction. Throughout the paper $\Omega$ will denote an open subset of $\mathbb{R}^n$. Stanford Colombeau generalized functions on $\Omega$ are defined as equivalence classes $u = [(u_{\varepsilon})_{\varepsilon \in (0,1]}]$ of nets of smooth functions $u_{\varepsilon} \in C^{\infty}(\Omega)$ (regularizations) subjected to asymptotic norm conditions with respect to $\varepsilon \in (0, 1]$ for their derivatives on compact sets.

The basic idea of classical Colombeau's theory of nonlinear generalized functions [1],[2] is regularization by sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter $\varepsilon$. Let $(u_{\varepsilon})_{\varepsilon \in (0,1]}$ with
$$ (U_{\varepsilon}(\tilde{\lambda}, t, s)f_{\varepsilon}(x))_{\varepsilon} = \left( \int_{\mathbb{R}^n} \left\{ \lim_{\delta(\Delta) \to 0} \mathbf{I}_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s, x, y) \right\} f_{\varepsilon}(y) dy \right)_{\varepsilon} = \\ \left( \lim_{\delta(\Delta) \to 0} E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s)f_{\varepsilon}(x) \right)_{\varepsilon} \qquad (0.8.36) $$

in the sense of convergence in $L_2(\mathbb{R}^n)$, i.e. one can interchange the order of operations $\lim_{\delta(\Delta) \to 0}$ and $\int dy$ if $f_{\varepsilon}(x) \in L_2(\mathbb{R}^n)$.

**Theorem.0.8.4.** If $|t-s| < \delta(T)$, then for any integers $N \ge 0$ and $m \ge 0$, we have

$$ \left( \| k_{\varepsilon}(\tilde{\lambda}, t, s) - a^{(N)}(\tilde{\lambda}, t, s) \|_{m} \right)_{\varepsilon} \le ((\gamma_{\varepsilon})_{\varepsilon}) |\tilde{\lambda}|^{-N} |t-s|^{N+1}, \\ \left( \| k_{\varepsilon}(\tilde{\lambda}, s, t) - a^{(N)}_{\varepsilon}(\tilde{\lambda}, s, t) \|_{m} \right)_{\varepsilon} \le ((\gamma_{\varepsilon})_{\varepsilon}) |\tilde{\lambda}|^{-N} |t-s|^{N+1} \qquad (0.8.37) $$

with some constant $\gamma_{\varepsilon} = \gamma(m, N, T, \varepsilon)$. Here $(a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s))_{\varepsilon}$ is the generalized amplitude function of $(E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s))_{\varepsilon}$ of Eq. (0.8.23).

## I.1. Colombeau generalized functions.

We use [1],[2],[3],[4],[5] as standard references for the foundations and various applications of standard Colombeau theory. We briefly recall the basic Colombeau construction. Throughout the paper $\Omega$ will denote an open subset of $\mathbb{R}^n$. Stanford Colombeau generalized functions on $\Omega$ are defined as equivalence classes $u = [(u_{\varepsilon})_{\varepsilon \in (0,1]}]$ of nets of smooth functions $u_{\varepsilon} \in C^{\infty}(\Omega)$ (regularizations) subjected to asymptotic norm conditions with respect to $\varepsilon \in (0, 1]$ for their derivatives on compact sets.

The basic idea of classical Colombeau's theory of nonlinear generalized functions [1],[2] is regularization by sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter $\varepsilon$. Let $(u_{\varepsilon})_{\varepsilon \in (0,1]}$ with
$u_{\epsilon} \in C^{\infty}(M)$ for all $\epsilon \in \mathbb{R}_{+}$, where $M$ a separable, smooth orientable Hausdorff manifold of dimension $n$.

**Definition 1.1.1.** The classical Colombeau's algebra of generalized functions on $M$ is defined as the quotient:

$$G(M) \triangleq E_M(M)/N(M) \qquad (1.1.1)$$

of the space $E_M(M)$ of sequences of moderate growth modulo the space $N(M)$ of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates (where $X(M)$ denoting the space of smooth vector fields on $M$):

$$E_M(M) \triangleq \{(u_\epsilon)_\epsilon | \forall K(K \subsetneq M) \forall k(k \in \mathbb{N}) \exists N(N \in \mathbb{N}) \\ \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in X(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^{-N}) \text{ as } \epsilon \to 0 \right] \}, \quad (1.1.2)$$

$$N(M) \triangleq \{(u_\epsilon)_\epsilon | \forall K(K \subsetneq M), \forall k(k \in \mathbb{N}_0) \forall q(q \in N) \\ \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in X(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q) \text{ as } \epsilon \to 0 \right] \}. \quad (1.1.3)$$

**Remark 1.1.1.** In the definition the Landau symbol $a_\epsilon = O(\psi(\epsilon))$ appears, having the following meaning: $\exists C(C > 0) \exists \epsilon_0(\epsilon_0 \in (0,1]) \forall \epsilon(\epsilon < \epsilon_0)[a_\epsilon \le C\psi(\epsilon)]$.

**Definition 1.1.2.** Elements of $G(M)$ are denoted by:

$$u = \mathbf{cl}[(u_{\epsilon})_{\epsilon}] \triangleq (u_{\epsilon})_{\epsilon} + N(M). \qquad (1.1.4)$$

**Remark 1.1.2.** With componentwise operations $(\cdot, \pm)$ $G(M)$ is a fine sheaf of differential algebras with respect to the Lie derivative defined by $L_{\xi}u \triangleq \mathbf{cl}[(L_{\xi}u_{\epsilon})_{\epsilon}]$. The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in G(M)$ iff $u \circ \psi_{\alpha} \in G(\psi_{\alpha}(V_{\alpha}))$ for all charts $(V_{\alpha}, \psi_{\alpha})$, where on the open set $\psi_{\alpha}(V_{\alpha}) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in G(M)$ iff $u \circ \psi_{\alpha} \in G(\psi_{\alpha}(V_{\alpha}))$ for all charts $(V_{\alpha}, \psi_{\alpha})$, where on the open set $\psi_{\alpha}(V_{\alpha}) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

**Remark 1.1.3.** Smooth functions $f \in C^{\infty}(M)$ are embedded into $G(M)$ simply by the "constant" embedding $\sigma$, i.e., $\sigma(f) = \mathbf{cl}[(f)_{\epsilon}]$, hence $C^{\infty}(M)$ is a faithful subalgebra
$u_{\epsilon} \in C^{\infty}(M)$ for all $\epsilon \in \mathbb{R}_{+}$, where $M$ a separable, smooth orientable Hausdorff manifold of dimension $n$.

**Definition 1.1.1.** The classical Colombeau's algebra of generalized functions on $M$ is defined as the quotient:

$$G(M) \triangleq E_M(M)/N(M) \qquad (1.1.1)$$

of the space $E_M(M)$ of sequences of moderate growth modulo the space $N(M)$ of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates (where $X(M)$ denoting the space of smooth vector fields on $M$):

$$E_M(M) \triangleq \{(u_\epsilon)_\epsilon | \forall K(K \subsetneq M) \forall k(k \in \mathbb{N}) \exists N(N \in \mathbb{N}) \\ \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in X(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^{-N}) \text{ as } \epsilon \to 0 \right] \}, \quad (1.1.2)$$

$$N(M) \triangleq \{(u_\epsilon)_\epsilon | \forall K(K \subsetneq M), \forall k(k \in \mathbb{N}_0) \forall q(q \in N) \\ \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in X(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q) \text{ as } \epsilon \to 0 \right] \}. \quad (1.1.3)$$

**Remark 1.1.1.** In the definition the Landau symbol $a_\epsilon = O(\psi(\epsilon))$ appears, having the following meaning: $\exists C(C > 0) \exists \epsilon_0(\epsilon_0 \in (0,1]) \forall \epsilon(\epsilon < \epsilon_0)[a_\epsilon \le C\psi(\epsilon)]$.

**Definition 1.1.2.** Elements of $G(M)$ are denoted by:

$$u = \mathbf{cl}[(u_{\epsilon})_{\epsilon}] \triangleq (u_{\epsilon})_{\epsilon} + N(M). \qquad (1.1.4)$$

**Remark 1.1.2.** With componentwise operations $(\cdot, \pm)$ $G(M)$ is a fine sheaf of differential algebras with respect to the Lie derivative defined by $L_{\xi}u \triangleq \mathbf{cl}[(L_{\xi}u_{\epsilon})_{\epsilon}]$. The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in G(M)$ iff $u \circ \psi_{\alpha} \in G(\psi_{\alpha}(V_{\alpha}))$ for all charts $(V_{\alpha}, \psi_{\alpha})$, where on the open set $\psi_{\alpha}(V_{\alpha}) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in G(M)$ iff $u \circ \psi_{\alpha} \in G(\psi_{\alpha}(V_{\alpha}))$ for all charts $(V_{\alpha}, \psi_{\alpha})$, where on the open set $\psi_{\alpha}(V_{\alpha}) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

**Remark 1.1.3.** Smooth functions $f \in C^{\infty}(M)$ are embedded into $G(M)$ simply by the "constant" embedding $\sigma$, i.e., $\sigma(f) = \mathbf{cl}[(f)_{\epsilon}]$, hence $C^{\infty}(M)$ is a faithful subalgebra
of G(M).

### I.1.1. Point Values of Generalized Functions on M. Generalized Numbers.

Within the classical distribution theory, distributions cannot be characterized by their point values in any way similar to classical functions. On the other hand, there is a very natural and direct way of obtaining the point values of the elements of Colombeau's algebra: points are simply inserted into representatives. The objects so obtained are sequences of numbers, and as such are not the elements in the field $\mathbb{R}$ or $\mathbb{C}$. Instead, they are the representatives of Colombeau's generalized numbers [6],[7],[8]. We give the exact definition of these "numbers".

**Definition 1.1.3.** Inserting $p \in M$ into $u \in G(M)$ yields a well defined element of the ring of constants (also called generalized numbers) $K$ (corresponding to $K = \mathbb{R}$ resp. $\mathbb{C}$), defined as the set of moderate nets of numbers ($|r_\epsilon| \in K^{(0,1]}$ with $|r_\epsilon| = O(\epsilon^{-N})$ for some $N$) modulo negligible nets ($|r_\epsilon| = O(\epsilon^m)$ for each $m$); componentwise insertion of points of $M$ into elements of $G(M)$ yields well-defined generalized numbers, i.e., elements of the ring of constants:

$$ K = E_c(M)/N_c(M) \qquad (1.1.8) $$

with $K = \tilde{\mathbb{R}}$ or $K = \tilde{\mathbb{C}}$ for $K = \mathbb{R}$ or $K = \mathbb{C}$, where

$$ \begin{aligned} E_c(M) &= \{ (r_\epsilon)_\epsilon \in \mathbf{K}^I \mid \exists n(n \in \mathbb{N}) \text{ such that } |r_\epsilon| = O(\epsilon^{-n}) \text{ as } \epsilon \to 0 \} \\ N_c(M) &= \{ (r_\epsilon)_\epsilon \in \mathbf{K}^I \mid \forall m(m \in \mathbb{N}) \text{ such that } |r_\epsilon| = O(\epsilon^m) \text{ as } \epsilon \to 0 \} \\ I &= (0, 1]. \end{aligned} \qquad (1.1.9) $$

Generalized functions on $M$ are characterized by their generalized point values, i.e., by their values on points in $\tilde{M}_c$, the space of equivalence classes of compactly supported nets $(p_\epsilon)_\epsilon \in M^{(0,1]}$ with respect to the relation $p_\epsilon \sim p'_\epsilon :\Leftrightarrow d_h(p_\epsilon, p'_\epsilon) = O(\epsilon^m)$ for all $m$, where $d_h$ denotes the distance on $M$ induced by any Riemannian metric.

**Definition 1.1.3.** For $u \in G(M)$ and $x_0 \in M$, the point value of $u$ at the point $x_0, u(x_0)$, is defined as the class of $(u_\epsilon(x_0))_\epsilon$ in $K$.

**Definition 1.1.4.** We say that an element $r \in K$ is strictly nonzero if there exists a representative $(r_\epsilon)_\epsilon$ and a $q \in \mathbb{N}$ such that $|r_\epsilon| \ge \epsilon^q$ for $\epsilon$ sufficiently small. If $r$ is
of G(M).

### I.1.1. Point Values of Generalized Functions on M. Generalized Numbers.

Within the classical distribution theory, distributions cannot be characterized by their point values in any way similar to classical functions. On the other hand, there is a very natural and direct way of obtaining the point values of the elements of Colombeau's algebra: points are simply inserted into representatives. The objects so obtained are sequences of numbers, and as such are not the elements in the field $\mathbb{R}$ or $\mathbb{C}$. Instead, they are the representatives of Colombeau's generalized numbers [6],[7],[8]. We give the exact definition of these "numbers".

**Definition 1.1.3.** Inserting $p \in M$ into $u \in G(M)$ yields a well defined element of the ring of constants (also called generalized numbers) $K$ (corresponding to $K = \mathbb{R}$ resp. $\mathbb{C}$), defined as the set of moderate nets of numbers ($|r_\epsilon| \in K^{(0,1]}$ with $|r_\epsilon| = O(\epsilon^{-N})$ for some $N$) modulo negligible nets ($|r_\epsilon| = O(\epsilon^m)$ for each $m$); componentwise insertion of points of $M$ into elements of $G(M)$ yields well-defined generalized numbers, i.e., elements of the ring of constants:

$$ K = E_c(M)/N_c(M) \qquad (1.1.8) $$

with $K = \tilde{\mathbb{R}}$ or $K = \tilde{\mathbb{C}}$ for $K = \mathbb{R}$ or $K = \mathbb{C}$, where

$$ \begin{aligned} E_c(M) &= \{ (r_\epsilon)_\epsilon \in \mathbf{K}^I \mid \exists n(n \in \mathbb{N}) \text{ such that } |r_\epsilon| = O(\epsilon^{-n}) \text{ as } \epsilon \to 0 \} \\ N_c(M) &= \{ (r_\epsilon)_\epsilon \in \mathbf{K}^I \mid \forall m(m \in \mathbb{N}) \text{ such that } |r_\epsilon| = O(\epsilon^m) \text{ as } \epsilon \to 0 \} \\ I &= (0, 1]. \end{aligned} \qquad (1.1.9) $$

Generalized functions on $M$ are characterized by their generalized point values, i.e., by their values on points in $\tilde{M}_c$, the space of equivalence classes of compactly supported nets $(p_\epsilon)_\epsilon \in M^{(0,1]}$ with respect to the relation $p_\epsilon \sim p'_\epsilon :\Leftrightarrow d_h(p_\epsilon, p'_\epsilon) = O(\epsilon^m)$ for all $m$, where $d_h$ denotes the distance on $M$ induced by any Riemannian metric.

**Definition 1.1.3.** For $u \in G(M)$ and $x_0 \in M$, the point value of $u$ at the point $x_0, u(x_0)$, is defined as the class of $(u_\epsilon(x_0))_\epsilon$ in $K$.

**Definition 1.1.4.** We say that an element $r \in K$ is strictly nonzero if there exists a representative $(r_\epsilon)_\epsilon$ and a $q \in \mathbb{N}$ such that $|r_\epsilon| \ge \epsilon^q$ for $\epsilon$ sufficiently small. If $r$ is
strictly nonzero, then it is also invertible with the inverse $[(1/r_\epsilon)_\epsilon]$. The converse is true as well.

Treating the elements of Colombeau algebras as a generalization of classical functions, the question arises whether the definition of point values can be extended in such a way that each element is characterized by its values. Such an extension is indeed possible.

**Definition 1.1.5.** Let $\Omega$ be an open subset of $\mathbb{R}^n$. On a set $\hat{\Omega}$:

$$ \hat{\Omega} = \{(x_\epsilon)_\epsilon \in \Omega^l | \exists p(p > 0) [ |x_\epsilon| = O(\epsilon^p) ] \} = \\ \{(x_\epsilon)_\epsilon \in \Omega^l | \exists p(p > 0) \exists \epsilon_0 (\epsilon_0 > 0) [ |x_\epsilon| \le \epsilon^p, \text{ for } 0 < \epsilon < \epsilon_0 ] \} \quad (1.1.10) $$

we introduce an equivalence relation:

$$ (x_{\epsilon})_{\epsilon} \sim (y_{\epsilon})_{\epsilon} \Leftrightarrow \forall q(q > 0) \forall \epsilon(\epsilon > 0) [ |x_{\epsilon} - y_{\epsilon}| \le \epsilon^q, \text{ for } 0 < \epsilon < \epsilon_0 ] \quad (1.1.11) $$

and denote by $\tilde{\Omega} = \hat{\Omega} / \sim$ the set of generalized points. The set of points with compact support is

$$ \tilde{\Omega}_c = \\ \{ \tilde{x} = \mathbf{cl}[(x_\epsilon)_\epsilon] \in \tilde{\Omega} | \exists K \subset \Omega \exists \epsilon_0 (\epsilon_0 > 0) [ x_\epsilon \in K \text{ for } 0 < \epsilon < \epsilon_0 ] \} \quad (1.1.12) $$

**Definition 1.1.6.** A generalized function $u \in G(M)$ is called associated to zero, $u \approx 0$ on $\Omega \subseteq M$ in $L$. *Schwartz's sense* if one (hence any) representative $(u_\epsilon)_\epsilon$ converges to zero weakly,i.e.

$$ w - \lim_{\epsilon \to 0} u_\epsilon = 0 \quad (1.1.13) $$

We shall often write:
strictly nonzero, then it is also invertible with the inverse $[(1/r_\epsilon)_\epsilon]$. The converse is true as well.

Treating the elements of Colombeau algebras as a generalization of classical functions, the question arises whether the definition of point values can be extended in such a way that each element is characterized by its values. Such an extension is indeed possible.

**Definition 1.1.5.** Let $\Omega$ be an open subset of $\mathbb{R}^n$. On a set $\hat{\Omega}$:

$$ \hat{\Omega} = \{(x_{\epsilon})_{\epsilon} \in \Omega^l | \exists p(p > 0) [ |x_{\epsilon}| = O(\epsilon^p) ] \} = \\ \{(x_{\epsilon})_{\epsilon} \in \Omega^l | \exists p(p > 0) \exists \epsilon_0 (\epsilon_0 > 0) [ |x_{\epsilon}| \le \epsilon^p, \text{ for } 0 < \epsilon < \epsilon_0 ] \} \quad (1.1.10) $$

we introduce an equivalence relation:

$$ (x_{\epsilon})_{\epsilon} \sim (y_{\epsilon})_{\epsilon} \Leftrightarrow \forall q(q > 0) \forall \epsilon(\epsilon > 0) [ |x_{\epsilon} - y_{\epsilon}| \le \epsilon^q, \text{ for } 0 < \epsilon < \epsilon_0 ] \quad (1.1.11) $$

and denote by $\tilde{\Omega} = \hat{\Omega}/\sim$ the set of generalized points. The set of points with compact support is

$$ \tilde{\Omega}_c = \\ \{ \tilde{x} = \text{cl}[(x_\epsilon)_\epsilon] \in \tilde{\Omega} | \exists K \subset \Omega \exists \epsilon_0 (\epsilon_0 > 0) [ x_\epsilon \in K \text{ for } 0 < \epsilon < \epsilon_0 ] \} \quad (1.1.12) $$

**Definition 1.1.6.** A generalized function $u \in G(M)$ is called associated to zero, $u \approx 0$ on $\Omega \subseteq M$ in $L$. *Schwartz's sense* if one (hence any) representative $(u_\epsilon)_\epsilon$ converges to zero weakly,i.e.

$$ w \lim_{\epsilon \to 0} u_\epsilon = 0 \quad (1.1.13) $$

We shall often write:
$$ u \approx 0. \qquad (1.1.14) \\ \text{Sch} $$

### 1.1.2. Colombeau algebra of compactly supported generalized functions and Colombeau algebra of tempered generalized functions.

**Definition 1.1.7.** We define the spaces $E_{c,M}(\Omega)$ and $N_c(\Omega)$ as the sets of all nets $(u_\epsilon)_\epsilon$ in $E_M(\Omega)$ and $N(\Omega)$ respectively such that

$$ \exists K(K \in \Omega) \forall \epsilon(\epsilon \in (0,1]) [\mathbf{supp}(u_\epsilon) \subseteq \Omega]. \qquad (1.1.15) $$

**Definition 1.1.8.** The Colombeau algebra $G_c(\Omega)$ is isomorphic to the factor space $E_{c,M}(\Omega)/N_c(\Omega)$ by canonical map

$$ l: G_c(\Omega) \to E_{c,M}(\Omega)/N_c(\Omega) : (u_\epsilon)_\epsilon + N(\Omega) \to (\psi_u u_\epsilon)_\epsilon + N_c(\Omega), \qquad (1.1.16) $$

where $\psi_u = \psi u$, $\psi \in C_c^\infty(\Omega)$ and $\psi$ is identically 1 in a neighborhood of $\text{supp}(u)$.

**Definition 1.1.9.** We call the elements of the set
$$ u \approx 0. \qquad (1.1.14) \\ \text{Sch} $$

### I.1.2. Colombeau algebra of compactly supported generalized functions and Colombeau algebra of tempered generalized functions.

**Definition 1.1.7.** We define the spaces $E_{c,M}(\Omega)$ and $N_c(\Omega)$ as the sets of all nets $(u_\epsilon)_\epsilon$ in $E_M(\Omega)$ and $N(\Omega)$ respectively such that

$$ \exists K(K \in \Omega) \forall \epsilon(\epsilon \in (0,1]) [\mathbf{supp}(u_\epsilon) \subseteq \Omega]. \qquad (1.1.15) $$

**Definition 1.1.8.** The Colombeau algebra $G_c(\Omega)$ is isomorphic to the factor space $E_{c,M}(\Omega)/N_c(\Omega)$ by canonical map

$$ l: G_c(\Omega) \to E_{c,M}(\Omega)/N_c(\Omega) : (u_\epsilon)_\epsilon + N(\Omega) \to (\psi_u u_\epsilon)_\epsilon + N_c(\Omega), \qquad (1.1.16) $$

where $\psi_u = \psi u$, $\psi \in C_c^\infty(\Omega)$ and $\psi$ is identically 1 in a neighborhood of $\text{supp}(u)$.

**Definition 1.1.9.** We call the elements of the set
$$E_{\tau}(\mathbb{R}^n) \triangleq$$

$$\{(u_{\varepsilon})_{\varepsilon} \in E_{\tau}[\mathbb{R}^n] | \forall \alpha \in \mathbb{N}^n \exists N \in \mathbb{N}$$

$$\left[ \sup_{x \in \mathbb{R}^n} (1 + |x|)^{-N} |\partial^\alpha u_\varepsilon(x)| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right], \qquad (1.1.17)$$

$$E_{\tau}[\mathbb{R}^n] \triangleq O_M(\mathbb{R}^n)^{(0,1]}$$

τ -moderate.

We call the elements of the set

$$N_{\tau}(\mathbb{R}^n) \triangleq$$

$$\{(u_{\varepsilon})_{\varepsilon} \in E_{\tau}[\mathbb{R}^n] | \forall \alpha \in \mathbb{N}^n \exists N \in \mathbb{N} \forall q \in \mathbb{N}$$

(1.1.18)

$$\left[ \sup_{x \in \mathbb{R}^n} (1 + |x|)^{-N} |\partial^\alpha u_\varepsilon(x)| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right],$$

τ-negligible.

**Definition 1.1.10.** [2]. The Colombeau algebra of tempered generalized functions is defined as the quotient

$$G_{\tau}(\mathbb{R}^n) \triangleq E_{\tau}(\mathbb{R}^n)/N_{\tau}(\mathbb{R}^n). \qquad (1.1.19)$$

**Theorem 1.1.1.** $(u_\varepsilon)_\varepsilon \in E_\tau(\mathbb{R}^n)$ is $\tau$-negligible if and only if

$$\exists N(N \in \mathbb{N}) \forall q(q \in \mathbb{N}) \left[ \sup_{x \in \mathbb{R}^n} (1 + |x|)^{-N} |\partial^\alpha u_\varepsilon(x)| = O(\varepsilon^q) \text{ as } \varepsilon \to 0 \right]. \qquad (1.1.20)$$
$$E_{\tau}(\mathbb{R}^n) \triangleq$$

$$\{(u_{\varepsilon})_{\varepsilon} \in E_{\tau}[\mathbb{R}^n] | \forall \alpha \in \mathbb{N}^n \exists N \in \mathbb{N}$$

$$\left[ \sup_{x \in \mathbb{R}^n} (1 + |x|)^{-N} |\partial^\alpha u_\varepsilon(x)| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right], \qquad (1.1.17)$$

$$E_{\tau}[\mathbb{R}^n] \triangleq O_M(\mathbb{R}^n)^{(0,1]}$$

τ -moderate.

We call the elements of the set

$$N_{\tau}(\mathbb{R}^n) \triangleq$$

$$\{(u_{\varepsilon})_{\varepsilon} \in E_{\tau}[\mathbb{R}^n] | \forall \alpha \in \mathbb{N}^n \exists N \in \mathbb{N} \forall q \in \mathbb{N}$$

(1.1.18)

$$\left[ \sup_{x \in \mathbb{R}^n} (1 + |x|)^{-N} |\partial^\alpha u_\varepsilon(x)| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right],$$

τ-negligible.

**Definition 1.1.10.** [2]. The Colombeau algebra of tempered generalized functions is defined as the quotient

$$G_{\tau}(\mathbb{R}^n) \triangleq E_{\tau}(\mathbb{R}^n)/N_{\tau}(\mathbb{R}^n). \qquad (1.1.19)$$

**Theorem 1.1.1.** $(u_{\varepsilon})_{\varepsilon} \in E_{\tau}(\mathbb{R}^n)$ is $\tau$-negligible if and only if

$$\exists N(N \in \mathbb{N}) \forall q(q \in \mathbb{N}) \left[ \sup_{x \in \mathbb{R}^n} (1 + |x|)^{-N} |\partial^\alpha u_\varepsilon(x)| = O(\varepsilon^q) \text{ as } \varepsilon \to 0 \right]. \qquad (1.1.20)$$
### I.1.3. Algebra of generalized functions based on $\mathcal{L}(\mathbb{R}^n)$. Colombeau algebra $G_{\tau, \mathcal{L}}(\mathbb{R}^n)$.

**Definition 1.1.11.** Let $\mathcal{L}[\mathbb{R}^n] \triangleq \mathcal{L}(\mathbb{R}^n)^{(0,1]}$. The elements of

$$ E_{\mathcal{L}}[\mathbb{R}^n] \triangleq \\ \{(u_{\varepsilon})_{\varepsilon} \in \mathcal{L}[\mathbb{R}^n] | \forall \alpha \forall \beta (\alpha, \beta \in \mathbb{N}^n) \\ \exists N(N \in \mathbb{N}) \left[ \sup_{x \in \mathbb{R}^n} |x^\alpha \partial^\beta u_\varepsilon(x)| O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right] \} \quad (1.1.21) $$

are called $\mathcal{L}$-moderate. The elements of

$$ N_{\mathcal{L}}[\mathbb{R}^n] \triangleq \\ \{(u_{\varepsilon})_{\varepsilon} \in \mathcal{L}[\mathbb{R}^n] | \forall \alpha \forall \beta (\alpha, \beta \in \mathbb{N}^n) \\ \forall q(q \in \mathbb{N}) \in \mathbb{N} \left[ \sup_{x \in \mathbb{R}^n} |x^\alpha \partial^\beta u_\varepsilon(x)| O(\varepsilon^q) \text{ as } \varepsilon \to 0 \right] \} \quad (1.1.22) $$

are called $\mathcal{L}$-negligible.

**Definition 1.1.12.** The algebra of generalized functions based on $\mathcal{L}(\mathbb{R}^n)$ is defined as the factor space

$$ G_{\mathcal{L}}(\mathbb{R}^n) \triangleq E_{\mathcal{L}}[\mathbb{R}^n]/N_{\mathcal{L}}[\mathbb{R}^n]. \quad (1.1.23) $$
### I.1.3. Algebra of generalized functions based on $\mathcal{L}(\mathbb{R}^n)$. Colombeau algebra $G_{\tau, \mathcal{L}}(\mathbb{R}^n)$.

**Definition 1.1.11.** Let $\mathcal{L}[\mathbb{R}^n] \triangleq \mathcal{L}(\mathbb{R}^n)^{(0,1]}$. The elements of

$$ E_{\mathcal{L}}[\mathbb{R}^n] \triangleq \\ \{(u_{\varepsilon})_{\varepsilon} \in \mathcal{L}[\mathbb{R}^n] | \forall \alpha \forall \beta (\alpha, \beta \in \mathbb{N}^n) \\ \exists N(N \in \mathbb{N}) \left[ \sup_{x \in \mathbb{R}^n} |x^\alpha \partial^\beta u_\varepsilon(x)| O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right] \} \quad (1.1.21) $$

are called $\mathcal{L}$-moderate. The elements of

$$ N_{\mathcal{L}}[\mathbb{R}^n] \triangleq \\ \{(u_{\varepsilon})_{\varepsilon} \in \mathcal{L}[\mathbb{R}^n] | \forall \alpha \forall \beta (\alpha, \beta \in \mathbb{N}^n) \\ \forall q(q \in \mathbb{N}) \in \mathbb{N} \left[ \sup_{x \in \mathbb{R}^n} |x^\alpha \partial^\beta u_\varepsilon(x)| O(\varepsilon^q) \text{ as } \varepsilon \to 0 \right] \} \quad (1.1.22) $$

are called $\mathcal{L}$-negligible.

**Definition 1.1.12.** The algebra of generalized functions based on $\mathcal{L}(\mathbb{R}^n)$ is defined as the factor space

$$ G_{\mathcal{L}}(\mathbb{R}^n) \triangleq E_{\mathcal{L}}[\mathbb{R}^n]/N_{\mathcal{L}}[\mathbb{R}^n]. \quad (1.1.23) $$
**Definition 1.1.13.** By factorizing $E_τ(\mathbb{R}^n)$ with respect to the ideal $N_ℓ[\mathbb{R}^n]$ Thus we obtain the quotient algebra

$$ G_{\tau, \mathcal{L}}(\mathbb{R}^n) \triangleq E_τ(\mathbb{R}^n)/N_\mathcal{L}[\mathbb{R}^n]. \qquad (1.1.24) $$

It is clearly that $G_\mathcal{L}(\mathbb{R}^n)$ is contained in $G_{\tau, \mathcal{L}}(\mathbb{R}^n)$.

### 1.1.4. Regular Colombeau algebras.

**Definition 1.1.14.** In the setting of Colombeau theory, the notion of regularity is based on the subalgebra $G^\infty(\Omega)$ of regular generalized functions in $G(\Omega)$. It is defined by those elements which have a representative $(u_ε)_ε$ satisfying

$$ \forall K(K \in \Omega) \exists N(N \in \mathbb{N}) \forall \alpha(\alpha \in \mathbb{N}^n) \left[ \sup_{x \in K} |\partial^\alpha u_\epsilon(x)| = O(\epsilon^{-N}) \text{ as } \epsilon \to 0 \right]. \qquad (1.1.25) $$

**Remark 1.1.4.** Note that (1.1.25) is obtained by interchanging the quantifiers in the definition of $E_M(\Omega)$. It is well known that

$$ G^\infty(\Omega) \cap D'(\Omega) = C^\infty(\Omega). \qquad (1.1.26) $$

**Definition 1.1.15.** The Colombeau algebra $G_c^\infty(\Omega)$ of regular generalized functions with compact support is defined by intersection

$$ G_c^\infty(\Omega) \triangleq G^\infty(\Omega) \cap G_c(\Omega). \qquad (1.1.27) $$
**Definition 1.1.13.** By factorizing $E_τ(\mathbb{R}^n)$ with respect to the ideal $N_ℓ[\mathbb{R}^n]$ Thus we obtain the quotient algebra

$$ G_{\tau, \mathcal{L}}(\mathbb{R}^n) \triangleq E_τ(\mathbb{R}^n)/N_ℓ[\mathbb{R}^n]. \qquad (1.1.24) $$

It is clearly that $G_ℓ(\mathbb{R}^n)$ is contained in $G_{\tau, \mathcal{L}}(\mathbb{R}^n)$.

### 1.1.4. Regular Colombeau algebras.

**Definition 1.1.14.** In the setting of Colombeau theory, the notion of regularity is based on the subalgebra $G^∞(\Omega)$ of regular generalized functions in $G(\Omega)$. It is defined by those elements which have a representative $(u_ε)_{ε}$ satisfying

$$ \forall K(K \in \Omega) \exists N(N \in \mathbb{N}) \forall \alpha(\alpha \in \mathbb{N}^n) \left[ \sup_{x \in K} |\partial^\alpha u_\epsilon(x)| = O(\epsilon^{-N}) \text{ as } \epsilon \to 0 \right]. \qquad (1.1.25) $$

**Remark 1.1.4.** Note that (1.1.25) is obtained by interchanging the quantifiers in the definition of $E_M(\Omega)$. It is well known that

$$ G^∞(\Omega) \cap D'(\Omega) = C^∞(\Omega). \qquad (1.1.26) $$

**Definition 1.1.15.** The Colombeau algebra $G_c^∞(\Omega)$ of regular generalized functions with compact support is defined by intersection

$$ G_c^∞(\Omega) \triangleq G^∞(\Omega) \cap G_c(\Omega). \qquad (1.1.27) $$
**Remark 1.1.5.** It well known that $G_c^\infty(\Omega) \cap E'(\Omega) = C_c^\infty(\Omega)$.

**Definition 1.1.16.** The Colombeau algebra $G_\ell^\infty(\Omega)$ is given by the quotient

$$G_\ell^\infty(\mathbb{R}^n) \triangleq E_\ell^\infty(\mathbb{R}^n)/N_\ell(\mathbb{R}^n), \qquad (1.1.28)$$

where

$$E_\ell^\infty[\mathbb{R}^n] \triangleq \\ \{(u_\epsilon)_\epsilon \in \mathcal{L}[\mathbb{R}^n] | \exists N \in \mathbb{N} \ \forall \alpha \forall \beta (\alpha, \beta \in \mathbb{N}^n) \qquad (1.1.29) \\ \left[ \sup_{x \in \mathbb{R}^n} |x^\alpha \partial^\beta u_\epsilon(x)| O(\epsilon^{-N}) \text{ as } \epsilon \to 0 \right] \}. $$

**I.1.5.**Integration in Colombeau theory

**Definition 1.1.17.** Let $K \subset \Omega$ be a compact set and $u \in G(M)$. The linear map from $G(M)$ into $\tilde{\mathbb{C}}$ via formula

$$u \to \int_K u(x)dx \triangleq \left( \int_K u_\epsilon(x)dx \right)_\epsilon + N \qquad (1.1.30)$$

is called the Colombeau integral of $u$. Colombeau integral is $\tilde{\mathbb{C}}$ linear by definition,i.e., $\forall \alpha_1, \alpha_2 \in \tilde{\mathbb{C}}, \forall u_1, u_2 \in G(M)$

$$\int_K [\alpha_1 u_1(x) + \alpha_2 u_2(x)] dx = \alpha_1 \int_K u_1(x) dx + \alpha_2 \int_K u_2(x) dx \qquad (1.1.31)$$
**Remark 1.1.5.** It well known that $G_c^\infty(\Omega) \cap E'(\Omega) = C_c^\infty(\Omega)$.

**Definition 1.1.16.** The Colombeau algebra $G_{\mathcal{L}}^{\infty}(\Omega)$ is given by the quotient

$$G_{\mathcal{L}}^{\infty}(\mathbb{R}^n) \triangleq E_{\mathcal{L}}^{\infty}(\mathbb{R}^n)/N_{\mathcal{L}}(\mathbb{R}^n), \qquad (1.1.28)$$

where

$$E_{\mathcal{L}}^{\infty}[\mathbb{R}^n] \triangleq \\ \{(u_{\varepsilon})_{\varepsilon} \in \mathcal{L}[\mathbb{R}^n] | \exists N(N \in \mathbb{N}) \forall \alpha \forall \beta (\alpha, \beta \in \mathbb{N}^n) \\ \qquad \left[ \sup_{x \in \mathbb{R}^n} |x^\alpha \partial^\beta u_{\varepsilon}(x)| O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right] \}. \qquad (1.1.29)$$

**I.1.5.**Integration in Colombeau theory

**Definition 1.1.17.** Let $K \subset \Omega$ be a compact set and $u \in G(M)$. The linear map from $G(M)$ into $\tilde{\mathbb{C}}$ via formula

$$u \to \int_K u(x) dx \triangleq \left( \int_K u_{\varepsilon}(x) dx \right)_{\varepsilon} + N \qquad (1.1.30)$$

is called the Colombeau integral of $u$. Colombeau integral is $\tilde{\mathbb{C}}$ linear by definition, i.e., $\forall \alpha_1, \alpha_2 \in \tilde{\mathbb{C}}, \forall u_1, u_2 \in G(M)$

$$\int_K [\alpha_1 u_1(x) + \alpha_2 u_2(x)] dx = \alpha_1 \int_K u_1(x) dx + \alpha_2 \int_K u_2(x) dx \qquad (1.1.31)$$
**Definition.1.1.18.** Let $u \in G(\Omega)$ with compact support $K \subset \Omega$. We define

$$ \int_{\Omega} u(x)dx \triangleq \int_{\hat{K}} u(x)dx, \qquad (1.1.32) $$

where $\hat{K}$ is any compact subset of $\Omega$ containing $K$ in its interior.

**Theorem.1.1.2.** (1) If $u, v \in G(\Omega)$ and either $u$ or $v$ has compact support then for any $\alpha \in \mathbb{N}^n$

$$ \int_{\Omega} u(x) \partial^{\alpha} v(x) dx = (-1)^{|\alpha|} \int_{\Omega} (\partial^{\alpha} u(x)) v(x) dx \qquad (1.1.33) $$

(2) If $u \in G_{\ell}(\mathbb{R}^n)$ then integral

$$ \int_{\mathbb{R}^n} u(x) dx \triangleq \mathbf{cl} \left[ \left( \int_{\mathbb{R}^n} u_{\varepsilon}(x) dx \right)_{\varepsilon} \right] \qquad (1.1.34) $$

is a well-defined element of $\tilde{\mathbb{C}}$. Moreover, the following properties hold:

(i)

$$ \int_{\mathbb{R}^n} (\iota g)(x) dx = \mathbf{cl} \left[ \left( \int_{\mathbb{R}^n} g(x) dx \right)_{\varepsilon} \right] \qquad (1.1.34) $$

(ii)

$$ \int_{\mathbb{R}^n} u(x) \partial^\alpha v(x) dx = (-1)^{|\alpha|} \int_{\mathbb{R}^n} (\partial^\alpha u(x)) v(x) dx \qquad (1.1.35) $$

for all $u, v \in G_{\ell}(\mathbb{R}^n)$ and $\alpha \in \mathbb{N}^n$.
**Definition.1.1.18.** Let $u \in G(\Omega)$ with compact support $K \subset \Omega$. We define

$$ \int_{\Omega} u(x)dx \triangleq \int_{\hat{K}} u(x)dx, \qquad (1.1.32) $$

where $\hat{K}$ is any compact subset of $\Omega$ containing $K$ in its interior.

**Theorem.1.1.2.** (1) If $u, v \in G(\Omega)$ and either $u$ or $v$ has compact support then for any $\alpha \in \mathbb{N}^n$

$$ \int_{\Omega} u(x)\partial^{\alpha}v(x)dx = (-1)^{|\alpha|} \int_{\Omega} (\partial^{\alpha}u(x))v(x)dx \qquad (1.1.33) $$

(2) If $u \in G_{\ell}(\mathbb{R}^n)$ then integral

$$ \int_{\mathbb{R}^n} u(x)dx \triangleq \mathbf{cl} \left[ \left( \int_{\mathbb{R}^n} u_{\varepsilon}(x)dx \right)_{\varepsilon} \right] \qquad (1.1.34) $$

is a well-defined element of $\tilde{\mathbb{C}}$. Moreover, the following properties hold:

(i)

$$ \int_{\mathbb{R}^n} (\iota g)(x)dx = \mathbf{cl} \left[ \left( \int_{\mathbb{R}^n} g(x)dx \right)_{\varepsilon} \right] \qquad (1.1.34) $$

(ii)

$$ \int_{\mathbb{R}^n} u(x)\partial^{\alpha}v(x)dx = (-1)^{|\alpha|} \int_{\mathbb{R}^n} (\partial^{\alpha}u(x))v(x)dx \qquad (1.1.35) $$

for all $u, v \in G_{\ell}(\mathbb{R}^n)$ and $\alpha \in \mathbb{N}^n$.
**Definition.1.1.19.** Let $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n) \cup G_{\tau}(\mathbb{R}^n)$ and $\varphi$ be a mollifier in $\mathcal{L}(\mathbb{R}^n)$. We define the $\hat{\varphi}$-integral of $u$ as the complex generalized number by

$$ \int_{\mathbb{R}^n} u(x) d_{\hat{\varphi}} x \triangleq \int_{\mathbb{R}^n} u_{\hat{\varphi}}(x) dx = \mathbf{cl} \left[ \left( \int_{\mathbb{R}^n} u_{\varepsilon}(x) \hat{\varphi}_{\varepsilon}(x) dx \right) \right], \quad (1.1.36) $$

where $u_{\hat{\varphi}}(x) \triangleq (u_{\varepsilon}(x)\hat{\varphi}_{\varepsilon}(x))_{\varepsilon} + N_{\mathcal{L}}(\mathbb{R}^n)$.

**Theorem.1.1.3.** (1) If $u \in G_c(\mathbb{R}^n) \subseteq G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ then

$$ \int_{\Omega} u(x) dx = \int_{\mathbb{R}^n} u(x) d_{\hat{\varphi}} x. \quad (1.1.37) $$

(2) If $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and $v \in G_{\mathcal{L}}^{\infty}(\mathbb{R}^n) \subseteq G_{r, \mathcal{L}}(\mathbb{R}^n)$ then

$$ \int_{\mathbb{R}^n} u(x)v(x)dx = \int_{\mathbb{R}^n} u(x)v(x)d_{\hat{\varphi}}x. \quad (1.1.38) $$

(3) If $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and $v \in G_{\mathcal{L}}^{\infty}(\mathbb{R}^n)$ then for all $\alpha \in \mathbb{N}^{\alpha}$

$$ \int_{\mathbb{R}^n} (\partial^\alpha u(x)) v(x) dx = (-1)^{|\alpha|} \int_{\mathbb{R}^n} u(x) (\partial^\alpha v(x)) d_{\hat{\varphi}} x. \quad (1.1.39) $$

**Definition.1.1.20.** Let $u, v$ be generalized functions in $G(\Omega)$. $u$ and $v$ are said to be equal in the generalized distributions sense ($u =_{g,d.} v$ for short) iff

$$ \int_{\mathbb{R}^n} [u(x) - v(x)][l(f)](x)dx = 0. \quad (1.1.40) $$
**Definition.1.1.19.** Let $u \in G_{\tau, \ell}(\mathbb{R}^n) \cup G_{\tau}(\mathbb{R}^n)$ and $\varphi$ be a mollifier in $\mathcal{L}(\mathbb{R}^n)$. We define the $\hat{\varphi}$-integral of $u$ as the complex generalized number by

$$ \int_{\mathbb{R}^n} u(x) d_{\hat{\varphi}} x \triangleq \int_{\mathbb{R}^n} u_{\hat{\varphi}}(x) dx = \mathbf{cl} \left[ \left( \int_{\mathbb{R}^n} u_{\varepsilon}(x) \hat{\varphi}_{\varepsilon}(x) dx \right) \right], \quad (1.1.36) $$

where $u_{\hat{\varphi}}(x) \triangleq (u_{\varepsilon}(x)\hat{\varphi}_{\varepsilon}(x))_{\varepsilon} + N_{\ell}(\mathbb{R}^n)$.

**Theorem.1.1.3.** (1) If $u \in G_c(\mathbb{R}^n) \subseteq G_{\tau, \ell}(\mathbb{R}^n)$ then

$$ \int_{\Omega} u(x) dx = \int_{\mathbb{R}^n} u(x) d_{\hat{\varphi}} x. \quad (1.1.37) $$

(2) If $u \in G_{\tau, \ell}(\mathbb{R}^n)$ and $v \in G_{\ell}^{\infty}(\mathbb{R}^n) \subseteq G_{r, \ell}(\mathbb{R}^n)$ then

$$ \int_{\mathbb{R}^n} u(x)v(x)dx = \int_{\mathbb{R}^n} u(x)v(x)d_{\hat{\varphi}}x. \quad (1.1.38) $$

(3) If $u \in G_{\tau, \ell}(\mathbb{R}^n)$ and $v \in G_{\ell}^{\infty}(\mathbb{R}^n)$ then for all $\alpha \in \mathbb{N}^{\alpha}$

$$ \int_{\mathbb{R}^n} (\partial^\alpha u(x))v(x)dx = (-1)^{|\alpha|} \int_{\mathbb{R}^n} u(x)(\partial^\alpha v(x))d_{\hat{\varphi}}x. \quad (1.1.39) $$

**Definition.1.1.20.** Let $u, v$ be generalized functions in $G(\Omega)$. $u$ and $v$ are said to be equal in the generalized distributions sense ($u =_{g,d.} v$ for short) iff

$$ \int_{\mathbb{R}^n} [u(x) - v(x)][l(f)](x)dx = 0. \quad (1.1.40) $$
for all $f \in C_c^\infty(\Omega)$.

**Definition.1.1.21.** Let $u, v$ be generalized functions in $G_{\tau, \mathcal{L}}(\Omega)$. $u$ and $v$ are said to be equal in the generalized distributions sense ($u =_{\text{g.t.d.}} v$ for short)

$$ \int_{\mathbb{R}^n} [u(x) - v(x)][l(f)](x)dx = 0. \qquad (1.1.41) $$

iff for all $f \in \mathcal{L}(\mathbb{R}^n)$.

**Theorem.1.1.4.** Let us consider the expression

$$ (\hat{ku})(x) = \int_{\Omega} k(x,y)u(y)dy = 0. \qquad (1.1.42) $$

(1) If $k \in G(\Omega \times \Omega)$ then (1.1.42) defines a $\tilde{C}$-linear map $\hat{k} : G_c(\Omega) \to G(\Omega)$ : $u \to \hat{ku}$, where $\hat{ku}$ is the generalized function with representative

$$ ((\hat{k}_\varepsilon u_\varepsilon)(x))_\varepsilon = \left( \int_{\Omega} k_\varepsilon(x,y)u_\varepsilon(y)dy \right)_\varepsilon. \qquad (1.1.43) $$

(2) If $k \in G^\infty(\Omega \times \Omega)$ then $\hat{k}$ maps $G_c(\Omega)$ into $G^\infty(\Omega)$.

(3) If $k \in G_c(\Omega \times \Omega)$ then $\hat{k}$ maps $G(\Omega)$ into $G_c(\Omega)$.

(4) If $k \in G_c^\infty(\Omega \times \Omega)$ then $\hat{k}$ maps $G(\Omega)$ into $G_c^\infty(\Omega)$.

(5) If $k \in G(\Omega \times \Omega)$ and $\pi_1, \pi_2 : \text{supp}(k) \to \Omega$ are proper then $\text{supp}(\hat{ku}) \in \Omega$ for all $u \in G_c(\Omega)$ and $k$ can be uniquely extended to a $\tilde{C}$-linear map from $G(\Omega)$ into $G(\Omega)$ such that for all $u \in G(\Omega)$ and $v \in G_c(\Omega)$
for all $f \in C_c^\infty(\Omega)$.

**Definition.1.1.21.** Let $u, v$ be generalized functions in $G_{\tau, \mathcal{L}}(\Omega)$. $u$ and $v$ are said to be equal in the generalized distributions sense ($u =_{\text{g.t.d.}} v$ for short)

$$ \int_{\mathbb{R}^n} [u(x) - v(x)][l(f)](x)dx = 0. \qquad (1.1.41) $$

iff for all $f \in \mathcal{L}(\mathbb{R}^n)$.

**Theorem.1.1.4.** Let us consider the expression

$$ (\hat{ku})(x) = \int_{\Omega} k(x,y)u(y)dy = 0. \qquad (1.1.42) $$

(1) If $k \in G(\Omega \times \Omega)$ then (1.1.42) defines a $\tilde{C}$-linear map $\hat{k} : G_c(\Omega) \to G(\Omega)$ : $u \to \hat{ku}$, where $\hat{ku}$ is the generalized function with representative

$$ ((\hat{k}_\varepsilon u_\varepsilon)(x))_\varepsilon = \left( \int_{\Omega} k_\varepsilon(x,y)u_\varepsilon(y)dy \right)_\varepsilon. \qquad (1.1.43) $$

(2) If $k \in G^\infty(\Omega \times \Omega)$ then $\hat{k}$ maps $G_c(\Omega)$ into $G^\infty(\Omega)$.

(3) If $k \in G_c(\Omega \times \Omega)$ then $\hat{k}$ maps $G(\Omega)$ into $G_c(\Omega)$.

(4) If $k \in G_c^\infty(\Omega \times \Omega)$ then $\hat{k}$ maps $G(\Omega)$ into $G_c^\infty(\Omega)$.

(5) If $k \in G(\Omega \times \Omega)$ and $\pi_1, \pi_2 : \text{supp}(k) \to \Omega$ are proper then $\text{supp}(\hat{ku}) \in \Omega$ for all $u \in G_c(\Omega)$ and $k$ can be uniquely extended to a $\tilde{C}$-linear map from $G(\Omega)$ into $G(\Omega)$ such that for all $u \in G(\Omega)$ and $v \in G_c(\Omega)$
$$ \int_{\Omega} (\hat{k}u)(x)v(x)dx = \int_{\Omega} v(y)(\hat{k}\hat{k}v)(y)dy, \qquad (1.1.44) $$

$$ (\hat{k}v)(y) = \int_{\Omega} \hat{k}(x,y)v(x)dx. $$

(6) If $k \in G^{\infty}(\Omega \times \Omega)$ and $\pi_1, \pi_2: \text{supp}(k) \to \Omega$ are proper then the extension defined above maps $G(\Omega)$ into $G^{\infty}(\Omega)$.

(7) Let us now consider the expression

$$ (\hat{k}u)(y) = \int_{\mathbb{R}^n} \hat{k}(x,y)u(y)dy. \qquad (1.1.45) $$

If $k \in G_{\tau}(\mathbb{R}^{2n})$ has a representative $(k_{\varepsilon})_{\varepsilon}$ such that $(k_{\varepsilon})_{\varepsilon}$ satisfying the condition

$$ \forall \alpha(\alpha \in \mathbb{N}^n) \forall s(s \in \mathbb{N}) \exists N(N \in \mathbb{N}) $$

$$ \left[ \sup_{x \in \mathbb{R}^n} (1 + |x|)^{-N} \sup_{y \in \mathbb{R}^n, \beta \le s} (1 + |y|)^{-s} \left| \partial_x^\alpha \partial_y^\beta k_\varepsilon(x,y) \right| = O(\varepsilon^{-N}) \quad \text{as } \varepsilon \to 0 \right], \qquad (1.1.46) $$

then $\hat{k}$ maps $G_{\tau}(\mathbb{R}^n)$ into $G_{\tau}(\mathbb{R}^n)$.

(8) If $k \in G_{\ell}(\mathbb{R}^{2n})$ then $\hat{k}$ maps $G_{\tau}(\mathbb{R}^n)$ into $G_{\ell}(\mathbb{R}^n)$.

(9) If $k \in G_{\ell}^{\infty}(\mathbb{R}^{2n})$ then $\hat{k}$ maps $G_{\tau}(\mathbb{R}^n)$ into $G_{\ell}^{\infty}(\mathbb{R}^n)$.

### I.1.6. Fourier transform in Colombeau theory

**Definition.1.1.22.** Let $u \in G_{\ell}(\mathbb{R}^n)$. The Fourier transform $F_u$ and the inverse Fourier transform $F^*u$ of $u$ are defined by the corresponding transformations at the level of
$$ \int_{\Omega} (\hat{k}u)(x)v(x)dx = \int_{\Omega} v(y)(\hat{k}\hat{k}v)(y)dy, \qquad (1.1.44) $$

$$ (\hat{k}v)(y) = \int_{\Omega} \hat{k}(x,y)v(x)dx. $$

(6) If $k \in G^{\infty}(\Omega \times \Omega)$ and $\pi_1, \pi_2: \text{supp}(k) \to \Omega$ are proper then the extension defined above maps $G(\Omega)$ into $G^{\infty}(\Omega)$.

(7) Let us now consider the expression

$$ (\hat{k}u)(y) = \int_{\mathbb{R}^n} \hat{k}(x,y)u(y)dy. \qquad (1.1.45) $$

If $k \in G_{\tau}(\mathbb{R}^{2n})$ has a representative $(k_{\varepsilon})_{\varepsilon}$ such that $(k_{\varepsilon})_{\varepsilon}$ satisfying the condition

$$ \forall \alpha(\alpha \in \mathbb{N}^n) \forall s(s \in \mathbb{N}) \exists N(N \in \mathbb{N}) $$

$$ \left[ \sup_{x \in \mathbb{R}^n} (1 + |x|)^{-N} \sup_{y \in \mathbb{R}^n, \beta \le s} (1 + |y|)^{-s} \left| \partial_x^\alpha \partial_y^\beta k_\varepsilon(x,y) \right| = O(\varepsilon^{-N}) \quad \text{as } \varepsilon \to 0 \right], \qquad (1.1.46) $$

then $\hat{k}$ maps $G_{\tau}(\mathbb{R}^n)$ into $G_{\tau}(\mathbb{R}^n)$.

(8) If $k \in G_{\ell}(\mathbb{R}^{2n})$ then $\hat{k}$ maps $G_{\tau}(\mathbb{R}^n)$ into $G_{\ell}(\mathbb{R}^n)$.

(9) If $k \in G_{\ell}^{\infty}(\mathbb{R}^{2n})$ then $\hat{k}$ maps $G_{\tau}(\mathbb{R}^n)$ into $G_{\ell}^{\infty}(\mathbb{R}^n)$.

### I.1.6. Fourier transform in Colombeau theory

**Definition.1.1.22.** Let $u \in G_{\ell}(\mathbb{R}^n)$. The Fourier transform $F_u$ and the inverse Fourier transform $F^*u$ of $u$ are defined by the corresponding transformations at the level of
representatives.

Thus $\mathcal{F}u$ and $\mathcal{F}^*u$ map $G_\mathcal{L}(\mathbb{R}^n)$ into itself and extend the classical Fourier transform on $\mathcal{L}(\mathbb{R}^n)$.

**Theorem.1.1.5.** For all $u, v \in G_\mathcal{L}(\mathbb{R}^n)$ and $\alpha \in \mathbb{N}^n$
(1)

$$ \int_{\mathbb{R}^n} (\mathcal{F}u)(x)v(v)dx = \int_{\mathbb{R}^n} u(x)(\mathcal{F}v)(v)dx, \qquad (1.1.47) $$

$$ \int_{\mathbb{R}^n} (\mathcal{F}^*u)(x)v(v)dx = \int_{\mathbb{R}^n} u(x)(\mathcal{F}^*v)(v)dx. $$

(2)

$$ u = \mathcal{F}[\mathcal{F}^*u] = \mathcal{F}^*[\mathcal{F}u]. \qquad (1.1.48) $$

(3)

$$ \mathcal{F}[\iota(y^\alpha)u] = i^{|\alpha|} \partial^\alpha \mathcal{F}[u], \qquad (1.1.49) $$

$$ \mathcal{F}^*[\iota(y^\alpha)u] = (-i)^{|\alpha|} \partial^\alpha \mathcal{F}^*[u]. $$

(4)

$$ (-i)^{|\alpha|} \mathcal{F}[\partial^\alpha u] = \iota(y^\alpha) \mathcal{F}u, \qquad (1.1.50) $$

$$ i^{|\alpha|} \mathcal{F}^*[\partial^\alpha u] = \iota(y^\alpha) \mathcal{F}^*u, $$

where $\iota(y^\alpha)$ is intended as the embedding of $y^\alpha$ in $G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and the product $\iota(y^\alpha)u \in G_\mathcal{L}(\mathbb{R}^n)$.

**Remark 1.1.6.** Note that the generalized function $\iota(y^\alpha)$ has $y^\alpha$ as a representative since $(\phi_\varepsilon * y^\alpha)(x) = x^\alpha$.
representatives.

Thus $\mathcal{F}u$ and $\mathcal{F}^*u$ map $G_\ell(\mathbb{R}^n)$ into itself and extend the classical Fourier transform on $\mathcal{L}(\mathbb{R}^n)$.

**Theorem.1.1.5.** For all $u, v \in G_\ell(\mathbb{R}^n)$ and $\alpha \in \mathbb{N}^n$
(1)

$$ \int_{\mathbb{R}^n} (\mathcal{F}u)(x)v(v)dx = \int_{\mathbb{R}^n} u(x)(\mathcal{F}v)(v)dx, \qquad (1.1.47) $$

$$ \int_{\mathbb{R}^n} (\mathcal{F}^*u)(x)v(v)dx = \int_{\mathbb{R}^n} u(x)(\mathcal{F}^*v)(v)dx. $$

(2)

$$ u = \mathcal{F}[\mathcal{F}^*u] = \mathcal{F}^*[\mathcal{F}u]. \qquad (1.1.48) $$

(3)

$$ \mathcal{F}[\iota(y^\alpha)u] = i^{|\alpha|} \partial^\alpha \mathcal{F}[u], \qquad (1.1.49) $$

$$ \mathcal{F}^*[\iota(y^\alpha)u] = (-i)^{|\alpha|} \partial^\alpha \mathcal{F}^*[u]. $$

(4)

$$ (-i)^{|\alpha|} \mathcal{F}[\partial^\alpha u] = \iota(y^\alpha) \mathcal{F}u, \qquad (1.1.50) $$

$$ i^{|\alpha|} \mathcal{F}^*[\partial^\alpha u] = \iota(y^\alpha) \mathcal{F}^*u, $$

where $\iota(y^\alpha)$ is intended as the embedding of $y^\alpha$ in $G_{\tau, \ell}(\mathbb{R}^n)$ and the product $\iota(y^\alpha)u \in G_\ell(\mathbb{R}^n)$.

**Remark 1.1.6.** Note that the generalized function $\iota(y^\alpha)$ has $y^\alpha$ as a representative since $(\phi_\epsilon * y^\alpha)(x) = x^\alpha$.
**Definition.1.1.23.** Let $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and $\varphi$ be a mollifier in $\mathcal{L}(\mathbb{R}^n)$. We define the $\hat{\varphi}$-Fourier transform of $u$ as the generalized function in $G_{\mathcal{L}}(\mathbb{R}^n)$ given by

$$ \mathcal{F}_{\hat{\varphi}}[u] \triangleq \mathcal{F}[u_{\hat{\varphi}}] \qquad (1.1.51) $$

and the inverse $\hat{\varphi}$-Fourier transform of $u$ as

$$ \mathcal{F}_{\hat{\varphi}}^{*}[u] \triangleq \mathcal{F}^{*}[u_{\hat{\varphi}}]. \qquad (1.1.52) $$

**Remark 1.1.7.** Note that $\mathcal{F}_{\hat{\varphi}}[u]$ and $\mathcal{F}_{\hat{\varphi}}^{*}[u]$ have representatives of the form

$$ \begin{gathered} \int_{\mathbb{R}^n} (\exp[-iy\xi])u_\varepsilon(y)\hat{\varphi}_\varepsilon(y)dy, \qquad (1.1.53) \\ \int_{\mathbb{R}^n} (\exp[iy\xi])u_\xi(\xi)\hat{\varphi}_\xi(\xi)dy \end{gathered} $$

respectively, with $d\xi = (2\pi)^{-n}d\xi$.

**Theorem.1.1.6.** (1) $\mathcal{F}_{\hat{\varphi}}[u] = \mathcal{F}[u]$ on $G_{\mathcal{L}}^{\infty}(\mathbb{R}^n)$ and $G_c(\Omega)$.

(2) If $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and $v \in G_{\mathcal{L}}^{\infty}(\mathbb{R}^n)$ then

$$ \int_{\mathbb{R}^n} \mathcal{F}_{\hat{\varphi}}[u](x)v(x)dx = \int_{\mathbb{R}^n} u(x)\mathcal{F}_{\hat{\varphi}}[v](x)dx. \qquad (1.1.54) $$

(3) If $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and $v \in G_{\mathcal{L}}^{\infty}(\mathbb{R}^n)$ then

$$ \int_{\mathbb{R}^n} \mathcal{F}_{\hat{\varphi}}^{*}[u](x)v(x)dx = \int_{\mathbb{R}^n} u(x)\mathcal{F}_{\hat{\varphi}}^{*}[v](x)dx. \qquad (1.1.55) $$
**Definition.1.1.23.** Let $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and $\varphi$ be a mollifier in $\mathcal{L}(\mathbb{R}^n)$. We define the $\hat{\varphi}$-Fourier transform of $u$ as the generalized function in $G_{\mathcal{L}}(\mathbb{R}^n)$ given by

$$ \mathcal{F}_{\hat{\varphi}}[u] \triangleq \mathcal{F}[u_{\hat{\varphi}}] \qquad (1.1.51) $$

and the inverse $\hat{\varphi}$-Fourier transform of $u$ as

$$ \mathcal{F}_{\hat{\varphi}}^*[u] \triangleq \mathcal{F}^*[u_{\hat{\varphi}}]. \qquad (1.1.52) $$

**Remark 1.1.7.** Note that $\mathcal{F}_{\hat{\varphi}}[u]$ and $\mathcal{F}_{\hat{\varphi}}^*[u]$ have representatives of the form

$$ \int_{\mathbb{R}^n} (\exp[-iy\xi])u_{\varepsilon}(y)\hat{\varphi}_{\varepsilon}(y)dy, \qquad (1.1.53) \\ \int_{\mathbb{R}^n} (\exp[iy\xi])u_{\varepsilon}(\xi)\hat{\varphi}_{\varepsilon}(\xi)dy $$

respectively, with $d\xi = (2\pi)^{-n}d\xi$.

**Theorem.1.1.6.** (1) $\mathcal{F}_{\hat{\varphi}}[u] = \mathcal{F}[u]$ on $G_{\mathcal{L}}^{\infty}(\mathbb{R}^n)$ and $G_c(\Omega)$.

(2) If $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and $v \in G_{\mathcal{L}}^{\infty}(\mathbb{R}^n)$ then

$$ \int_{\mathbb{R}^n} \mathcal{F}_{\hat{\varphi}}[u](x)v(x)dx = \int_{\mathbb{R}^n} u(x)\mathcal{F}_{\hat{\varphi}}[v](x)dx. \qquad (1.1.54) $$

(3) If $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and $v \in G_{\mathcal{L}}^{\infty}(\mathbb{R}^n)$ then

$$ \int_{\mathbb{R}^n} \mathcal{F}_{\hat{\varphi}}^*[u](x)v(x)dx = \int_{\mathbb{R}^n} u(x)\mathcal{F}_{\hat{\varphi}}^*[v](x)dx. \qquad (1.1.55) $$
(4) If $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ then

$$ \mathcal{F}_{\hat{\phi}}[\iota(y^{\alpha})u] = i^{|\alpha|} \partial^{\alpha} \mathcal{F}_{\hat{\phi}}[u], \qquad (1.1.56) $$

$$ \mathcal{F}_{\hat{\phi}}^{*}[\iota(y^{\alpha})] = (-i)^{|\alpha|} \partial^{\alpha} \mathcal{F}_{\hat{\phi}}^{*}[u]. $$

(5) If $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and $v \in G_{\mathcal{L}}^{\infty}(\mathbb{R}^n)$ then

$$ (-i)^{|\alpha|} \int_{\mathbb{R}^n} \mathcal{F}_{\hat{\phi}}[\partial^\alpha u](x)v(x)dx = \int_{\mathbb{R}^n} (\iota(y^\alpha)\mathcal{F}_{\hat{\phi}}[u])(x)v(x)dx, \qquad (1.1.57) $$

$$ i^{|\alpha|} \int_{\mathbb{R}^n} \mathcal{F}_{\hat{\phi}}^*[\partial^\alpha u](x)v(x)dx = \int_{\mathbb{R}^n} (\iota(y^\alpha)\mathcal{F}_{\hat{\phi}}^*[u])(x)v(x)dx. $$

### I.1.7. Topological $\tilde{\mathbb{C}}$-modules

**Definition.1.1.24.** An abelian group $(G,+)$ is a $\tilde{\mathbb{C}}$-module if there is given a (product) map $\tilde{\mathbb{C}} \times G \to G : (\lambda, u) \to \lambda u$ such that for all $\lambda, \mu \in \tilde{\mathbb{C}}$ and $u, v \in G$:

$$ \begin{align} (\lambda + \mu)u &= \lambda u + \mu u, \notag \\ \mu(u + v) &= \mu u + \mu v, \tag{1.1.58} \end{align} $$

$$ \lambda_1(\lambda_2)u = (\lambda_1 \lambda_2)u, $$

$$ 1u = u. $$
(4) If $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ then

$$ \mathcal{F}_{\hat{\phi}}[\iota(y^{\alpha})u] = i^{|\alpha|} \partial^{\alpha} \mathcal{F}_{\hat{\phi}}[u], \qquad (1.1.56) $$

$$ \mathcal{F}_{\hat{\phi}}^{*}[\iota(y^{\alpha})] = (-i)^{|\alpha|} \partial^{\alpha} \mathcal{F}_{\hat{\phi}}^{*}[u]. $$

(5) If $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and $v \in G_{\mathcal{L}}^{\infty}(\mathbb{R}^n)$ then

$$ (-i)^{|\alpha|} \int_{\mathbb{R}^n} \mathcal{F}_{\hat{\phi}}[\partial^\alpha u](x)v(x)dx = \int_{\mathbb{R}^n} (\iota(y^\alpha)\mathcal{F}_{\hat{\phi}}[u])(x)v(x)dx, \qquad (1.1.57) $$

$$ i^{|\alpha|} \int_{\mathbb{R}^n} \mathcal{F}_{\hat{\phi}}^*[\partial^\alpha u](x)v(x)dx = \int_{\mathbb{R}^n} (\iota(y^\alpha)\mathcal{F}_{\hat{\phi}}^*[u])(x)v(x)dx. $$

### I.1.7. Topological $\tilde{\mathbb{C}}$-modules

**Definition.1.1.24.** An abelian group $(G,+)$ is a $\tilde{\mathbb{C}}$-module if there is given a (product) map $\tilde{\mathbb{C}} \times G \to G : (\lambda, u) \to \lambda u$ such that for all $\lambda, \mu \in \tilde{\mathbb{C}}$ and $u, v \in G$:

$$ \begin{align} (\lambda + \mu)u &= \lambda u + \mu u, \notag \\ \mu(u + v) &= \mu u + \mu v, \tag{1.1.58} \end{align} $$

$$ \lambda_1(\lambda_2)u = (\lambda_1\lambda_2)u, $$

$$ 1u = u. $$
**Definition.1.1.25.** Let us now introduce the function $v[(u_\epsilon)_\epsilon]$ on $E(M)$:

$$ v : E(M) \to (-\infty, +\infty] : (u_\epsilon)_\epsilon \to \sup\{\gamma \in \mathbb{R} \mid \|u_\epsilon\| = O(\epsilon^\gamma) \text{ as } \epsilon \to 0\} \quad (1.1.59) $$

It satisfies the following properties:

(1) $v[(u_\epsilon)_\epsilon] = -\infty$ iff $(u_\epsilon)_\epsilon \in N(M)$,

(2) $v[(u_\epsilon)_\epsilon(v_\epsilon)_\epsilon] \ge v[(u_\epsilon)_\epsilon] + v[(v_\epsilon)_\epsilon]$,

(3) $v[(u_\epsilon)_\epsilon + v(v_\epsilon)_\epsilon] \ge \min\{v[(u_\epsilon)_\epsilon], v[(v_\epsilon)_\epsilon]\}$.

**Definition.1.1.26.** Let us define the valuation of the complex generalized numbers

$$ u = \mathrm{cl}[(u_\epsilon)_\epsilon] $$

$$ v_{\tilde{\zeta}}(u) \triangleq v[(u_\epsilon)_\epsilon] \quad (1.1.60) $$

Note that all the previous properties hold for the elements of $\tilde{\mathbb{C}}$.

**Definition.1.1.27.** We let

$$ |\cdot|_e \triangleq \tilde{\mathbb{C}} \to [0, +\infty) : u \to |u|_e = \exp[-v_{\tilde{\zeta}}(u)] \quad (1.1.61) $$

**Remark 1.1.8.** Note that the properties of the valuation on $\tilde{\mathbb{C}}$ makes the coarsest topology on $\tilde{\mathbb{C}}$ such that the map $|\cdot|_e$ is continuous compatible with the ring structure.

**Definition.1.1.28.** Let $G$ be a $\tilde{\mathbb{C}}$-module. A valuation on $G$ is a function $v : G \to (-\infty, +\infty]$ such that for all $\lambda \in \tilde{\mathbb{C}}$ and $u, v \in G$:

(1) $v(0) = +\infty$,

(2) $v(\lambda u) \ge v_{\tilde{\zeta}}(\lambda) + v(u)$,

(2') $v(\lambda u) \ge v_{\tilde{\zeta}}(\lambda) + v(u)$ for all $\lambda = \mathrm{cl}[(c\epsilon^a)_\epsilon], c \in \mathbb{C}, a \in \mathbb{R}$,
**Definition.1.1.25.** Let us now introduce the function $v[(u_\epsilon)_\epsilon]$ on $E(M)$:

$$ v : E(M) \to (-\infty, +\infty] : (u_\epsilon)_\epsilon \to \sup\{\gamma \in \mathbb{R} \mid \|u_\epsilon\| = O(\epsilon^\gamma) \text{ as } \epsilon \to 0\} \quad (1.1.59) $$

It satisfies the following properties:

(1) $v[(u_\epsilon)_\epsilon] = -\infty$ iff $(u_\epsilon)_\epsilon \in \mathcal{N}(M)$,

(2) $v[(u_\epsilon)_\epsilon(v_\epsilon)_\epsilon] \ge v[(u_\epsilon)_\epsilon] + v[(v_\epsilon)_\epsilon]$,

(3) $v[(u_\epsilon)_\epsilon + v(v_\epsilon)_\epsilon] \ge \min\{v[(u_\epsilon)_\epsilon], v[(v_\epsilon)_\epsilon]\}$.

**Definition.1.1.26.** Let us define the valuation of the complex generalized numbers

$$ u = \mathrm{cl}[(u_\epsilon)_\epsilon] $$

$$ v_{\tilde{\zeta}}(u) \triangleq v[(u_\epsilon)_\epsilon] \quad (1.1.60) $$

Note that all the previous properties hold for the elements of $\tilde{\mathbb{C}}$.

**Definition.1.1.27.** We let

$$ |\cdot|_e \triangleq \tilde{\mathbb{C}} \to [0, +\infty) : u \to |u|_e = \exp[-v_{\tilde{\zeta}}(u)] \quad (1.1.61) $$

**Remark 1.1.8.** Note that the properties of the valuation on $\tilde{\mathbb{C}}$ makes the coarsest topology on $\tilde{\mathbb{C}}$ such that the map $|\cdot|_e$ is continuous compatible with the ring structure.

**Definition.1.1.28.** Let $G$ be a $\tilde{\mathbb{C}}$-module. A valuation on $G$ is a function $v : G \to (-\infty, +\infty]$ such that for all $\lambda \in \tilde{\mathbb{C}}$ and $u, v \in G$:

(1) $v(0) = +\infty$,

(2) $v(\lambda u) \ge v_{\tilde{\zeta}}(\lambda) + v(u)$,

(2') $v(\lambda u) \ge v_{\tilde{\zeta}}(\lambda) + v(u)$ for all $\lambda = \mathrm{cl}[(c\epsilon^a)_\epsilon], c \in \mathbb{C}, a \in \mathbb{R}$,
$$ (3) \quad v(u + v) \geq \min\{v(u), v(v)\}. $$

**Definition.1.1.29.** An ultra-pseudo-seminorm on $G$ is a function

$\wp : G \to [0, +\infty)$ such that for all $\lambda \in \tilde{\mathbb{C}}$ and $u, v \in G$:

$$ (1) \quad \wp(0) = 0, $$

$$ (2) \quad \wp(\lambda u) \leq |\lambda|_{\mathbb{e}} \wp(u), $$

$$ (2') \quad \wp(\lambda u) = |\lambda|_{\mathbb{e}} \wp(u) \text{ for all } \lambda = \mathbf{cl}[(c\varepsilon^a)_{\varepsilon}], c \in \mathbb{C}, a \in \mathbb{R}, $$

$$ (3) \quad \wp(u + v) \leq \max\{\wp(u), \wp(v)\}. $$

**Remark 1.1.9.** (1) We note that a typical example of an ultra-pseudo-seminorm obtained by means of a valuation $v(\cdot)$ on $G$ is given via canonical formula

$$ \wp(u) = \exp[-v(u)]. \qquad (1.1.62) $$

An ultra-pseudo-norm is an ultra-pseudo-seminorm $\wp(\cdot)$ such that $\wp(0) = 0$ implies $u = 0$.

(2) The function $|\cdot|_{\mathbb{e}}$ introduced in (1.1.61) is an ultra-pseudo-norm on $\tilde{\mathbb{C}}$.

**Definition.1.1.30.** [9] (1) A subset $A$ of a $\tilde{\mathbb{C}}$-module $G$ is $\tilde{\mathbb{C}}$-absorbent if for all $u \in G$ there exists $a \in \mathbb{R}$ such that $u \in (\mathbf{cl}[(\varepsilon^b)_{\varepsilon}])A$ for all $b \leq a$.

(2) $A \subseteq G$ is $\tilde{\mathbb{C}}$-balanced if $\lambda A \subseteq A$ for all $\lambda \in \tilde{\mathbb{C}}$ with $|\lambda|_{\mathbb{e}} < 1$.

(3) $A \subseteq G$ is $\tilde{\mathbb{C}}$-convex if $A + A \subseteq A$ and $(\mathbf{cl}[(\varepsilon^b)_{\varepsilon}])A \subseteq A$ all $b \geq 0$.

We now present another nontrivial example of a valuation on a $\tilde{\mathbb{C}}$-module $G$.

**Definition.1.1.31.** [9]. A locally convex topological $\tilde{\mathbb{C}}$-module is a topological $\tilde{\mathbb{C}}$-module which has a base of $\tilde{\mathbb{C}}$-convex neighborhoods of the origin.

**Theorem.1.1.7.** [9]. Every locally convex topological $\tilde{\mathbb{C}}$-module $G$ has a base of absolutely convex and absorbent neighborhoods of the origin.

**Theorem.1.1.8.** [9]. Let $A$ be an absolutely convex and absorbent subset of a $\tilde{\mathbb{C}}$-module $G$. Then

$$ v_A(u) = \sup\{\gamma \in \mathbb{R}|u \in (\mathbf{cl}[(\varepsilon^\gamma)])A\} \qquad (1.1.63) $$

is a valuation on $G$. Moreover, for $\wp_A(u) = \exp[-v_A(u)]$ and $\eta > 0$ the chain of
$$ (3) \quad v(u + v) \geq \min\{v(u), v(v)\}. $$

**Definition.1.1.29.** An ultra-pseudo-seminorm on $G$ is a function

$\wp : G \to [0, +\infty)$ such that for all $\lambda \in \tilde{\mathbb{C}}$ and $u, v \in G$:

$$ (1) \quad \wp(0) = 0, $$

$$ (2) \quad \wp(\lambda u) \leq |\lambda|_{\mathfrak{e}} \wp(u), $$

$$ (2') \quad \wp(\lambda u) = |\lambda|_{\mathfrak{e}} \wp(u) \text{ for all } \lambda = \mathbf{cl}[(c\varepsilon^a)_\varepsilon], c \in \mathbb{C}, a \in \mathbb{R}, $$

$$ (3) \quad \wp(u + v) \leq \max\{\wp(u), \wp(v)\}. $$

**Remark 1.1.9.** (1) We note that a typical example of an ultra-pseudo-seminorm obtained by means of a valuation $v(\cdot)$ on $G$ is given via canonical formula

$$ \wp(u) = \exp[-v(u)]. \qquad (1.1.62) $$

An ultra-pseudo-norm is an ultra-pseudo-seminorm $\wp(\cdot)$ such that $\wp(0) = 0$ implies $u = 0$.

(2) The function $|\cdot|_{\mathfrak{e}}$ introduced in (1.1.61) is an ultra-pseudo-norm on $\tilde{\mathbb{C}}$.

**Definition.1.1.30.** [9] (1) A subset $A$ of a $\tilde{\mathbb{C}}$-module $G$ is $\tilde{\mathbb{C}}$-absorbent if for all $u \in G$ there exists $a \in \mathbb{R}$ such that $u \in (\mathbf{cl}[(\varepsilon^b)_\varepsilon])A$ for all $b \leq a$.

(2) $A \subseteq G$ is $\tilde{\mathbb{C}}$-balanced if $\lambda A \subseteq A$ for all $\lambda \in \tilde{\mathbb{C}}$ with $|\lambda|_{\mathfrak{e}} < 1$.

(3) $A \subseteq G$ is $\tilde{\mathbb{C}}$-convex if $A + A \subseteq A$ and $(\mathbf{cl}[(\varepsilon^b)_\varepsilon])A \subseteq A$ all $b \geq 0$.

We now present another nontrivial example of a valuation on a $\tilde{\mathbb{C}}$-module $G$.

**Definition.1.1.31.** [9]. A locally convex topological $\tilde{\mathbb{C}}$-module is a topological $\tilde{\mathbb{C}}$-module which has a base of $\tilde{\mathbb{C}}$-convex neighborhoods of the origin.

**Theorem.1.1.7.** [9]. Every locally convex topological $\tilde{\mathbb{C}}$-module $G$ has a base of absolutely convex and absorbent neighborhoods of the origin.

**Theorem.1.1.8.** [9]. Let $A$ be an absolutely convex and absorbent subset of a $\tilde{\mathbb{C}}$-module $G$. Then

$$ v_A(u) = \sup\{\gamma \in \mathbb{R}|u \in (\mathbf{cl}[(\varepsilon^\gamma)])A\} \qquad (1.1.63) $$

is a valuation on $G$. Moreover, for $\wp_A(u) = \exp[-v_A(u)]$ and $\eta > 0$ the chain of
inclusion

$$ \{u \in G | \wp_A(u) < \eta\} \subseteq (\mathbf{cl}[(\varepsilon^{-\ln\eta})_\varepsilon])A \subseteq \{u \in G | \wp_A(u) \le \eta\} \quad (1.1.64) $$

holds.

### 1.1.9. Definition and basic properties of $G_E$ and $\tilde{G}_E$

**Definition 1.1.33.** Let $E$ be a locally convex topological vector space topologized through the family of seminorms $\{p_i\}_{i \in I}$. The elements of

$$ M_E \triangleq \{(u_\varepsilon)_\varepsilon \in E^{(0,1]} | \forall i(i \in I) \exists N(N \in \mathbb{N}) [p_i(u_\varepsilon) = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0]\}, \quad (1.1.65) $$

are called $E$-moderate. The elements of

$$ N_E \triangleq \{(u_\varepsilon)_\varepsilon \in E^{(0,1]} | \forall i(i \in I) \exists q(q \in \mathbb{N}) [p_i(u_\varepsilon) = O(\varepsilon^q) \text{ as } \varepsilon \to 0]\}, \quad (1.1.66) $$

are called $E$-negligible. We define the space of generalized functions based on $E$ as the factor space

$$ G_E \triangleq M_E/N_E. \quad (1.1.67) $$

We note that the definition of $G_E$ does not depend on the family of seminorms which determines the locally convex topology of $E$. We adopt the notation $u = \mathbf{cl}[(u_\varepsilon)_\varepsilon]$ for the class $u$ of $(u_\varepsilon)_\varepsilon$ in $G_E$ and we embed $E$ into $G_E$ via the constant embedding $e \mapsto \mathbf{cl}[(e_\varepsilon)_\varepsilon]$. By the properties of seminorms on $E$ we may define the
inclusion

$$ \{u \in G | \wp_A(u) < \eta\} \subseteq (\mathbf{cl}[(\varepsilon^{-\ln\eta})_\varepsilon])A \subseteq \{u \in G | \wp_A(u) \le \eta\} \quad (1.1.64) $$

holds.

### 1.1.9. Definition and basic properties of $G_E$ and $\tilde{G}_E$

**Definition 1.1.33.** Let $E$ be a locally convex topological vector space topologized through the family of seminorms $\{p_i\}_{i \in I}$. The elements of

$$ M_E \triangleq \{(u_\varepsilon)_\varepsilon \in E^{(0,1]} | \forall i(i \in I) \exists N(N \in \mathbb{N}) [p_i(u_\varepsilon) = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0]\}, \quad (1.1.65) $$

are called $E$-moderate. The elements of

$$ N_E \triangleq \{(u_\varepsilon)_\varepsilon \in E^{(0,1]} | \forall i(i \in I) \exists q(q \in \mathbb{N}) [p_i(u_\varepsilon) = O(\varepsilon^q) \text{ as } \varepsilon \to 0]\}, \quad (1.1.66) $$

are called $E$-negligible. We define the space of generalized functions based on $E$ as the factor space

$$ G_E \triangleq M_E/N_E. \quad (1.1.67) $$

We note that the definition of $G_E$ does not depend on the family of seminorms which determines the locally convex topology of $E$. We adopt the notation $u = \mathbf{cl}[(u_\varepsilon)_\varepsilon]$ for the class $u$ of $(u_\varepsilon)_\varepsilon$ in $G_E$ and we embed $E$ into $G_E$ via the constant embedding $e \mapsto \mathbf{cl}[(e_\varepsilon)_\varepsilon]$. By the properties of seminorms on $E$ we may define the
product between complex generalized numbers and elements of $G_E$ via the map

$$ \tilde{\mathbb{C}} \times G_E \to G_E : (\mathbf{cl}[(\lambda_\varepsilon)_\varepsilon], \mathbf{cl}[(u_\varepsilon)_\varepsilon]) \to \mathbf{cl}[(\lambda_\varepsilon u_\varepsilon)_\varepsilon], \quad (1.1.68) $$

which equips $G_E$ with the structure of a $\tilde{\mathbb{C}}$-module.

Since the growth in $\varepsilon$ of an $E$-moderate net is estimated in terms of any seminorm $p_i$ of $E$, it is natural to introduce the $p_i$-valuation of $(u_\varepsilon)_\varepsilon \in M_E$ as

$$ \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon] = \sup\{b \in \mathbb{R}| p_i(u_\varepsilon) = O(\varepsilon^b) \text{ as } \varepsilon \to 0\} \quad (1.1.69) $$

Note that

$$ \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon] = \tilde{\mathbf{v}}_{\varepsilon}[(p_i(u_\varepsilon))_\varepsilon] \quad (1.1.70) $$

where the function $\tilde{\mathbf{v}}_\varepsilon$ in (1.1.70) gives the valuation on $\tilde{\mathbb{C}}$.

**Theorem.1.1.9.** The function $\mathbf{v}_{p_i}$ maps $M_E$ into $(-\infty, +\infty]$ and the following properties hold:

(i) $\mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon] = +\infty$ for all $i \in I$ iff $(u_\varepsilon)_\varepsilon \in N_E$,

(ii) $\mathbf{v}_{p_i}[(\lambda_\varepsilon u_\varepsilon)_\varepsilon] \ge \mathbf{v}[(\lambda_\varepsilon)_\varepsilon] + \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon]$ for all $(\lambda_\varepsilon)_\varepsilon \in E_M, (u_\varepsilon)_\varepsilon \in M_E$,

(ii)' $\mathbf{v}_{p_i}[(\lambda_\varepsilon u_\varepsilon)_\varepsilon] = \mathbf{v}[(\lambda_\varepsilon)_\varepsilon] + \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon]$ for all $(\lambda_\varepsilon)_\varepsilon = (c\varepsilon^b)_\varepsilon, c \in \mathbb{C}, (u_\varepsilon)_\varepsilon \in M_E$,

(iii) $\mathbf{v}_{p_i}[(w_\varepsilon + u_\varepsilon)_\varepsilon] \ge \inf\{\mathbf{v}_{p_i}[(w_\varepsilon)_\varepsilon], \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon]\}$.

Property (i) combined with (iii) shows that $\mathbf{v}_{p_i}[(w_\varepsilon)_\varepsilon] = \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon]$ iff $\mathbf{v}_{p_i}[(w_\varepsilon - u_\varepsilon)_\varepsilon]$ is $E$-negligible.

Therefore we can use (1.1.70) for defining the $p_i$-valuation $\mathbf{v}_{p_i}[u] = \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon]$ of a generalized function $u = \mathbf{cl}[(u_\varepsilon)_\varepsilon] \in G_E$. Thus

$$ \wp_i[u] = \exp[-\mathbf{v}_{p_i}[u]] \quad (1.1.71) $$

is an ultra-pseudo-seminorm on the $\tilde{\mathbb{C}}$-module $G_E$.

**Theorem.1.1.10.** Let $E$ be a locally convex topological vector space.

(i) If $E$ is topologized through an increasing sequence $\{p_i\}_{i \in \mathbb{N}}$ of seminorms and
product between complex generalized numbers and elements of $G_E$ via the map

$$ \tilde{\mathbb{C}} \times G_E \to G_E : (\mathbf{cl}[(\lambda_\varepsilon)_\varepsilon], \mathbf{cl}[(u_\varepsilon)_\varepsilon]) \to \mathbf{cl}[(\lambda_\varepsilon u_\varepsilon)_\varepsilon], \quad (1.1.68) $$

which equips $G_E$ with the structure of a $\tilde{\mathbb{C}}$-module.

Since the growth in $\varepsilon$ of an $E$-moderate net is estimated in terms of any seminorm $p_i$ of $E$, it is natural to introduce the $p_i$-valuation of $(u_\varepsilon)_\varepsilon \in M_E$ as

$$ \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon] = \sup\{b \in \mathbb{R}| p_i(u_\varepsilon) = O(\varepsilon^b) \text{ as } \varepsilon \to 0\} \quad (1.1.69) $$

Note that

$$ \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon] = \mathbf{v}_{\tilde{\mathbb{C}}}[(p_i(u_\varepsilon))_\varepsilon] \quad (1.1.70) $$

where the function $\mathbf{v}_{\tilde{\mathbb{C}}}$ in (1.1.70) gives the valuation on $\tilde{\mathbb{C}}$.

**Theorem.1.1.9.** The function $\mathbf{v}_{p_i}$ maps $M_E$ into $(-\infty, +\infty]$ and the following properties hold:

(i) $\mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon] = +\infty$ for all $i \in I$ iff $(u_\varepsilon)_\varepsilon \in N_E$,

(ii) $\mathbf{v}_{p_i}[(\lambda_\varepsilon u_\varepsilon)_\varepsilon] \ge \mathbf{v}[(\lambda_\varepsilon)_\varepsilon] + \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon]$ for all $(\lambda_\varepsilon)_\varepsilon \in E_M, (u_\varepsilon)_\varepsilon \in M_E$,

(ii)' $\mathbf{v}_{p_i}[(\lambda_\varepsilon u_\varepsilon)_\varepsilon] = \mathbf{v}[(\lambda_\varepsilon)_\varepsilon] + \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon]$ for all $(\lambda_\varepsilon)_\varepsilon = (c\varepsilon^b)_\varepsilon, c \in \mathbb{C}, (u_\varepsilon)_\varepsilon \in M_E$,

(iii) $\mathbf{v}_{p_i}[(w_\varepsilon + u_\varepsilon)_\varepsilon] \ge \inf\{\mathbf{v}_{p_i}[(w_\varepsilon)_\varepsilon], \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon]\}$.

Property (i) combined with (iii) shows that $\mathbf{v}_{p_i}[(w_\varepsilon)_\varepsilon] = \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon]$ iff $\mathbf{v}_{p_i}[(w_\varepsilon - u_\varepsilon)_\varepsilon]$ is $E$-negligible.

Therefore we can use (1.1.70) for defining the $p_i$-valuation $\mathbf{v}_{p_i}[u] = \mathbf{v}_{p_i}[(u_\varepsilon)_\varepsilon]$ of a generalized function $u = \mathbf{cl}[(u_\varepsilon)_\varepsilon] \in G_E$. Thus

$$ \wp_i[u] = \exp[-\mathbf{v}_{p_i}[u]] \quad (1.1.71) $$

is an ultra-pseudo-seminorm on the $\tilde{\mathbb{C}}$-module $G_E$.

**Theorem.1.1.10.** Let $E$ be a locally convex topological vector space.

(i) If $E$ is topologized through an increasing sequence $\{p_i\}_{i \in \mathbb{N}}$ of seminorms and
$$N_E = \{(u_\epsilon)_\epsilon \in M_E | \forall q \in \mathbb{N}[p_0(u_\epsilon) = O(\epsilon^q) \text{ as } \epsilon \to 0]\} \quad (1.1.72)$$

then each $\wp_i$ is an ultra-pseudo-norm on $G_E$.

(ii) If $E$ has a countable base of neighborhoods of the origin then $G_E$ with the sharp topology is metrizable.

**Definition 1.1.34.** Let $E$ be an algebra on $\mathbb{C}$ with a family of seminorms $\{p_i\}_{i \in I}$ on $E$.

We recall that a $\tilde{\mathbb{C}}$-module $G_E$ is a $\tilde{\mathbb{C}}$-algebra $\tilde{G}_E$ if there is given a multiplication $G_E \times G_E \to G_E : (u, w) \to uw$ such that $\forall u, w, v \in \tilde{G}_E, \lambda \in \tilde{\mathbb{C}} :$

(i) $u(vw) = (uw)v,$

(ii) $u(w + v) = uw + uv,$

(iii) $(w + v)u = wu + vu,$

(iv) $\lambda(uv) = (\lambda u)v = u(\lambda v),$

(v) $\forall i [i \in I] \text{ there exist finite subsets } I_0 \subset I, I'_0 \subset I \text{ and a constant } c_i > 0 \text{ such that} \\ \text{for all } u, v \in E$

$$p_i(uv) \le c_i \left( \max_{j \in I_0} p_j(u) \right) \left( \max_{j \in I'_0} p_j(v) \right) \quad (1.1.73)$$

Then $\tilde{G}_E$ with the sharp topology determined by the ultra-pseudo-seminorms $\{\wp_i\}_{i \in I}$ is a locally convex topological $\tilde{\mathbb{C}}$-module and a topological $\tilde{\mathbb{C}}$-algebra since from (1.1.73) it follows that $\forall i \in I, \forall u, v \in \tilde{G}_E :$

$$\wp_i(uv) \le c_i \left( \max_{j \in I_0} \wp_j(u) \right) \left( \max_{j \in I'_0} \wp_j(v) \right) \quad (1.1.74)$$

Let us consider examples of the Colombeau algebras $\tilde{G}_E$ obtained as $\tilde{\mathbb{C}}$-modules $G_E$.

Particular choices of $E$ in Definition 1.1.33 give us well known algebras of generalized functions and the corresponding sharp topologies. This is of course the case for $E = \mathbb{C}, G_E = \tilde{\mathbb{C}}$ and $\tilde{G}_E = \tilde{\mathbb{C}}$ which is an ultra-pseudo-normed $\tilde{\mathbb{C}}$-module and
$$N_E = \{(u_\epsilon)_\epsilon \in M_E | \forall q \in \mathbb{N}[p_0(u_\epsilon) = O(\epsilon^q) \text{ as } \epsilon \to 0]\} \quad (1.1.72)$$

then each $\wp_i$ is an ultra-pseudo-norm on $G_E$.

(ii) If $E$ has a countable base of neighborhoods of the origin then $G_E$ with the sharp topology is metrizable.

**Definition 1.1.34.** Let $E$ be an algebra on $\mathbb{C}$ with a family of seminorms $\{p_i\}_{i \in I}$ on $E$.

We recall that a $\tilde{\mathbb{C}}$-module $G_E$ is a $\tilde{\mathbb{C}}$-algebra $\tilde{G}_E$ if there is given a multiplication $G_E \times G_E \to G_E : (u, w) \to uw$ such that $\forall u, w, v \in \tilde{G}_E, \lambda \in \tilde{\mathbb{C}} :$

(i) $u(vw) = (uw)v,$

(ii) $u(w + v) = uw + uv,$

(iii) $(w + v)u = wu + vu,$

(iv) $\lambda(uv) = (\lambda u)v = u(\lambda v),$

(v) $\forall i [i \in I] \text{ there exist finite subsets } I_0 \subset I, I'_0 \subset I \text{ and a constant } c_i > 0 \text{ such that} \\ \text{for all } u, v \in E$

$$p_i(uv) \le c_i \left( \max_{j \in I_0} p_j(u) \right) \left( \max_{j \in I'_0} p_j(v) \right) \quad (1.1.73)$$

Then $\tilde{G}_E$ with the sharp topology determined by the ultra-pseudo-seminorms $\{\wp_i\}_{i \in I}$ is a locally convex topological $\tilde{\mathbb{C}}$-module and a topological $\tilde{\mathbb{C}}$-algebra since from (1.1.73) it follows that $\forall i \in I, \forall u, v \in \tilde{G}_E :$

$$\wp_i(uv) \le c_i \left( \max_{j \in I_0} \wp_j(u) \right) \left( \max_{j \in I'_0} \wp_j(v) \right) \quad (1.1.74)$$

Let us consider examples of the Colombeau algebras $\tilde{G}_E$ obtained as $\tilde{\mathbb{C}}$-modules $G_E$.

Particular choices of $E$ in Definition 1.1.33 give us well known algebras of generalized functions and the corresponding sharp topologies. This is of course the case for $E = \mathbb{C}, G_E = \tilde{\mathbb{C}}$ and $\tilde{G}_E = \tilde{\mathbb{C}}$ which is an ultra-pseudo-normed $\tilde{\mathbb{C}}$-module and
more precisely a topological $\tilde{\mathbb{C}}$-algebra.

Consider now an open subset $\Omega \subset \mathbb{R}^n$. $E = E(\Omega) = C^\infty(\Omega)$ topologized through the family of seminorms $p_{K_{i,j}}(f) = \sup_{x \in K_{i,j}, |\alpha| \le j} |\partial^\alpha f(x)|$, where $K_0 K_1 \dots K_i \dots$ is a countable and exhausting sequence of compact subsets of $\Omega$, provides $\tilde{G}_E = G_E(\Omega)$.

$G_E(\Omega)$ endowed with the sharp topology determined by $\{\wp_{K_{i,j}}(\cdot)\}_{i,j \in \mathbb{N}}$ is a Fréchet $\tilde{\mathbb{C}}$-module and a topological $\tilde{\mathbb{C}}$-algebra. More examples of Fréchet $\tilde{\mathbb{C}}$-modules which are also topological $\tilde{\mathbb{C}}$-algebras are given by $G_E$ when $E$ is $L(\mathbb{R}^n)$ or $W^{\infty,p}(\mathbb{R}^n), p \in [1, +\infty]$

In this way we construct the algebras $G_L(\mathbb{R}^n)$ and $G_{p,p}(\mathbb{R}^n)$ respectively, whose sharp topologies are obtained from $p_k(f) = \sup_{x \in \mathbb{R}^n, |\alpha| \le k} (1+x^k) |\partial^\alpha f(x)|$ and $p_k(f) = \sup_{|\alpha| \le k} |\partial^\alpha f(x)|$.

### 1.1.10. Definition and basic properties of $G_B$ and $\tilde{G}_B$.

**Definition 1.1.35.** Let $B$ be an Banach space topologized through the norm $\|\cdot\|$. The elements of

$$ M_E \triangleq \{ (u_\varepsilon)_\varepsilon \in B^{(0,1]} | \forall i \in I, \exists N \in \mathbb{N} \, [ \|u_\varepsilon\| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 ] \}, \quad (1.1.75) $$

are called $B$-moderate. The elements of

The elements of

$$ N_B \triangleq \{ (u_\varepsilon)_\varepsilon \in B^{(0,1]} | \forall i \in I, \exists q \in \mathbb{N} \, [ p_i(u_\varepsilon) = O(\varepsilon^q) \text{ as } \varepsilon \to 0 ] \}, \quad (1.1.76) $$

are called $B$-negligible. We define the space of generalized functions
more precisely a topological $\tilde{\mathbb{C}}$-algebra.

Consider now an open subset $\Omega \subset \mathbb{R}^n$. $E = E(\Omega) = C^\infty(\Omega)$ topologized through the family of seminorms $p_{K_{i,j}}(f) = \sup_{x \in K_{i,j}, |\alpha| \le j} |\partial^\alpha f(x)|$, where $K_0 K_1 \dots K_i \dots$ is a countable and exhausting sequence of compact subsets of $\Omega$, provides $\tilde{G}_E = G_E(\Omega)$.

$G_E(\Omega)$ endowed with the sharp topology determined by $\{\wp_{K_{i,j}}(\cdot)\}_{i,j \in \mathbb{N}}$ is a Fréchet $\tilde{\mathbb{C}}$-module and a topological $\tilde{\mathbb{C}}$-algebra. More examples of Fréchet $\tilde{\mathbb{C}}$-modules which are also topological $\tilde{\mathbb{C}}$-algebras are given by $G_E$ when $E$ is $L(\mathbb{R}^n)$ or $W^{\infty,p}(\mathbb{R}^n), p \in [1, +\infty]$.

In this way we construct the algebras $G_L(\mathbb{R}^n)$ and $G_{p,p}(\mathbb{R}^n)$ respectively, whose sharp topologies are obtained from $p_k(f) = \sup_{x \in \mathbb{R}^n, |\alpha| \le k} (1+x^k) |\partial^\alpha f(x)|$ and $p_k(f) = \sup_{|\alpha| \le k} |\partial^\alpha f(x)|$.

### I.1.10. Definition and basic properties of $G_B$ and $\tilde{G}_B$.

**Definition 1.1.35.** Let $B$ be an Banach space topologized through the norm $\|\cdot\|$. The elements of

$$ M_E \triangleq \{(u_\varepsilon)_\varepsilon \in B^{(0,1]} | \forall i(i \in I) \exists N(N \in \mathbb{N}) [\|u_\varepsilon\| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0]\}, \quad (1.1.75) $$

are called $B$-moderate. The elements of

The elements of

$$ N_B \triangleq \{(u_\varepsilon)_\varepsilon \in B^{(0,1]} | \forall i(i \in I) \exists q(q \in \mathbb{N}) [p_i(u_\varepsilon) = O(\varepsilon^q) \text{ as } \varepsilon \to 0]\}, \quad (1.1.76) $$

are called $B$-negligible. We define the space of generalized functions
based on B as the factor space

$$ G_B \triangleq M_B/N_B. \qquad (1.1.77) $$

We adopt the notation $u = \mathbf{cl}[(u_\varepsilon)_\varepsilon]$ for the class $u$ of $(u_\varepsilon)_\varepsilon$ in $G_B$ and we embed $B$

into $G_B$ via the constant embedding $e \mapsto \mathbf{cl}[(e_\varepsilon)_\varepsilon], e_\varepsilon = e$. By the properties of norm on $B$ we may define the product between complex generalized numbers and elements of $G_B$ via the map

$$ \tilde{\mathbb{C}} \times G_B \to G_B : (\mathbf{cl}[(\lambda_\varepsilon)_\varepsilon], \mathbf{cl}[(u_\varepsilon)_\varepsilon]) \to \mathbf{cl}[(\lambda_\varepsilon u_\varepsilon)_\varepsilon], \qquad (1.1.78) $$

which equips $G_B$ with the structure of a $\tilde{\mathbb{C}}$-module.

Since the growth in $\varepsilon$ of an $B$-moderate net is estimated in terms of norm $\|\cdot\|$ of $B$, it is natural to introduce the $\|\cdot\|$-valuation of $(u_\varepsilon)_\varepsilon \in M_B$ as

$$ \mathbf{v}_B[(u_\varepsilon)_\varepsilon] = \sup\{b \in \mathbb{R} \mid \|u_\varepsilon\| = O(\varepsilon^b) \text{ as } \varepsilon \to 0\} \qquad (1.1.79) $$

Note that

$$ \mathbf{v}_B[(u_\varepsilon)_\varepsilon] = \tilde{\mathbf{v}}_\varepsilon[(\|u_\varepsilon\|)_\varepsilon] \qquad (1.1.80) $$

where the function $\tilde{\mathbf{v}}_\varepsilon$ in (1.1.80) gives the valuation on $\tilde{\mathbb{C}}$.

**Theorem.1.1.11.** The function $\mathbf{v}_B$ maps $M_B$ into $(-\infty, +\infty]$ and the following properties hold:

(i) $\mathbf{v}_B[(u_\varepsilon)_\varepsilon] = +\infty$ iff $(u_\varepsilon)_\varepsilon \in N_B$,

(ii) $\mathbf{v}_B[(\lambda_\varepsilon u_\varepsilon)_\varepsilon] \ge \tilde{\mathbf{v}}_\varepsilon[(\lambda_\varepsilon)_\varepsilon] + \mathbf{v}_B[(u_\varepsilon)_\varepsilon]$ for all $(\lambda_\varepsilon)_\varepsilon \in B_M, (u_\varepsilon)_\varepsilon \in M_B$,
based on B as the factor space

$$G_B \triangleq M_B/N_B. \qquad (1.1.77)$$

We adopt the notation $u = \mathbf{cl}[(u_\epsilon)_\epsilon]$ for the class $u$ of $(u_\epsilon)_\epsilon$ in $G_B$ and we embed $B$

into $G_B$ via the constant embedding $e \mapsto \mathbf{cl}[(e_\epsilon)_\epsilon], e_\epsilon = e$. By the properties of norm on $B$ we may define the product between complex generalized numbers and elements of $G_B$ via the map

$$\tilde{\mathbb{C}} \times G_B \to G_B : (\mathbf{cl}[(\lambda_\epsilon)_\epsilon], \mathbf{cl}[(u_\epsilon)_\epsilon]) \to \mathbf{cl}[(\lambda_\epsilon u_\epsilon)_\epsilon], \qquad (1.1.78)$$

which equips $G_B$ with the structure of a $\tilde{\mathbb{C}}$-module.

Since the growth in $\epsilon$ of an $B$-moderate net is estimated in terms of norm $\|\cdot\|$ of $B$, it is natural to introduce the $\|\cdot\|$-valuation of $(u_\epsilon)_\epsilon \in M_B$ as

$$v_B[(u_\epsilon)_\epsilon] = \sup\{b \in \mathbb{R} \mid \|u_\epsilon\| = O(\epsilon^b) \text{ as } \epsilon \to 0\} \qquad (1.1.79)$$

Note that

$$v_B[(u_\epsilon)_\epsilon] = v_{\tilde{\mathbb{C}}}[(\|u_\epsilon\|)_\epsilon] \qquad (1.1.80)$$

where the function $v_{\tilde{\mathbb{C}}}$ in (1.1.80) gives the valuation on $\tilde{\mathbb{C}}$.

**Theorem.1.1.11.** The function $v_B$ maps $M_B$ into $(-\infty, +\infty]$ and the following properties hold:

(i) $v_B[(u_\epsilon)_\epsilon] = +\infty$ iff $(u_\epsilon)_\epsilon \in N_B$,

(ii) $v_B[(\lambda_\epsilon u_\epsilon)_\epsilon] \ge v_{\tilde{\mathbb{C}}}[(\lambda_\epsilon)_\epsilon] + v_B[(u_\epsilon)_\epsilon]$ for all $(\lambda_\epsilon)_\epsilon \in B_M, (u_\epsilon)_\epsilon \in M_B$,
(ii') $v_B[(\lambda_\epsilon u_\epsilon)_\epsilon] = v_{\tilde{\mathbb{C}}}[(\lambda_\epsilon)_\epsilon] + v_B[(u_\epsilon)_\epsilon]$ for all $(\lambda_\epsilon)_\epsilon = (c\epsilon^b)_\epsilon, c \in \mathbb{C}, (u_\epsilon)_\epsilon \in M_B,$

(iii) $v_B[(w_\epsilon + u_\epsilon)_\epsilon] \geq \inf\{v_B[(w_\epsilon)_\epsilon], v_B[(u_\epsilon)_\epsilon]\}$.

Property (i) combined with (iii) shows that $v_B[(w_\epsilon)_\epsilon] = v_B[(u_\epsilon)_\epsilon]$ iff $v_B[(w_\epsilon - u_\epsilon)_\epsilon]$ is $B$-negligible.

Therefore we can use (1.1.70) for defining the $\|\cdot\|$-valuation $v_B[u] = v_B[\mathbf{cl}[(u_\epsilon)_\epsilon]]$ of a generalized function $u = \mathbf{cl}[(u_\epsilon)_\epsilon] \in G_B$. Thus

$$ \wp_B[u] = \exp[-v_B[u]] \qquad (1.1.81) $$

is an ultra-pseudo-seminorm on the $\tilde{\mathbb{C}}$-module $G_B$.

**Theorem.1.1.12.** Let $B$ be an Banach space.

If $B$ is topologized through of norm $\|\cdot\|$ and

$$ N_B = \{(u_\epsilon)_\epsilon \in M_B | \forall q \in \mathbb{N}[\|u_\epsilon\| = O(\epsilon^q) \text{ as } \epsilon \to 0]\} \qquad (1.1.82) $$

then $\wp_B$ is an ultra-pseudo-norm on $G_B$.

**Definition 1.1.36.** Let $B$ be an Banach algebra on $\mathbb{C}$ with norm $\|\cdot\|$ on $B$.

We recall that a $\tilde{\mathbb{C}}$-module $G_B$ is a $\tilde{\mathbb{C}}$-algebra $\tilde{G}_E$ if there is given a multiplication $G_B \times G_B \to G_B : (u, w) \to uw$ such that $\forall u, w, v \in \tilde{G}_B, \lambda \in \tilde{\mathbb{C}} :$

(i) $u(wv) = (uw)v,$

(ii) $u(w + v) = uw + uv,$

(iii) $(w + v)u = wu + vu,$

(iv) $\lambda(uv) = (\lambda u)v = u(\lambda v),$

Note that by definition of Banach algebra $B$ for all $u, v \in B$

$$ \|uv\| \leq \|u\|\|v\|. \qquad (1.1.83) $$

Therefore $\tilde{G}_B$ with the sharp topology determined by the ultra-pseudo-seminorm

$\wp_B$

is a locally convex topological $\tilde{\mathbb{C}}$-module and a topological $\tilde{\mathbb{C}}$-algebra since from (1.1.83) it follows that $\forall u, v \in \tilde{G}_B :$
(ii') $v_B[(\lambda_\epsilon u_\epsilon)_\epsilon] = v_{\tilde{C}}[(\lambda_\epsilon)_\epsilon] + v_B[(u_\epsilon)_\epsilon]$ for all $(\lambda_\epsilon)_\epsilon = (c\epsilon^b)_\epsilon, c \in \mathbb{C}, (u_\epsilon)_\epsilon \in M_B,$

(iii) $v_B[(w_\epsilon + u_\epsilon)_\epsilon] \geq \inf\{v_B[(w_\epsilon)_\epsilon], v_B[(u_\epsilon)_\epsilon]\}$.

Property (i) combined with (iii) shows that $v_B[(w_\epsilon)_\epsilon] = v_B[(u_\epsilon)_\epsilon]$ iff $v_B[(w_\epsilon - u_\epsilon)_\epsilon]$ is $B$-negligible.

Therefore we can use (1.1.70) for defining the $\|\cdot\|$-valuation $v_B[u] = v_B[cl[(u_\epsilon)_\epsilon]]$ of a generalized function $u = cl[(u_\epsilon)_\epsilon] \in G_B$. Thus

$$ \wp_B[u] = \exp[-v_B[u]] \qquad (1.1.81) $$

is an ultra-pseudo-seminorm on the $\tilde{C}$-module $G_B$.

**Theorem.1.1.12.** Let $B$ be an Banach space.

If $B$ is topologized through of norm $\|\cdot\|$ and

$$ N_B = \{(u_\epsilon)_\epsilon \in M_B | \forall q \in \mathbb{N}[\|u_\epsilon\| = O(\epsilon^q) \text{ as } \epsilon \to 0]\} \qquad (1.1.82) $$

then $\wp_B$ is an ultra-pseudo-norm on $G_B$.

**Definition 1.1.36.** Let $B$ be an Banach algebra on $\mathbb{C}$ with norm $\|\cdot\|$ on $B$.

We recall that a $\tilde{C}$-module $G_B$ is a $\tilde{C}$-algebra $\tilde{G}_E$ if there is given a multiplication $G_B \times G_B \to G_B : (u, w) \to uw$ such that $\forall u, w, v \in \tilde{G}_B, \lambda \in \tilde{C}$:

(i) $u(wv) = (uw)v,$

(ii) $u(w + v) = uw + uv,$

(iii) $(w + v)u = wu + vu,$

(iv) $\lambda(uv) = (\lambda u)v = u(\lambda v),$

Note that by definition of Banach algebra $B$ for all $u, v \in B$

$$ \|uv\| \leq \|u\|\|v\|. \qquad (1.1.83) $$

Therefore $\tilde{G}_B$ with the sharp topology determined by the ultra-pseudo-seminorm

$\wp_B$

is a locally convex topological $\tilde{C}$-module and a topological $\tilde{C}$-algebra since from (1.1.83) it follows that $\forall u, v \in \tilde{G}_B:$
$$ \wp_B(uv) \le \wp_B(u) \wp_B(v). \qquad (1.1.84) $$

Let us consider examples of the Colombeau algebras $\tilde{G}_B$ obtained as $\tilde{\mathbb{C}}$-modules $G_E$.

Particular choices of $B$ in Definition 1.1.35 give us well known algebras of generalized functions and the corresponding sharp topologies. This is of course the case for $B = \mathbb{C}, G_B = \tilde{\mathbb{C}}$ and $\tilde{G}_B = \tilde{\mathbb{C}}$ which is an ultra-pseudo-normed $\tilde{\mathbb{C}}$-module and more precisely a topological $\tilde{\mathbb{C}}$-algebra.

**Definition 1.1.35. (I)** Consider now an open subset $\Omega \subseteq \mathbb{R}^n$. Suppose that $u \in L_p(\Omega)$ and there exist weak derivatives $\partial^\alpha u(x)$ for any $\alpha$ with $|\alpha| \le l$ (all derivatives up to order $l$), such that $\partial^\alpha u \in L_p(\Omega)$. Then we say that $u \in W_p^l(\Omega)$. We introduce the (standard) norm in $W_p^l(\Omega)$:

$$ \|u\|_{W_p^l(\Omega)} = \left( \int_{\Omega} \sum_{|\alpha| \le l} |\partial^\alpha u(x)| d^n x \right)^{1/p} . \qquad (1.1.85) $$

We recall that (1) the norm $\sum_{|\alpha| \le l} \|\partial^\alpha u(x)\|_{p,\Omega}$ is equivalent to the standard norm.

(2) $W_p^l(\Omega)$ is complete, i.e. $W_p^l(\Omega)$ is a Banach space.

(3) If $p = 2$, the space $W_2^l(\Omega)$ is a Hilbert space with the inner product

$$ (u, w)_{W_p^l(\Omega)} = \left( \int_{\Omega} \sum_{|\alpha| \le l} \partial^\alpha u(x) \overline{\partial^\alpha w(x)} d^n x \right). \qquad (1.1.86) $$

Let $B = W_p^l(\Omega)$ topologized through by norm $\|u\|_{W_p^l(\Omega)}$ provides $G_{W_p^l(\Omega)} \cdot G_{W_p^l(\Omega)}$ endowed with the sharp topology determined by $\wp_{W_p^l(\Omega)}(\cdot)$ is a $\tilde{\mathbb{C}}$-module.

(II) Let $\omega(x)$ be a weight on $\mathbb{R}^n$ i.e., a locally integrable function on $\mathbb{R}^n$ such that $\omega(x) > 0$ for a.e. $x \in \mathbb{R}^n$. Let $\Omega \subseteq \mathbb{R}^n$ be open, $1 \le p < 1$, and $l$ a nonnegative integer. The weighted Sobolev spaces $W_p^l(\Omega, \omega)$ consists of all functions $u$ with weak derivatives $\partial^\alpha u(x)$ for any $\alpha$ with $|\alpha| \le l$ satisfying
$$ \wp_B(uv) \le \wp_B(u) \wp_B(v). \qquad (1.1.84) $$

Let us consider examples of the Colombeau algebras $\tilde{G}_B$ obtained as $\tilde{\mathbb{C}}$-modules $G_E$.

Particular choices of $B$ in Definition 1.1.35 give us well known algebras of generalized functions and the corresponding sharp topologies. This is of course the case for $B = \mathbb{C}$, $G_B = \tilde{\mathbb{C}}$ and $\tilde{G}_B = \tilde{\mathbb{C}}$ which is an ultra-pseudo-normed $\tilde{\mathbb{C}}$-module and more precisely a topological $\tilde{\mathbb{C}}$-algebra.

**Definition 1.1.35. (I)** Consider now an open subset $\Omega \subseteq \mathbb{R}^n$. Suppose that $u \in L_p(\Omega)$ and there exist weak derivatives $\partial^\alpha u(x)$ for any $\alpha$ with $|\alpha| \le l$ (all derivatives up to order $l$), such that $\partial^\alpha u \in L_p(\Omega)$. Then we say that $u \in W_p^l(\Omega)$. We introduce the (standard) norm in $W_p^l(\Omega)$:

$$ \|u\|_{W_p^l(\Omega)} = \left( \int_{\Omega} \sum_{|\alpha| \le l} |\partial^\alpha u(x)| d^n x \right)^{1/p} . \qquad (1.1.85) $$

We recall that (1) the norm $\sum_{|\alpha| \le l} \|\partial^\alpha u(x)\|_{p,\Omega}$ is equivalent to the standard norm.

(2) $W_p^l(\Omega)$ is complete, i.e. $W_p^l(\Omega)$ is a Banach space.

(3) If $p = 2$, the space $W_2^l(\Omega)$ is a Hilbert space with the inner product

$$ (u, w)_{W_p^l(\Omega)} = \left( \int_{\Omega} \sum_{|\alpha| \le l} \partial^\alpha u(x) \overline{\partial^\alpha w(x)} d^n x \right). \qquad (1.1.86) $$

Let $B = W_p^l(\Omega)$ topologized through by norm $\|u\|_{W_p^l(\Omega)}$ provides $G_{W_p^l(\Omega)} \cdot G_{W_p^l(\Omega)}$ endowed with the sharp topology determined by $\wp_{W_p^l(\Omega)}(\cdot)$ is a $\tilde{\mathbb{C}}$-module.

(II) Let $\omega(x)$ be a weight on $\mathbb{R}^n$ i.e., a locally integrable function on $\mathbb{R}^n$ such that $\omega(x) > 0$ for a.e. $x \in \mathbb{R}^n$. Let $\Omega \subseteq \mathbb{R}^n$ be open, $1 \le p < 1$, and $l$ a nonnegative integer. The weighted Sobolev spaces $W_p^l(\Omega, \omega)$ consists of all functions $u$ with weak derivatives $\partial^\alpha u(x)$ for any $\alpha$ with $|\alpha| \le l$ satisfying
$$ \|u\|_{W_p^l(\Omega, \omega)} = \left( \int_{\Omega} \sum_{|\alpha| \le l} |\partial^\alpha u(x)| \omega(x) d^n x \right)^{1/p} < \infty. \quad (1.1.86') $$

Let $B = W_p^l(\Omega, \omega)$ topologized through by norm $\|u\|_{W_p^l(\Omega, \omega)}$ provides $G_{W_p^l(\Omega, \omega)} \cdot G_{W_p^l(\Omega, \omega)}$ endowed with the sharp topology determined by $\wp_{W_p^l(\Omega, \omega)}(\cdot)$ is a $\tilde{\mathbb{C}}$-module.

### I.1.11. Colombeau Sobolev spaces. Definition and basic properties of $(W_p^{l,k,\varepsilon}(\Omega))_{\varepsilon \in (0,1]}$.

**Definition 1.1.36.** $(C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon \in (0,1]} = (C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}$ is the linear space of all functions $u(x)$ in $\mathbb{R}^n$ such that:

(1) $u(x)$ and $\partial^\alpha u(x)$ $|\alpha| \le l$ are continuous in $\mathbb{R}^n$,

(2) $\sum_{|\alpha| \le l} \sup_{x \in \mathbb{R}^n} |\partial^\alpha u(x)| \le \infty$ and the $\varepsilon$-norm $\|u\|_{C^{l,k,\varepsilon}}$:

$$ \|u\|_{C^{l,k,\varepsilon}} = \sum_{|\alpha| \le l} \sup_{x \in \mathbb{R}^n} \left| \partial^\alpha \left[ \frac{u(x)}{1 + \varepsilon^k \|x\|^{2k}} \right] \right| d^n x \quad (1.1.87) $$

is finite for any $\varepsilon \in (0,1]$. If $l = 0$, we denote $(C^{0,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon} = (C^{k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}$.

**Definition 1.1.37.** We can use (1.1.87) for defining the $\tilde{\mathbb{R}}$-valued norm $(\|u\|_{C^{l,k,\varepsilon}})_{\varepsilon}$:

$$ \|u\|_{l,k,\tilde{\mathbb{R}}} = \mathbf{cl}[(\|u\|_{C^{l,k,\varepsilon}})_{\varepsilon}] \quad (1.1.88) $$

of a function $u \in (C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon \in (0,1]}$.

**Definition 1.1.38.** We can use (1.1.88) for defining the valuation $v(C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}[u]$:

$$ v(C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}[u] = v_{\tilde{\mathbb{R}}}[\mathbf{cl}[(\|u\|_{C^{l,k,\varepsilon}})_{\varepsilon}]] \quad (1.1.89) $$
$$ \|u\|_{W_p^l(\Omega, \omega)} = \left( \int_{\Omega} \sum_{|\alpha| \le l} |\partial^\alpha u(x)| \omega(x) d^n x \right)^{1/p} < \infty. \quad (1.1.86') $$

Let $B = W_p^l(\Omega, \omega)$ topologized through by norm $\|u\|_{W_p^l(\Omega, \omega)}$ provides $G_{W_p^l(\Omega, \omega)} \cdot G_{W_p^l(\Omega, \omega)}$ endowed with the sharp topology determined by $\wp_{W_p^l(\Omega, \omega)}(\cdot)$ is a $\tilde{\mathbb{C}}$-module.

### I.1.11. Colombeau Sobolev spaces. Definition and basic properties of $(W_p^{l,k,\varepsilon}(\Omega))_{\varepsilon \in (0,1]}$.

**Definition 1.1.36.** $(C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon \in (0,1]} = (C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}$ is the linear space of all functions $u(x)$ in $\mathbb{R}^n$ such that:

(1) $u(x)$ and $\partial^\alpha u(x)$ $|\alpha| \le l$ are continuous in $\mathbb{R}^n$,

(2) $\sum_{|\alpha| \le l} \sup_{x \in \mathbb{R}^n} |\partial^\alpha u(x)| \le \infty$ and the $\varepsilon$-norm $\|u\|_{C^{l,k,\varepsilon}}$:

$$ \|u\|_{C^{l,k,\varepsilon}} = \sum_{|\alpha| \le l} \sup_{x \in \mathbb{R}^n} \left| \partial^\alpha \left[ \frac{u(x)}{1 + \varepsilon^k \|x\|^{2k}} \right] \right| d^n x \quad (1.1.87) $$

is finite for any $\varepsilon \in (0,1]$. If $l = 0$, we denote $(C^{0,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon} = (C^{k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}$.

**Definition 1.1.37.** We can use (1.1.87) for defining the $\tilde{\mathbb{R}}$-valued norm $(\|u\|_{C^{l,k,\varepsilon}})_{\varepsilon}$:

$$ \|u\|_{l,k,\tilde{\mathbb{R}}} = \mathbf{cl}[(\|u\|_{C^{l,k,\varepsilon}})_{\varepsilon}] \quad (1.1.88) $$

of a function $u \in (C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon \in (0,1]}$.

**Definition 1.1.38.** We can use (1.1.88) for defining the valuation $v(C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}[u]$:

$$ v(C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}[u] = v_{\tilde{\mathbb{R}}}[\mathbf{cl}[(\|u\|_{C^{l,k,\varepsilon}})_{\varepsilon}]] \quad (1.1.89) $$
of a function $u \in (C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon \in (0,1]}$.

**Definition 1.1.39.** We can use (1.1.89) for defining the ultra pseudo-norm
$$ \wp(C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}[u]: $$

$$ \wp(C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}[u] = \exp\left[-\mathbf{v}(C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}[u]\right] \quad (1.1.90) $$

of a function $u \in (C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon \in (0,1]}$.

**Definition 1.1.40.** Consider now an open subset $\Omega \subset \mathbb{R}^n$. $(W_p^{l,k,\varepsilon}(\Omega))_{\varepsilon \in (0,1]} = (W_p^{l,k,\varepsilon}(\Omega))_{\varepsilon} ((W_p^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon \in (0,1]} = (W_p^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon})$ is the linear space of all functions $u(x)$ in $\Omega$ (in $\mathbb{R}^n$) with weak derivatives $\partial^\alpha u(x) |_{\alpha| \le l}$ satisfying
$$ \sum_{|\alpha| \le l} \sup_{x \in \mathbb{R}^n} |\partial^\alpha u(x)| \le \infty \quad \text{and the } \varepsilon\text{-norm } \|u\|_{W_p^{l,k,\varepsilon}(\Omega)} (\|u\|_{W_p^{l,k,\varepsilon}(\mathbb{R}^n)}) : $$

$$ \|u\|_{W_p^{l,k,\varepsilon}(\Omega)} = \left( \int_{\Omega} \sum_{|\alpha| \le l} \left| \partial^\alpha \left[ \frac{u(x)}{1 + \varepsilon^k \|x\|^{2k}} \right] \right|^p d^n x \right)^{1/p}, \quad (1.1.91) $$

$$ \|u\|_{W_p^{l,k,\varepsilon}(\mathbb{R}^n)} = \left( \int_{\mathbb{R}^n} \sum_{|\alpha| \le l} \left| \partial^\alpha \left[ \frac{u(x)}{1 + \varepsilon^k \|x\|^{2k}} \right] \right|^p d^n x \right)^{1/p} $$

is finite for any $\varepsilon \in (0,1]$.

**Definition 1.1.41.** We can use (1.1.91) for defining the $\tilde{\mathbb{R}}$-valued norm
$$ (\|u\|_{W_p^{l,k,\varepsilon}(\Omega)})_{\varepsilon}: $$

$$ \|u\|_{l,k,p,(\Omega),\tilde{\mathbb{R}}} = \mathbf{cl}\left[ (\|u\|_{W_p^{l,k,\varepsilon}(\Omega)})_{\varepsilon} \right], \quad (1.1.92) $$

$$ \|u\|_{l,k,p,(\Omega),\tilde{\mathbb{R}},\tilde{\mathbb{R}}} = \mathbf{cl}\left[ (\|u\|_{W_p^{l,k,\varepsilon}(\mathbb{R}^n)})_{\varepsilon} \right] $$

of a function $u \in (W_p^{l,k,\varepsilon}(\Omega))_{\varepsilon \in (0,1]}$ $(u \in (W_p^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon \in (0,1]})$.

**Definition 1.1.42.** We can use (1.1.92) for defining the valuation $\mathbf{v}(W_p^{l,k,\varepsilon}(\Omega))_{\varepsilon}[u]$
of a function $u \in (C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon \in (0,1]}$.

**Definition 1.1.39.** We can use (1.1.89) for defining the ultra pseudo-norm

$\wp(C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}$ [u] :

$$ \wp(C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}[u] = \exp[-v(C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon}[u]] \quad (1.1.90) $$

of a function $u \in (C^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon \in (0,1]}$.

**Definition 1.1.40.** Consider now an open subset $\Omega \subset \mathbb{R}^n$. $(W_p^{l,k,\varepsilon}(\Omega))_{\varepsilon \in (0,1]} = (W_p^{l,k,\varepsilon}(\Omega))_{\varepsilon}((W_p^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon \in (0,1]} = (W_p^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon})$ is the linear space of all functions $u(x)$ in $\Omega$ (in $\mathbb{R}^n$) with weak derivatives $\partial^\alpha u(x) |_{\alpha| \le l}$ satisfying $\sum_{|\alpha| \le l} \sup_{x \in \mathbb{R}^n} |\partial^\alpha u(x)| \le \infty$ and the $\varepsilon$-norm $\|u\|_{W_p^{l,k,\varepsilon}(\Omega)} (\|u\|_{W_p^{l,k,\varepsilon}(\mathbb{R}^n)}) :$

$$ \|u\|_{W_p^{l,k,\varepsilon}(\Omega)} = \left( \int_{\Omega} \sum_{|\alpha| \le l} \left| \partial^\alpha \left[ \frac{u(x)}{1 + \varepsilon^k \|x\|^{2k}} \right] \right|^p d^n x \right)^{1/p}, \quad (1.1.91) $$

$$ \|u\|_{W_p^{l,k,\varepsilon}(\mathbb{R}^n)} = \left( \int_{\mathbb{R}^n} \sum_{|\alpha| \le l} \left| \partial^\alpha \left[ \frac{u(x)}{1 + \varepsilon^k \|x\|^{2k}} \right] \right|^p d^n x \right)^{1/p} $$

is finite for any $\varepsilon \in (0,1]$.

**Definition 1.1.41.** We can use (1.1.91) for defining the $\tilde{\mathbb{R}}$-valued norm

($\|u\|_{W_p^{l,k,\varepsilon}(\Omega)}$) :

$$ \|u\|_{l,k,p,(\Omega),\tilde{\mathbb{R}}} = \mathbf{cl} \left[ (\|u\|_{W_p^{l,k,\varepsilon}(\Omega)})_{\varepsilon} \right], \quad (1.1.92) $$

$$ \|u\|_{l,k,p,(\Omega),\tilde{\mathbb{R}},\tilde{\mathbb{R}}} = \mathbf{cl} \left[ (\|u\|_{W_p^{l,k,\varepsilon}(\mathbb{R}^n)})_{\varepsilon} \right] $$

of a function $u \in (W_p^{l,k,\varepsilon}(\Omega))_{\varepsilon \in (0,1]}$ ($u \in (W_p^{l,k,\varepsilon}(\mathbb{R}^n))_{\varepsilon \in (0,1]}$).

**Definition 1.1.42.** We can use (1.1.92) for defining the valuation $v(W_p^{l,k,\varepsilon}(\Omega))_{\varepsilon}$ [u]
$$ (\mathbf{v}(w_p^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon}[u]) : $$

$$ \mathbf{v}(w_p^{l,k,\epsilon}(\Omega))_{\epsilon}[u] = \mathbf{v}_{\tilde{\mathbb{R}}}[\mathrm{cl}[(\|u\|_{w_p^{l,k,\epsilon}(\Omega)})_{\epsilon}]] \quad (1.1.93) $$

$$ \mathbf{v}(C^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon}[u] = \mathbf{v}_{\tilde{\mathbb{R}}}[\mathrm{cl}[(\|u\|_{w_p^{l,k,\epsilon}(\mathbb{R}^n)})_{\epsilon}]] $$

of a function $u \in (W_p^{l,k,\epsilon}(\Omega))_{\epsilon \in (0,1]}$ ($u \in (W_p^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon \in (0,1]}$).

**Definition 1.1.43.** We can use (1.1.93) for defining the pseudo-norm

$$ \mathcal{S}(w_p^{l,k,\epsilon}(\Omega))_{\epsilon}[u] = \mathcal{S}(w_p^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon}[u] : $$

$$ \mathcal{S}(w_p^{l,k,\epsilon}(\Omega))_{\epsilon}[u] = \exp[-\mathbf{v}(w_p^{l,k,\epsilon}(\Omega))_{\epsilon}[u]], \quad (1.1.94) $$

$$ \mathcal{S}(w_p^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon}[u] = \exp[-\mathbf{v}(w_p^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon}[u]] $$

of a function $u \in (W_p^{l,k,\epsilon}(\Omega))_{\epsilon \in (0,1]}$ ($u \in (W_p^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon \in (0,1]}$).

**Definition 1.1.44.** $(W_p^{l,\epsilon}(\mathbb{R}^n, \omega(x)))_{\epsilon \in (0,1]}$, $\omega(x) = \exp(-\epsilon x^2)$ is the linear space of all functions $u(x)$ in $\mathbb{R}^n$ with weak derivatives $\partial^a u(x) |_{\alpha} \le l \sum_{|\alpha| \le l} \sup_{\mathbb{R}^n} |\partial^a u(x)| \le \infty$ and the $\epsilon$-norm

$$ \|u\|_{W_p^{l,\epsilon}(\mathbb{R}^n, \omega)} = \left( \int_{\mathbb{R}^n} |\partial^a u(x)| e^{-\epsilon x^2} d^n x \right)^{1/p}, \quad (1.1.95) $$

is finite for any $\epsilon \in (0,1]$.

**Definition 1.1.**
$$ (\mathbf{v}(W_p^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon}[u]) : $$

$$ \mathbf{v}(W_p^{l,k,\epsilon}(\Omega))_{\epsilon}[u] = \mathbf{v}_{\tilde{\mathbb{R}}}[\mathrm{cl}[(\|u\|_{W_p^{l,k,\epsilon}(\Omega)})_{\epsilon}]] \quad (1.1.93) $$

$$ \mathbf{v}(C^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon}[u] = \mathbf{v}_{\tilde{\mathbb{R}}}[\mathrm{cl}[(\|u\|_{W_p^{l,k,\epsilon}(\mathbb{R}^n)})_{\epsilon}]] $$

of a function $u \in (W_p^{l,k,\epsilon}(\Omega))_{\epsilon \in (0,1]}$ ($u \in (W_p^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon \in (0,1]}$).

**Definition 1.1.43.** We can use (1.1.93) for defining the pseudo-norm

$$ \mathcal{S}(W_p^{l,k,\epsilon}(\Omega))_{\epsilon}[u] = \mathcal{S}(W_p^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon}[u] : $$

$$ \mathcal{S}(W_p^{l,k,\epsilon}(\Omega))_{\epsilon}[u] = \exp[-\mathbf{v}(W_p^{l,k,\epsilon}(\Omega))_{\epsilon}[u]], \quad (1.1.94) $$

$$ \mathcal{S}(W_p^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon}[u] = \exp[-\mathbf{v}(W_p^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon}[u]] $$

of a function $u \in (W_p^{l,k,\epsilon}(\Omega))_{\epsilon \in (0,1]}$ ($u \in (W_p^{l,k,\epsilon}(\mathbb{R}^n))_{\epsilon \in (0,1]}$).

**Definition 1.1.44.** $(W_p^{l,\epsilon}(\mathbb{R}^n, \omega(x)))_{\epsilon \in (0,1]}$, $\omega(x) = \exp(-\epsilon x^2)$ is the linear space of all functions $u(x)$ in $\mathbb{R}^n$ with weak derivatives $\partial^a u(x) |_{\alpha} \le l \sum_{|\alpha| \le l} \sup_{\mathbb{R}^n} |\partial^a u(x)| \le \infty$ and the $\epsilon$-norm

$$ \|u\|_{W_p^{l,\epsilon}(\mathbb{R}^n, \omega)} = \left( \int_{\mathbb{R}^n} |\partial^a u(x)| e^{-\epsilon x^2} d^n x \right)^{1/p}, \quad (1.1.95) $$

is finite for any $\epsilon \in (0,1]$.

**Definition 1.1.**
## I.2. Pseudodifferential operators with coefficients are distributions.

Let us consider partiel differential equations of the form

$$ 
\begin{gathered}
\frac{\partial u_{\varepsilon}(t,x)}{\partial t} + A_{\varepsilon}(t,x,D)u_{\varepsilon}(t,x) = f(t,x), \\
u(0,x) = u_0(x), \\
x \in \Omega \subseteq \mathbb{R}^n.
\end{gathered}
\qquad (1.2.1)
$$

We assume that

$$ A(t,x,D) = \sum_{|a| \le m} a_a(t,x) D^a \qquad (1.2.2) $$

with coefficients $a_a(t,x)$ are distributions. Thus we may form the regularized operator

$$ A_{\varepsilon}(t,x,D) = \sum_{|a| \le m} a_a(t,x) * \varphi_{\varepsilon}(x) D^{\alpha} \qquad (1.2.3) $$

where $\varphi_{\varepsilon}(x) \in C_c^{\infty}(\mathbb{R}^n)$ is a mollifier of the form

$$ 
\begin{gathered}
\varphi_{\varepsilon}(x) = \frac{1}{\varepsilon^n} \varphi\left(\frac{x}{\varepsilon}\right), \\
\int_{\mathbb{R}^n} \varphi(x) d^n x = 1.
\end{gathered}
\qquad (1.2.4)
$$

Thus Eq.(1.2.1) becomes
## I.2. Pseudodifferential operators with coefficients are distributions.

Let us consider partiel differential equations of the form

$$ 
\begin{gathered}
\frac{\partial u_{\varepsilon}(t,x)}{\partial t} + A_{\varepsilon}(t,x,D)u_{\varepsilon}(t,x) = f(t,x), \\
u(0,x) = u_0(x), \\
x \in \Omega \subseteq \mathbb{R}^n.
\end{gathered}
\qquad (1.2.1)
$$

We assume that

$$ A(t,x,D) = \sum_{|a| \le m} a_a(t,x) D^a \qquad (1.2.2) $$

with coefficients $a_a(t,x)$ are distributions. Thus we may form the regularized operator

$$ A_{\varepsilon}(t,x,D) = \sum_{|a| \le m} a_a(t,x) * \varphi_{\varepsilon}(x) D^{\alpha} \qquad (1.2.3) $$

where $\varphi_{\varepsilon}(x) \in C_c^{\infty}(\mathbb{R}^n)$ is a mollifier of the form

$$ 
\begin{gathered}
\varphi_{\varepsilon}(x) = \frac{1}{\varepsilon^n} \varphi\left(\frac{x}{\varepsilon}\right), \\
\int_{\mathbb{R}^n} \varphi(x) d^n x = 1.
\end{gathered}
\qquad (1.2.4)
$$

Thus Eq.(1.2.1) becomes
$$
\begin{align}
& \frac{\partial u_{\varepsilon}}{\partial t} + A_{\varepsilon}(t,x,D)u_{\varepsilon} = f_{\varepsilon}(t,x), \nonumber \\
& u_{\varepsilon}(0,x) = u_{0}(x) * \varphi_{\varepsilon}(x), \tag{1.2.5} \\
& x \in \Omega \subseteq \mathbb{R}^{n}. \nonumber
\end{align}
$$

**Definition 1.2.1.(1)** We say that a family of smooth functions $(f_\varepsilon)_{\varepsilon\in(0,1]}$ satisfying an estimate of the type (a) iff

$$
\forall \alpha(|\alpha| \in \mathbb{N}) \exists N(N \in \mathbb{N}) \forall \varepsilon (\varepsilon \in (0,1]) \left[ \sup_{x \in \mathbb{R}^n} |\partial^\alpha f(x)| = O(\varepsilon^{-N}) \right]. \quad (1.2.6)
$$

(2) We say that a family of smooth functions $(f_\varepsilon)_{\varepsilon\in(0,1]}$ satisfying an estimate of the type (b) iff

$$
\exists N(N \in \mathbb{N}) \forall \alpha(|\alpha| \in \mathbb{N}) \left[ \sup_{x \in \mathbb{R}^n} |\partial^\alpha f(x)| = O(\varepsilon^{-N}) \right]. \quad (1.2.7)
$$

(3) We say that a family of smooth functions $(f_\varepsilon)_{\varepsilon\in(0,1]}$ satisfying an estimate of the type (c) iff

$$
\forall \alpha(|\alpha| \in \mathbb{N}) \forall q(q \in \mathbb{N}) \left[ \sup_{x \in \mathbb{R}^n} |\partial^\alpha f(x)| = O(\varepsilon^q) \right]. \quad (1.2.8)
$$

**Remark 1.2.1.** The condition (2) signifies a regularity property of the family
$$ 
\begin{array}{l}
\displaystyle \frac{\partial u_{\varepsilon}}{\partial t} + A_{\varepsilon}(t,x,D)u_{\varepsilon} = f_{\varepsilon}(t,x), \\
\\
u_{\varepsilon}(0,x) = u_0(x) * \varphi_{\varepsilon}(x), \qquad (1.2.5) \\
\\
x \in \Omega \subseteq \mathbb{R}^n.
\end{array}
$$

**Definition 1.2.1. (1)** We say that a family of smooth functions $(f_{\varepsilon})_{\varepsilon \in (0,1]}$ satisfying an estimate of the type (a) iff

$$ \forall \alpha(|\alpha| \in \mathbb{N}) \exists N(N \in \mathbb{N}) \forall \varepsilon (\varepsilon \in (0,1]) \left[ \sup_{x \in \mathbb{R}^n} |\partial^\alpha f(x)| = O(\varepsilon^{-N}) \right]. \quad (1.2.6) $$

(2) We say that a family of smooth functions $(f_{\varepsilon})_{\varepsilon \in (0,1]}$ satisfying an estimate of the type (b) iff

$$ \exists N(N \in \mathbb{N}) \forall \alpha(|\alpha| \in \mathbb{N}) \left[ \sup_{x \in \mathbb{R}^n} |\partial^\alpha f(x)| = O(\varepsilon^{-N}) \right]. \quad (1.2.7) $$

(3) We say that a family of smooth functions $(f_{\varepsilon})_{\varepsilon \in (0,1]}$ satisfying an estimate of the type (c) iff

$$ \forall \alpha(|\alpha| \in \mathbb{N}) \forall q(q \in \mathbb{N}) \left[ \sup_{x \in \mathbb{R}^n} |\partial^\alpha f(x)| = O(\varepsilon^q) \right]. \quad (1.2.8) $$

**Remark 1.2.1.** The condition (2) signifies a regularity property of the family
$(f_ε)_{ε∈(0,1]}$

It is well known that if the regularizations $f_ε = f * φ_ε(x)$ of a distribution $f ∈ E'(\mathbb{R}^n)$ satisfy (1.2.7) then $f$ actually is an infinitely differentiable function.

**Remark 1.2.2.** A family of smooth functions $(f_ε)_{ε∈(0,1]}$ satisfying an estimate of the type (c) will be considered as asymptotically negligible.

**Remark 1.2.3.** The regularized coefficients of the operator $A_ε(t,x,D)$ given by Eq.(1.2.3) will satisfy an estimate of type (a) or type (b), at least locally on compact sets, and the action of

$A_ε(t,x,D)$ on nets $(u_ε)_{ε∈(0,1]}$ preserves the asymptotic properties (1.2.6) and (1.2.7).

Thus if $(u_ε)_{ε∈(0,1]}$ enjoys either of these properties, so does $(A_ε(t,x,D)u_ε)_{ε∈(0,1]}$.

## I.3. The generalized oscillatory integral.

In this section we argue the meaning and the most important properties of oscillatory integrals of the type

$$ \int_{K \times \mathbb{R}^p} \exp[i\varphi_\varepsilon(y, \xi)] a_\varepsilon(y, \xi) dy d\xi, \qquad (1.3.1) $$

where $K \in \Omega, \Omega$ an open subset of $\mathbb{R}^n$. The function $\varphi_\varepsilon(y, \xi)$ for any value of the parametr $\varepsilon \in (0,1]$ is assumed to be a phase function, i.e., it is smooth on $\Omega \times \mathbb{R}^p \setminus \{0\}$

real valued, positively homogeneous of degree 1 in $\xi$ and $\nabla\varphi_\varepsilon(y, \xi) \neq 0$ for all $y \in \Omega$,

$\xi \in \mathbb{R}^p \setminus \{0\}, \varepsilon \in (0,1]$. In the sequel of this paper we shall use the canonical square bracket notation

$$ S_{\rho, \delta}^{m}[\Omega \times \mathbb{R}^p] \qquad (1.3.3) $$
$(f_ε)_{ε∈(0,1]}$

It is well known that if the regularizations $f_ε = f * φ_ε(x)$ of a distribution $f ∈ E'(\mathbb{R}^n)$ satisfy (1.2.7) then $f$ actually is an infinitely differentiable function.

**Remark 1.2.2.** A family of smooth functions $(f_ε)_{ε∈(0,1]}$ satisfying an estimate of the type (c) will be considered as asymptotically negligible.

**Remark 1.2.3.** The regularized coefficients of the operator $A_ε(t,x,D)$ given by Eq. (1.2.3) will satisfy an estimate of type (a) or type (b), at least locally on compact sets, and the action of

$A_ε(t,x,D)$ on nets $(u_ε)_{ε∈(0,1]}$ preserves the asymptotic properties (1.2.6) and (1.2.7).

Thus if $(u_ε)_{ε∈(0,1]}$ enjoys either of these properties, so does $(A_ε(t,x,D)u_ε)_{ε∈(0,1]}$.

## I.3. The generalized oscillatory integral.

In this section we argue the meaning and the most important properties of oscillatory integrals of the type

$$ \int_{K \times \mathbb{R}^p} \exp[i\varphi_\varepsilon(y, \xi)] a_\varepsilon(y, \xi) dy d\xi, \qquad (1.3.1) $$

where $K \in \Omega, \Omega$ an open subset of $\mathbb{R}^n$. The function $\varphi_\varepsilon(y, \xi)$ for any value of the parametr $\varepsilon \in (0,1]$ is assumed to be a phase function, i.e., it is smooth on $\Omega \times \mathbb{R}^p \setminus \{0\}$

real valued, positively homogeneous of degree 1 in $\xi$ and $\nabla\varphi_\varepsilon(y, \xi) \neq 0$ for all $y \in \Omega$,

$\xi \in \mathbb{R}^p \setminus \{0\}, \varepsilon \in (0,1]$. In the sequel of this paper we shall use the canonical square bracket notation

$$ S_{\rho, \delta}^{m}[\Omega \times \mathbb{R}^p] \qquad (1.3.3) $$
for the space of nets $S_{\rho, \delta}^m(\Omega \times \mathbb{R}^p)^{(0,1]}$ where $S_{\rho, \delta}^m(\Omega \times \mathbb{R}^p), m \in \mathbb{N}, \rho, \delta \in [0, 1]$ is the usual space of Hörmander symbols.

**Definition1.3.1.** An element of $S_{\rho, \delta, M}^m(\Omega \times \mathbb{R}^p)$ is a net $(u_\epsilon)_\epsilon \in S_{\rho, \delta}^m[\Omega \times \mathbb{R}^p]$ such that

$$ \begin{gathered} \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^n) \forall K(K \in \Omega) \exists N(N \in \mathbb{N}) \exists \eta(\eta \in (0,1]) \exists c(c > 0) \\ \forall y(y \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0,1]) [\partial_{\xi}^{\alpha} \partial_{y}^{\beta} u_{\varepsilon}(y, \xi) \le c \langle \xi \rangle^{m-\rho|\alpha|+|\delta|\beta} \varepsilon^{-N}], \end{gathered} \quad (1.3.4) $$

$$ \langle \xi \rangle = \sqrt{1 + |\xi|^2}. $$

**Remark 1.3.1.** The subscript *M* underlines the moderate growth property, i. e., the bound of type $\varepsilon^{-N}$ as $\varepsilon \to 0$.

**Definition1.3.2.** An element of $N_{\rho, \delta}^m(\Omega \times \mathbb{R}^p)$ is a net $(u_\epsilon)_\epsilon \in S_{\rho, \delta}^m[\Omega \times \mathbb{R}^p]$ such that

$$ \begin{gathered} \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^n) \forall K(K \in \Omega) \exists q(q \in \mathbb{N}) \exists \eta(\eta \in (0,1]) \exists c(c > 0) \\ \forall y(y \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0,1]) [\partial_{\xi}^{\alpha} \partial_{y}^{\beta} u_{\varepsilon}(y, \xi) \le c \langle \xi \rangle^{m-\rho|\alpha|+|\delta|\beta} \varepsilon^{q}]. \end{gathered} \quad (1.3.5) $$

**Remark 1.3.2.** Nets of this type are called negligible.

**Definition1.3.3.** A generalized symbol of order *m* and type $(\rho, \delta)$ is an element of the factor space

$$ \tilde{S}_{\rho, \delta}^{m}(\Omega \times \mathbb{R}^{p}) \triangleq S_{\rho, \delta}^{m}(\Omega \times \mathbb{R}^{p}) / N_{\rho, \delta}^{m}(\Omega \times \mathbb{R}^{p}) \quad (1.3.6) $$

**Remark 1.3.3.** We denote an arbitrary representative of $v \in \tilde{S}_{\rho, \delta}^m[\Omega \times \mathbb{R}^p]$ by $(v_\epsilon)_{\epsilon \in (0,1]}$ or simply by $(v_\epsilon)_\epsilon$.

**Definition1.3.4.** An element $v \in \tilde{S}_{\rho, \delta}^m(\Omega \times \mathbb{R}^p)$ is called regular if it has a representative $(v_\epsilon)_\epsilon$ with the following property:
for the space of nets $S_{\rho, \delta}^m(\Omega \times \mathbb{R}^p)^{(0,1]}$ where $S_{\rho, \delta}^m(\Omega \times \mathbb{R}^p), m \in \mathbb{N}, \rho, \delta \in [0, 1]$ is the usual space of Hörmander symbols.

**Definition1.3.1.** An element of $S_{\rho, \delta, M}^m(\Omega \times \mathbb{R}^p)$ is a net $(u_\epsilon)_\epsilon \in S_{\rho, \delta}^m[\Omega \times \mathbb{R}^p]$ such that

$$ \begin{gathered} \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^n) \forall K(K \in \Omega) \exists N(N \in \mathbb{N}) \exists \eta(\eta \in (0,1]) \exists c(c > 0) \\ \forall y(y \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0,1]) [\partial_{\xi}^{\alpha} \partial_{y}^{\beta} u_{\varepsilon}(y, \xi) \le c \langle \xi \rangle^{m-\rho|\alpha|+|\delta|\beta} \varepsilon^{-N}], \end{gathered} \quad (1.3.4) $$

$$ \langle \xi \rangle = \sqrt{1 + |\xi|^2}. $$

**Remark 1.3.1.** The subscript *M* underlines the moderate growth property, i. e., the bound of type $\varepsilon^{-N}$ as $\varepsilon \to 0$.

**Definition1.3.2.** An element of $\mathbb{N}_{\rho, \delta}^m(\Omega \times \mathbb{R}^p)$ is a net $(u_\epsilon)_\epsilon \in S_{\rho, \delta}^m[\Omega \times \mathbb{R}^p]$ such that

$$ \begin{gathered} \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^n) \forall K(K \in \Omega) \exists q(q \in \mathbb{N}) \exists \eta(\eta \in (0,1]) \exists c(c > 0) \\ \forall y(y \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0,1]) [\partial_{\xi}^{\alpha} \partial_{y}^{\beta} u_{\varepsilon}(y, \xi) \le c \langle \xi \rangle^{m-\rho|\alpha|+|\delta|\beta} \varepsilon^{q}]. \end{gathered} \quad (1.3.5) $$

**Remark 1.3.2.** Nets of this type are called negligible.

**Definition1.3.3.** A generalized symbol of order *m* and type $(\rho, \delta)$ is an element of the factor space

$$ \tilde{S}_{\rho, \delta}^{m}(\Omega \times \mathbb{R}^{p}) \triangleq S_{\rho, \delta}^{m}(\Omega \times \mathbb{R}^{p}) / \mathbb{N}_{\rho, \delta}^{m}(\Omega \times \mathbb{R}^{p}) \quad (1.3.6) $$

**Remark 1.3.3.** We denote an arbitrary representative of $v \in \tilde{S}_{\rho, \delta}^m[\Omega \times \mathbb{R}^p]$ by $(v_\epsilon)_{\epsilon \in (0,1]}$ or simply by $(v_\epsilon)_\epsilon$.

**Definition1.3.4.** An element $v \in \tilde{S}_{\rho, \delta}^m(\Omega \times \mathbb{R}^p)$ is called regular if it has a representative $(v_\epsilon)_\epsilon$ with the following property:
$$ 
\begin{array}{l}
\forall K(K \in \Omega) \exists N(N \in \mathbb{N}) \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^n) \exists \eta(\eta \in (0,1]) \exists c(c > 0) \\
\forall y(y \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0,1]) [\partial_{\xi}^{\alpha} \partial_{y}^{\beta} u_{\varepsilon}(y, \xi) \le c \langle \xi \rangle^{m-\rho|\alpha|+|\delta|} \varepsilon^{-N}].
\end{array}
\qquad (1.3.7) $$

We denote by

$$ \tilde{S}_{\rho, \delta, \text{rg}}^{m}(\Omega \times \mathbb{R}^{p}) \qquad (1.3.8) $$

the subspace of regular elements of the space $\tilde{S}_{\rho, \delta}^{m}(\Omega \times \mathbb{R}^{p})$.

If the property (1.3.7) holds for one representative of $v$, it holds for every representative. Consequently, if $\tilde{S}_{\rho, \delta, \text{rg}}^{m}(\Omega \times \mathbb{R}^{p})$ is the space defined by (1.3.7) we can introduce

$\tilde{S}_{\rho, \delta, \text{rg}}^{m}(\Omega \times \mathbb{R}^{p})$ as the factor space:

$$ \tilde{S}_{\rho, \delta, \text{rg}}^{m}(\Omega \times \mathbb{R}^{p}) \triangleq S_{\rho, \delta, \text{rg}}^{m}(\Omega \times \mathbb{R}^{p}) / N_{\rho, \delta}^{m}(\Omega \times \mathbb{R}^{p}) \qquad (1.3.9) $$

**Remark 1.3.4.** Note that $(1) (v_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, \mathbf{M}}^{m}(\Omega \times \mathbb{R}^{p})$ implies $(\partial_{\xi}^{\alpha} \partial_{x}^{\beta} v_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, \mathbf{M}}^{m-\rho|\alpha|-|\delta|}(\Omega \times \mathbb{R}^{p})$, $(2) (v_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, \mathbf{M}}^{m_1}(\Omega \times \mathbb{R}^{p})$ and $(w_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, \mathbf{M}}^{m_2}(\Omega \times \mathbb{R}^{p})$ implies $(v_{\varepsilon} + w_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, \mathbf{M}}^{\max\{m_1, m_2\}}(\Omega \times \mathbb{R}^{p})$ and $(v_{\varepsilon} w_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, \mathbf{M}}^{m_1+m_2}(\Omega \times \mathbb{R}^{p})$. Since the results concerning derivatives and sums hold with $S_{\rho, \delta, \text{rg}}^{m}(\Omega \times \mathbb{R}^{p})$ and $N_{\rho, \delta}$ in place of $S_{\rho, \delta, \mathbf{M}}^{m}(\Omega \times \mathbb{R}^{p})$, we can define derivatives and sums on the corresponding factor spaces. Moreover, $(v_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, \mathbf{M}}^{m_1}(\Omega \times \mathbb{R}^{p})$ and $(w_{\varepsilon})_{\varepsilon} \in N_{\rho, \delta}^{m_2}(\Omega \times \mathbb{R}^{p})$ imply $(v_{\varepsilon} w_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta}^{m_1+m_2}(\Omega \times \mathbb{R}^{p})$, thus we obtain that the product is a well-defined map from the space $\tilde{S}_{\rho, \delta}^{m_1}(\Omega \times \mathbb{R}^{p}) \times \tilde{S}_{\rho, \delta}^{m_2}(\Omega \times \mathbb{R}^{p})$ into $\tilde{S}_{\rho, \delta}^{m_1+m_2}(\Omega \times \mathbb{R}^{p})$. Similarly, it is well-defined as a map from $\tilde{S}_{\rho, \delta, \text{rg}}^{m_1}(\Omega \times \mathbb{R}^{p}) \times \tilde{S}_{\rho, \delta, \text{rg}}^{m_2}(\Omega \times \mathbb{R}^{p})$ into $\tilde{S}_{\rho, \delta, \text{rg}}^{m_1+m_2}(\Omega \times \mathbb{R}^{p})$.

**Remark 1.3.5.** The classical space $S_{\rho, \delta}^{m_1}(\Omega \times \mathbb{R}^{p})$ is contained in $\tilde{S}_{\rho, \delta, \text{rg}}^{m}(\Omega \times \mathbb{R}^{p})$. We assume from now on that $\rho > 0$ and $\delta < 1$ and return to the meaning of the oscillatory integral
$$ 
\begin{array}{l}
\forall K(K \in \Omega) \exists N(N \in \mathbb{N}) \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^n) \exists \eta(\eta \in (0,1]) \exists c(c > 0) \\
\forall y(y \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0,1]) [\partial_{\xi}^{\alpha} \partial_{y}^{\beta} u_{\varepsilon}(y, \xi) \le c \langle \xi \rangle^{m-p|\alpha|+|\delta|} \varepsilon^{-N}].
\end{array}
\qquad (1.3.7)
$$

We denote by

$$ \tilde{\mathbf{S}}_{\rho, \delta, \mathrm{rg}}^{m}(\Omega \times \mathbb{R}^{p}) \qquad (1.3.8) $$

the subspace of regular elements of the space $\tilde{\mathbf{S}}_{\rho, \delta}^{m}(\Omega \times \mathbb{R}^{p})$.

If the property (1.3.7) holds for one representative of $v$, it holds for every representative. Consequently, if $\tilde{\mathbf{S}}_{\rho, \delta, \mathrm{rg}}^{m}(\Omega \times \mathbb{R}^{p})$ is the space defined by (1.3.7) we can introduce

$\tilde{\mathbf{S}}_{\rho, \delta, \mathrm{rg}}^{m}(\Omega \times \mathbb{R}^{p})$ as the factor space:

$$ \tilde{\mathbf{S}}_{\rho, \delta, \mathrm{rg}}^{m}(\Omega \times \mathbb{R}^{p}) \triangleq \mathbf{S}_{\rho, \delta, \mathrm{rg}}^{m}(\Omega \times \mathbb{R}^{p}) / \mathbf{N}_{\rho, \delta}^{m}(\Omega \times \mathbb{R}^{p}) \qquad (1.3.9) $$

**Remark 1.3.4.** Note that $(1) (v_{\varepsilon})_{\varepsilon} \in \mathbf{S}_{\rho, \delta, \mathbf{M}}^{m}(\Omega \times \mathbb{R}^{p})$ implies $(\partial_{\xi}^{\alpha} \partial_{x}^{\beta} v_{\varepsilon})_{\varepsilon} \in \mathbf{S}_{\rho, \delta, \mathbf{M}}^{m-\rho|\alpha|-|\delta|}(\Omega \times \mathbb{R}^{p})$, $(2) (v_{\varepsilon})_{\varepsilon} \in \mathbf{S}_{\rho, \delta, \mathbf{M}}^{m_1}(\Omega \times \mathbb{R}^{p})$ and $(w_{\varepsilon})_{\varepsilon} \in \mathbf{S}_{\rho, \delta, \mathbf{M}}^{m_2}(\Omega \times \mathbb{R}^{p})$ implies $(v_{\varepsilon} + w_{\varepsilon})_{\varepsilon} \in \mathbf{S}_{\rho, \delta, \mathbf{M}}^{\max\{m_1, m_2\}}(\Omega \times \mathbb{R}^{p})$ and $(v_{\varepsilon} w_{\varepsilon})_{\varepsilon} \in \mathbf{S}_{\rho, \delta, \mathbf{M}}^{m_1+m_2}(\Omega \times \mathbb{R}^{p})$. Since the results concerning derivatives and sums hold with $\mathbf{S}_{\rho, \delta, \mathrm{rg}}^{m}(\Omega \times \mathbb{R}^{p})$ and $N_{\rho, \delta}$ in place of $\mathbf{S}_{\rho, \delta, \mathbf{M}}^{m}(\Omega \times \mathbb{R}^{p})$, we can define derivatives and sums on the corresponding factor spaces. Moreover, $(v_{\varepsilon})_{\varepsilon} \in \mathbf{S}_{\rho, \delta, \mathbf{M}}^{m_1}(\Omega \times \mathbb{R}^{p})$ and $(w_{\varepsilon})_{\varepsilon} \in \mathbb{N}_{\rho, \delta}^{m_2}(\Omega \times \mathbb{R}^{p})$ imply $(v_{\varepsilon} w_{\varepsilon})_{\varepsilon} \in \mathbf{S}_{\rho, \delta}^{m_1+m_2}(\Omega \times \mathbb{R}^{p})$, thus we obtain that the product is a well-defined map from the space $\tilde{\mathbf{S}}_{\rho, \delta}^{m_1}(\Omega \times \mathbb{R}^{p}) \times \tilde{\mathbf{S}}_{\rho, \delta}^{m_2}(\Omega \times \mathbb{R}^{p})$ into $\tilde{\mathbf{S}}_{\rho, \delta}^{m_1+m_2}(\Omega \times \mathbb{R}^{p})$. Similarly, it is well-defined as a map from $\tilde{\mathbf{S}}_{\rho, \delta, \mathrm{rg}}^{m_1}(\Omega \times \mathbb{R}^{p}) \times \tilde{\mathbf{S}}_{\rho, \delta, \mathrm{rg}}^{m_2}(\Omega \times \mathbb{R}^{p})$ into $\tilde{\mathbf{S}}_{\rho, \delta, \mathrm{rg}}^{m_1+m_2}(\Omega \times \mathbb{R}^{p})$.

**Remark 1.3.5.** The classical space $\mathbf{S}_{\rho, \delta}^{m_1}(\Omega \times \mathbb{R}^{p})$ is contained in $\tilde{\mathbf{S}}_{\rho, \delta, \mathrm{rg}}^{m}(\Omega \times \mathbb{R}^{p})$. We assume from now on that $\rho > 0$ and $\delta < 1$ and return to the meaning of the oscillatory integral
$$ \int_{K \times \mathbb{R}^p} \exp[i\varphi_{\varepsilon}(y, \xi)] a_{\varepsilon}(y, \xi) dy d\xi. \qquad (1.3.10) $$

Obviously, if $(\varphi_{\varepsilon})_{\varepsilon}$, $(a_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, M}^{m_1}(\Omega \times \mathbb{R}^p)$ then (1.3.10) makes sense as an oscillatory integral for every $\varepsilon \in (0, 1]$. We recall now that given the phase function $\varphi_{\varepsilon}, \varepsilon \in (0, 1]$, there exists an operator

$$ L_{\varepsilon} = \sum_{i=1}^{p} a_{i,\varepsilon}(y, \xi) \frac{\partial}{\partial \xi_i} + \sum_{k=1}^{n} b_{k,\varepsilon}(y, \xi) \frac{\partial}{\partial y_k} + c_{\varepsilon}(y, \xi) \qquad (1.3.11) $$

such that $a_{i,\varepsilon}(y, \xi) \in S^0(\Omega \times \mathbb{R}^p)$, $b_{k,\varepsilon}(y, \xi) \in S^{-1}(\Omega \times \mathbb{R}^p)$, $c_{\varepsilon}(y, \xi) \in S^{-1}(\Omega \times \mathbb{R}^p)$, and such that $^tL \exp[i\varphi_{\varepsilon}] = \exp[i\varphi_{\varepsilon}]$, where $^tL$ is the formal adjoint. We note that for $m - js < -p$ and $\chi \in C_c^{\infty}(\mathbb{R}^p)$ identically equal to 1 in a neighborhood of the origin, the oscillatory integral $I_K(\varphi_{\varepsilon}, a_{\varepsilon})$ at fixed $\varepsilon$, can be defined by either of the two expressions on the right hand-side of Eq. (1.3.12)

$$ \begin{aligned} I_K(\varphi_{\varepsilon}, a_{\varepsilon}) &= \int_{K \times \mathbb{R}^p} \exp[i\varphi_{\varepsilon}(y, \xi)] a_{\varepsilon}(y, \xi) dy d\xi = \\ & \qquad \int_{K \times \mathbb{R}^p} \exp[i\varphi_{\varepsilon}(y, \xi)] L^j a_{\varepsilon}(y, \xi) dy d\xi = \\ & \qquad \lim_{h \to 0+} \int_{K \times \mathbb{R}^p} \exp[i\varphi_{\varepsilon}(y, \xi)] a_{\varepsilon}(y, \xi) \chi(h\xi) dy d\xi, \end{aligned} \qquad (1.3.12) $$

where the equalities hold for all $\varepsilon \in (0, 1]$.

**Theorem 1.3.1.** Let $s = \min\{\rho, 1 - \delta\}$ and $j \in \mathbb{N}$. Then the following statements

hold:

(1) if $(a_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, M}^{m}(\Omega \times \mathbb{R}^p)$ then $(L^j a_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, M}^{m-j\delta}(\Omega \times \mathbb{R}^p)$,

(2) statement (1) is valid with $S_{\rho, \delta, \text{rg}}^{m}(\Omega \times \mathbb{R}^p)$ in place of $S_{\rho, \delta, M}^{m}(\Omega \times \mathbb{R}^p)$.

**Theorem 1.3.2.** Let $K$ be a compact set contained in $\Omega$ and let $\varphi_{\varepsilon}, \varepsilon \in (0, 1]$ be a phase function on $\Omega \times \mathbb{R}^p$ and $\varphi = \text{cl}[(\varphi_{\varepsilon})_{\varepsilon}]$, $a = \text{cl}[(a_{\varepsilon})_{\varepsilon}]$ an element of $\tilde{S}_{\rho, \delta}^{m}(\Omega \times \mathbb{R}^p)$. Then the oscillatory integral
$$ \int_{K \times \mathbb{R}^p} \exp[i\varphi_{\varepsilon}(y, \xi)] a_{\varepsilon}(y, \xi) dy d\xi. \qquad (1.3.10) $$

Obviously, if $(\varphi_{\varepsilon})_{\varepsilon}$, $(a_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, M}^{m_1}(\Omega \times \mathbb{R}^p)$ then (1.3.10) makes sense as an oscillatory integral for every $\varepsilon \in (0, 1]$. We recall now that given the phase function $\varphi_{\varepsilon}, \varepsilon \in (0, 1]$, there exists an operator

$$ L_{\varepsilon} = \sum_{i=1}^{p} a_{i,\varepsilon}(y, \xi) \frac{\partial}{\partial \xi_i} + \sum_{k=1}^{n} b_{k,\varepsilon}(y, \xi) \frac{\partial}{\partial y_k} + c_{\varepsilon}(y, \xi) \qquad (1.3.11) $$

such that $a_{i,\varepsilon}(y, \xi) \in S^0(\Omega \times \mathbb{R}^p)$, $b_{k,\varepsilon}(y, \xi) \in S^{-1}(\Omega \times \mathbb{R}^p)$, $c_{\varepsilon}(y, \xi) \in S^{-1}(\Omega \times \mathbb{R}^p)$, and such that $^tL \exp[i\varphi_{\varepsilon}] = \exp[i\varphi_{\varepsilon}]$, where $^tL$ is the formal adjoint. We note that for $m - js < -p$ and $\chi \in C_c^{\infty}(\mathbb{R}^p)$ identically equal to 1 in a neighborhood of the origin, the oscillatory integral $I_K(\varphi_{\varepsilon}, a_{\varepsilon})$ at fixed $\varepsilon$, can be defined by either of the two expressions on the right hand-side of Eq. (1.3.12)

$$ \begin{align} I_K(\varphi_{\varepsilon}, a_{\varepsilon}) &= \int_{K \times \mathbb{R}^p} \exp[i\varphi_{\varepsilon}(y, \xi)] a_{\varepsilon}(y, \xi) dy d\xi = \nonumber \\ &\qquad \int_{K \times \mathbb{R}^p} \exp[i\varphi_{\varepsilon}(y, \xi)] L^j a_{\varepsilon}(y, \xi) dy d\xi = \tag{1.3.12} \\ &\qquad \lim_{h \to 0+} \int_{K \times \mathbb{R}^p} \exp[i\varphi_{\varepsilon}(y, \xi)] a_{\varepsilon}(y, \xi) \chi(h\xi) dy d\xi, \nonumber \end{align} $$

where the equalities hold for all $\varepsilon \in (0, 1]$.

**Theorem 1.3.1.** Let $s = \min\{\rho, 1 - \delta\}$ and $j \in \mathbb{N}$. Then the following statements hold:

(1) if $(a_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, M}^{m}(\Omega \times \mathbb{R}^p)$ then $(L^j a_{\varepsilon})_{\varepsilon} \in S_{\rho, \delta, M}^{m-j s}(\Omega \times \mathbb{R}^p)$,

(2) statement (1) is valid with $S_{\rho, \delta, \mathrm{rg}}^{m}(\Omega \times \mathbb{R}^p)$ in place of $S_{\rho, \delta, M}^{m}(\Omega \times \mathbb{R}^p)$.

**Theorem 1.3.2.** Let $K$ be a compact set contained in $\Omega$ and let $\varphi_{\varepsilon}, \varepsilon \in (0, 1]$ be a phase function on $\Omega \times \mathbb{R}^p$ and $\varphi = \mathbf{cl}[(\varphi_{\varepsilon})_{\varepsilon}]$, $a = \mathbf{cl}[(a_{\varepsilon})_{\varepsilon}]$ an element of $\tilde{S}_{\rho, \delta}^{m}(\Omega \times \mathbb{R}^p)$. Then the oscillatory integral
$$ I_K(\varphi, a) = \int_{K \times \mathbb{R}^p} \exp[i\varphi_{\varepsilon}(y, \xi)] a_{\varepsilon}(y, \xi) dy d\xi = \\ (I_K(\varphi_{\varepsilon}, a_{\varepsilon}))_{\varepsilon} + \mathbf{N} \qquad (1.3.13) $$

is a well-defined element of $\tilde{\mathbb{C}}$.

**Definition1.3.5.** Let $\varphi, a \in \tilde{S}_{\rho, \delta}^m(\Omega \times \mathbb{R}^p)$ with $\text{supp}_y(a)$. We define the generalized oscillatory integral

$$ I(\varphi, a) \triangleq \int_{\Omega \times \mathbb{R}^p} \exp[i\varphi(y, \xi)] a(y, \xi) dy d\xi \triangleq \\ \int_{K \times \mathbb{R}^p} \exp[i\varphi(y, \xi)] a(y, \xi) dy d\xi, \qquad (1.3.14) $$

where $K$ is any compact subset of $\Omega$ containing $\text{supp}_y(a)$ in its interior.

**Remark 1.3.6.** We note that definition 1.3.5 does not depend on the choice of $K$.

**Definition1.3.6.** Let us consider now phase functions and symbols depending on an additional parameter.

$$ I(\varphi, a)(x) = \int_{K \times \mathbb{R}^p} \exp[i\varphi(x, y, \xi)] a(x, y, \xi) dy d\xi, \qquad (1.3.15) $$

where $x \in \Omega'$, an open subset of $\mathbb{R}^{n'}$. If for any fixed $\tilde{x} \in \Omega'$, $\varphi(\tilde{x}, y, \xi) \in \tilde{S}_{\rho, \delta}^{m_1}(\Omega \times \mathbb{R}^p)$ is a phase function with respect to the variables $(y, \xi)$ and $a(\tilde{x}, y, \xi) \in \tilde{S}_{\rho, \delta}^{m_2}(\Omega \times \mathbb{R}^p)$ the oscillatory integral $I(\varphi, a)(x)$ defines a map from $\Omega'$ to $\tilde{\mathbb{C}}$.

**Remark 1.3.7.** Let for any $\varepsilon \in (0, 1]$, $\varphi_{\varepsilon}(x, y, \xi)$ be a real valued continuous function on $\Omega' \times \Omega \times \mathbb{R}^p$, smooth on $\Omega' \times \Omega \times \mathbb{R}^p \setminus \{0\}$ such that for all $x \in \Omega'$, such that for any $\varepsilon \in (0, 1]$, $\varphi(\tilde{x}, y, \xi)$ is a phase function with respect to the variables $(y, \xi)$. Obviously we have that for all $j \in \mathbb{N}$, $(\varphi_{\varepsilon}(x, y, \xi))_{\varepsilon} \in S_{\rho, \delta}^{m_1}(\Omega' \times \Omega \times \mathbb{R}^p)$ and $(a_{\varepsilon}(x, y, \xi))_{\varepsilon} \in S_{\rho, \delta}^{m_2}(\Omega' \times \Omega \times \mathbb{R}^p)$ implies $(L_x^j a_{\varepsilon}(x, y, \xi))_{\varepsilon} \in S_{\rho, \delta, \mathbb{M}}^{m_2-j\varepsilon}(\Omega' \times \Omega \times \mathbb{R}^p)$. The same result holds with $S_{\rho, \delta, \mathbb{R}^p}^m(\Omega' \times \Omega \times \mathbb{R}^p)$ in place of $S_{\rho, \delta, \mathbb{M}}^m(\Omega' \times \Omega \times \mathbb{R}^p)$.

**Theorem 1.3.3.** Let $\varphi(x, y, \xi) = \text{cl}[(\varphi_{\varepsilon}(x, y, \xi))_{\varepsilon}]$ be as in Remark 1.3.7.
$$
\begin{equation} \tag{1.3.13}
\begin{split}
I_K(\varphi, a) &= \int_{K \times \mathbb{R}^p} \exp[i\varphi_\varepsilon(y, \xi)] a_\varepsilon(y, \xi) dy d\xi \\
&= (I_K(\varphi_\varepsilon, a_\varepsilon))_\varepsilon + \mathbf{N}
\end{split}
\end{equation}
$$

is a well-defined element of $\tilde{\mathbb{C}}$.

**Definition1.3.5.** Let $\varphi, a \in \tilde{S}_{\rho, \delta}^m (\Omega \times \mathbb{R}^p)$ with $\text{supp}_y(a)$. We define the generalized oscillatory integral

$$
\begin{equation} \tag{1.3.14}
\begin{split}
I(\varphi, a) &\triangleq \int_{\Omega \times \mathbb{R}^p} \exp[i\varphi(y, \xi)] a(y, \xi) dy d\xi \triangleq \\
&\qquad \int_{K \times \mathbb{R}^p} \exp[i\varphi(y, \xi)] a(y, \xi) dy d\xi,
\end{split}
\end{equation}
$$

where $K$ is any compact subset of $\Omega$ containing $\text{supp}_y(a)$ in its interior.

**Remark 1.3.6.** We note that definition 1.3.5 does not depend on the choice of K.

**Definition1.3.6.** Let us consider now phase functions and symbols depending on an additional parameter.

$$
I(\varphi, a)(x) = \int_{K \times \mathbb{R}^p} \exp[i\varphi(x, y, \xi)] a(x, y, \xi) dy d\xi, \quad (1.3.15)
$$

where $x \in \Omega'$, an open subset of $\mathbb{R}^{n'}$. If for any fixed $\tilde{x} \in \Omega'$, $\varphi(\tilde{x}, y, \xi) \in \tilde{S}_{\rho, \delta}^{m_1}(\Omega \times \mathbb{R}^p)$ is a phase function with respect to the variables $(y, \xi)$ and $a(\tilde{x}, y, \xi) \in \tilde{S}_{\rho, \delta}^{m_2}(\Omega \times \mathbb{R}^p)$ the oscillatory integral $I(\varphi, a)(x)$ defines a map from $\Omega'$ to $\tilde{\mathbb{C}}$.

**Remark 1.3.7.** Let for any $\varepsilon \in (0, 1]$, $\varphi_\varepsilon(x, y, \xi)$ be a real valued continuous function on $\Omega' \times \Omega \times \mathbb{R}^p$, smooth on $\Omega' \times \Omega \times \mathbb{R}^p \setminus \{0\}$ such that for all $x \in \Omega'$, such that for any $\varepsilon \in (0, 1]$, $\varphi(\tilde{x}, y, \xi)$ is a phase function with respect to the variables $(y, \xi)$. Obviously we have that for all $j \in \mathbb{N}$, $(\varphi_\varepsilon(x, y, \xi))_\varepsilon \in S_{\rho, \delta}^{m_1}(\Omega' \times \Omega \times \mathbb{R}^p)$ and $(a_\varepsilon(x, y, \xi))_\varepsilon \in S_{\rho, \delta}^{m_2}(\Omega' \times \Omega \times \mathbb{R}^p)$ implies $(L_x^j a_\varepsilon(x, y, \xi))_\varepsilon \in S_{\rho, \delta, \mathbb{M}}^{m_2-j\varepsilon}(\Omega' \times \Omega \times \mathbb{R}^p)$. The same result holds with $S_{\rho, \delta, \mathbb{R}^p}^m(\Omega' \times \Omega \times \mathbb{R}^p)$ in place of $S_{\rho, \delta, \mathbb{M}}^m(\Omega' \times \Omega \times \mathbb{R}^p)$.

**Theorem 1.3.3.** Let $\varphi(x, y, \xi) = \text{cl}[(\varphi_\varepsilon(x, y, \xi))_\varepsilon]$ be as in Remark 1.3.7.
(1) If $a(x,y,\xi) \in \tilde{\mathbf{S}}_{\rho,\delta}^{m_2}(\Omega' \times \Omega \times \mathbb{R}^p)$ then for all $K \in \Omega$

$$W_K(x) = \int_{K \times \mathbb{R}^p} \exp[i\varphi(x,y,\xi)]a(x,y,\xi)dyd\xi \quad (1.3.16)$$

then $W_K(x) \in G(\Omega')$.

(2) If $\varphi(x,y,\xi) \in \tilde{\mathbf{S}}_{\rho,\delta,\text{rg}}^{m_1}(\Omega' \times \Omega \times \mathbb{R}^p)$ and $a(x,y,\xi) \in \tilde{\mathbf{S}}_{\rho,\delta,\text{rg}}^{m_2}(\Omega' \times \Omega \times \mathbb{R}^p)$ then $W_K(x) \in G^\infty(\Omega')$ for all $K \in \Omega$.

**Proof.** An arbitrary representative of $W_K(x)$ is given by the oscillatory integral

$$(W_{K,\varepsilon}(x))_{\varepsilon} = \left( \int_{K \times \mathbb{R}^p} \exp[i\varphi_{\varepsilon}(x,y,\xi)] a_{\varepsilon}(x,y,\xi) dy d\xi \right)_{\varepsilon} \quad (1.3.17)$$

From Remark 1.3.7. it follows that $(W_{K,\varepsilon}(x))_{\varepsilon} \in E(\Omega')$. At this point computing the $x$-derivatives of the expression $\exp[i\varphi_{\varepsilon}(x,y,\xi)]L_x^j a_{\varepsilon}(x,y,\xi)$ for $\xi \neq 0$, we conclude that

$$\begin{align}
& \forall \alpha(\alpha \in \mathbb{N}^{n'}) \forall K' (K' \in \Omega') \exists N(N \in \mathbb{N}) \exists \eta(\eta \in (0,1]) \forall x(x \in K') \nonumber \\
& \qquad \forall y(y \in K) \forall \xi(\xi \in \mathbb{R}^p \setminus \{0\}) \exists \varepsilon(\varepsilon \in (0,1]) \tag{1.3.18} \\
& \qquad \qquad \left[ \left| \partial_x^\alpha \left( \exp[i\varphi_\varepsilon(x,y,\xi)] L_x^j a_\varepsilon(x,y,\xi) \right) \right| \le \langle \xi \rangle^{m_1+m_2-js+|\alpha|} \varepsilon^{-N} \right]. \nonumber
\end{align}$$

Now if $m_1 + m_2 - js + |\alpha| < -p$ then we obtain for any $x \in K'$ and $\varepsilon \in (0,1]$ the inequality

$$|\partial_x^\alpha W_{K,\varepsilon}(x)| \le \varepsilon^{-N}. \quad (1.3.19)$$

Hence $(W_{K,\varepsilon}(x))_{\varepsilon} \in E_M(\Omega')$. Obviously if $(a_\varepsilon)_{\varepsilon} \in \mathbb{N}_{\rho,\delta}^m(\Omega' \times \Omega \times \mathbb{R}^p)$ then
(1) If $a(x,y,\xi) \in \tilde{\mathbf{S}}_{\rho,\delta}^{m_2}(\Omega' \times \Omega \times \mathbb{R}^p)$ then for all $K \in \Omega$

$$W_K(x) = \int_{K \times \mathbb{R}^p} \exp[i\varphi(x,y,\xi)]a(x,y,\xi)dyd\xi \quad (1.3.16)$$

then $W_K(x) \in G(\Omega')$.

(2) If $\varphi(x,y,\xi) \in \tilde{\mathbf{S}}_{\rho,\delta,\text{rg}}^{m_1}(\Omega' \times \Omega \times \mathbb{R}^p)$ and $a(x,y,\xi) \in \tilde{\mathbf{S}}_{\rho,\delta,\text{rg}}^{m_2}(\Omega' \times \Omega \times \mathbb{R}^p)$ then $W_K(x) \in G^\infty(\Omega')$ for all $K \in \Omega$.

**Proof.** An arbitrary representative of $W_K(x)$ is given by the oscillatory integral

$$(W_{K,\varepsilon}(x))_{\varepsilon} = \left( \int_{K \times \mathbb{R}^p} \exp[i\varphi_{\varepsilon}(x,y,\xi)] a_{\varepsilon}(x,y,\xi) dy d\xi \right)_{\varepsilon} \quad (1.3.17)$$

From Remark 1.3.7. it follows that $(W_{K,\varepsilon}(x))_{\varepsilon} \in E(\Omega')$. At this point computing the $x$-derivatives of the expression $\exp[i\varphi_{\varepsilon}(x,y,\xi)]L_x^j a_{\varepsilon}(x,y,\xi)$ for $\xi \neq 0$, we conclude that

$$\begin{align}
& \forall \alpha(\alpha \in \mathbb{N}^{n'}) \forall K' (K' \in \Omega') \exists N(N \in \mathbb{N}) \exists \eta(\eta \in (0,1]) \forall x(x \in K') \nonumber \\
& \qquad \forall y(y \in K) \forall \xi(\xi \in \mathbb{R}^p \setminus \{0\}) \exists \varepsilon(\varepsilon \in (0,1]) \tag{1.3.18} \\
& \qquad \qquad \left[ |\partial_x^\alpha (\exp[i\varphi_\varepsilon(x,y,\xi)] L_x^j a_\varepsilon(x,y,\xi))| \le \langle \xi \rangle^{m_1+m_2-js+|\alpha|} \varepsilon^{-N} \right]. \nonumber
\end{align}$$

Now if $m_1 + m_2 - js + |\alpha| < -p$ then we obtain for any $x \in K'$ and $\varepsilon \in (0,1]$ the inequality

$$|\partial_x^\alpha W_{K,\varepsilon}(x)| \le \varepsilon^{-N}. \quad (1.3.19)$$

Hence $(W_{K,\varepsilon}(x))_{\varepsilon} \in E_M(\Omega')$. Obviously if $(a_\varepsilon)_{\varepsilon} \in \mathbb{N}_{\rho,\delta}^m(\Omega' \times \Omega \times \mathbb{R}^p)$ then
$(W_{K,ε}(x))_ε ∈ \mathbf{N}(\Omega')$. If $φ ∈ \tilde{\mathbf{S}}_{\rho,δ,rg}^{m_1}(\Omega' × \Omega × \mathbb{R}^p)$ and $a ∈ \tilde{\mathbf{S}}_{\rho,δ,rg}^{m_2}(\Omega' × \Omega × \mathbb{R}^p)$ the exponent $N$ in (1.3.19) does not depend on the derivatives and then $W_K(x) ∈ G^∞(\Omega')$. This result completes the proof.

**Remark 1.3.8.** If $φ = \mathbf{cl}[(φ_ε)_ε]$ is a generalized phase function with respect to $(v, ξ)$ and there exists a compact subset $K$ of $\Omega$ such that $∀x(x ∈ Ω')[\mathrm{supp}_y(x,y,ξ) ⊆ K]$ then the oscillatory integral

$$ (W_{K,ε}(x))_ε = \left( \int_{\Omega \times \mathbb{R}^p} \exp[i\phi_ε(x,y,\xi)] a_ε(x,y,\xi) dy d\xi \right)_ε \quad (1.3.20) $$

defines a generalized function belonging to $G(\Omega')$.

**Remark 1.3.9.** We recall that for each generalized phase function $(φ_ε)_ε$

$$ C_{φ_ε} = \{(x, ξ) ∈ Ω × \mathbb{R}^p \setminus \{0\} | \nabla_ξ φ_ε(x, ξ) = 0, ε \neq 0\} \quad (1.3.21) $$

is a cone-shaped subset of $Ω × \mathbb{R}^p \setminus \{0\}$. Let $π: Ω × \mathbb{R}^p \setminus \{0\} → Ω$ be the projection onto $Ω$ and put $S_{φ_ε} = π(C_{φ_ε}), R_{φ_ε} = Ω \setminus S_{φ_ε}$. Interpreting $x ∈ Ω$ as a parameter we have from Theorem 1.3.3 that

$$ W(x) = $$

$$ \int_{\mathbb{R}^p} \exp[i\phi(x, \xi)] a(x, \xi) d\xi = \quad (1.3.22) $$

$$ \left( \int_{\mathbb{R}^p} \exp[i\phi_\epsilon(x, \xi)] a_\epsilon(x, \xi) d\xi \right)_\epsilon + \mathbf{N} = $$

$$ (W_\epsilon(x))_\epsilon + \mathbf{N}. $$

makes sense as an generalized oscillatory integral for $x ∈ R_{φ_ε}$.

(1) If $φ ∈ \tilde{\mathbf{S}}_{\rho,δ}^{m_1}(\Omega' × Ω × \mathbb{R}^p)$ and $a ∈ \tilde{\mathbf{S}}_{\rho,δ}^{m_2}(\Omega' × Ω × \mathbb{R}^p)$ then $W(x) ∈ G(R_φ)$, where $R_φ = \bigcap_{ε \in (0,1]} R_{φ_ε}$.

(2) If $φ ∈ \tilde{\mathbf{S}}_{\rho,δ}^{m_1}(\Omega' × Ω × \mathbb{R}^p)$ and $a ∈ \tilde{\mathbf{S}}_{\rho,δ}^{m_2}(\Omega' × Ω × \mathbb{R}^p)$ and $\mathrm{supp}_x a_ε ⊆ Ω \setminus Ω'_ε$ where $Ω'_ε$ is an open neighborhood of $R_{φ_ε}$ then $W(x)$ can be extended to a generalized function on $Ω$ with support contained in $Ω \setminus \cup_{ε \in (0,1]} Ω'_ε$. We just give some details concerning the proof of the assertion (2). Let
$(W_{K,\varepsilon}(x))_{\varepsilon} \in \mathbf{N}(\Omega')$. If $\varphi \in \tilde{\mathbf{S}}_{\rho,\delta,\mathrm{rg}}^{m_1}(\Omega' \times \Omega \times \mathbb{R}^p)$ and $a \in \tilde{\mathbf{S}}_{\rho,\delta,\mathrm{rg}}^{m_2}(\Omega' \times \Omega \times \mathbb{R}^p)$ the exponent $N$ in (1.3.19) does not depend on the derivatives and then $W_K(x) \in G^{\infty}(\Omega')$. This result completes the proof.

**Remark 1.3.8.** If $\varphi = \mathbf{cl}[(\varphi_{\varepsilon})_{\varepsilon}]$ is a generalized phase function with respect to $(\nu, \xi)$ and there exists a compact subset $K$ of $\Omega$ such that $\forall x(x \in \Omega')[\mathrm{supp}_y(x,y,\xi) \subseteq K]$ then the oscillatory integral

$$ (W_{K,\varepsilon}(x))_{\varepsilon} = \left( \int_{\Omega \times \mathbb{R}^p} \exp[i\varphi_{\varepsilon}(x,y,\xi)] a_{\varepsilon}(x,y,\xi) dy d\xi \right)_{\varepsilon} \quad (1.3.20) $$

defines a generalized function belonging to $G(\Omega')$.

**Remark 1.3.9.** We recall that for each generalized phase function $(\varphi_{\varepsilon})_{\varepsilon}$

$$ C_{\varphi_{\varepsilon}} = \{(x, \xi) \in \Omega \times \mathbb{R}^p \setminus \{0\} | \nabla_{\xi} \varphi_{\varepsilon}(x, \xi) = 0, \varepsilon \neq 0\} \quad (1.3.21) $$

is a cone-shaped subset of $\Omega \times \mathbb{R}^p \setminus \{0\}$. Let $\pi : \Omega \times \mathbb{R}^p \setminus \{0\} \to \Omega$ be the projection onto $\Omega$ and put $S_{\varphi_{\varepsilon}} = \pi(C_{\varphi_{\varepsilon}}), R_{\varphi_{\varepsilon}} = \Omega \setminus S_{\varphi_{\varepsilon}}$. Interpreting $x \in \Omega$ as a parameter we have from Theorem 1.3.3 that

$W(x) =$

$$ \int_{\mathbb{R}^p} \exp[i\varphi(x, \xi)] a(x, \xi) d\xi = \quad (1.3.22) $$

$$ \left( \int_{\mathbb{R}^p} \exp[i\varphi_{\varepsilon}(x, \xi)] a_{\varepsilon}(x, \xi) d\xi \right)_{\varepsilon} + \mathbf{N} = $$

$$ (W_{\varepsilon}(x))_{\varepsilon} + \mathbf{N}. $$

makes sense as an generalized oscillatory integral for $x \in R_{\varphi_{\varepsilon}}$.

(1) If $\varphi \in \tilde{\mathbf{S}}_{\rho,\delta}^{m_1}(\Omega' \times \Omega \times \mathbb{R}^p)$ and $a \in \tilde{\mathbf{S}}_{\rho,\delta}^{m_2}(\Omega' \times \Omega \times \mathbb{R}^p)$ then $W(x) \in G(R_{\varphi})$, where $R_{\varphi} = \bigcap_{\varepsilon \in (0,1]} R_{\varphi_{\varepsilon}}$.

(2) If $\varphi \in \tilde{\mathbf{S}}_{\rho,\delta}^{m_1}(\Omega' \times \Omega \times \mathbb{R}^p)$ and $a \in \tilde{\mathbf{S}}_{\rho,\delta}^{m_2}(\Omega' \times \Omega \times \mathbb{R}^p)$ and $\mathrm{supp}_x a_{\varepsilon} \subseteq \Omega \setminus \Omega'_{\varepsilon}$ where $\Omega'_{\varepsilon}$ is an open neighborhood of $R_{\varphi_{\varepsilon}}$ then $W(x)$ can be extended to a generalized function on $\Omega$ with support contained in $\Omega \setminus \bigcup_{\varepsilon \in (0,1]} \Omega'_{\varepsilon}$.

We just give some details concerning the proof of the assertion (2). Let
$\{\Omega'_{j,\epsilon}\}_{j \in \mathbb{N}\setminus\{0\}}$ be an open covering of $\Omega'_\epsilon$ such that $\Omega'_{j,\epsilon}$ is relatively compact and $\Omega'_{j,\epsilon} \subset \bar{\Omega}'_{j,\epsilon} \subset \Omega'_{j+1,\epsilon}$ for all $j$. Choosing now cut-off functions $\{\psi_{j,\epsilon}\}_{j \in \mathbb{N}\setminus\{0\}}$ such that $\psi_{j,\epsilon} \in C^\infty(\Omega'_\epsilon)$ and $\psi_{j,\epsilon} = 1$ in a neighborhood of $\bar{\Omega}'_{j,\epsilon}$, we observe that $((1 - \psi_{j,\epsilon}(x))a_\epsilon(x, \xi))_\epsilon$ is a representative of a identically equal to 0 on $\Omega'_{j,\epsilon}$ for all $\epsilon \in (0, 1]$. Hence we see that

$$ W_0(x) = \left( \int_{\mathbb{R}^p} \exp[i\varphi_\epsilon(x, \xi)] a_\epsilon(x, \xi) d\xi \right)_\epsilon + \mathbf{N}(R_\varphi), \\ W_j(x) = \left( \int_{\mathbb{R}^p} (1 - \psi_{j,\epsilon}(x)) \exp[i\varphi_\epsilon(x, \xi)] a_\epsilon(x, \xi) d\xi \Big|_{\Omega'_{j,\epsilon}} \right)_\epsilon + \mathbf{N}(\Omega'_{j,\epsilon}), \quad (1.3.23) $$

$j \ge 1$

is a coherent family of generalized functions which defines $W(x) \in G(\Omega)$ such that $W_0(x)$ and $W_j(x)$ are its restrictions to $R_\varphi$ and $\cup_{\epsilon \in (0,1]} \Omega'_{j,\epsilon}$ respectively.

## 1.4. Pseudodifferential operators with generalized symbol.

**Definition 1.4.1.** Let $m, \mu, \delta$, be real numbers, $\rho, \delta \in [0, 1]$. Let $\omega(\tau)$ be a real valued function $\omega : (0, 1] \to \mathbb{R}, \omega(\tau) > 0$ on the interval $(0, 1]$, such that for some $r$ in $\mathbb{R}$, for some $C > 0$ and for all $\epsilon \in (0, 1]$, $\omega(\epsilon) > C\epsilon^r$. We denote by $S_{\rho, \delta, \omega}^{m, \mu}(\Omega' \times \mathbb{R}^p)$, where $\Omega'$ an open subset of $\mathbb{R}^{n'}$, the set of all $(a_\epsilon)_\epsilon \in S_{\rho, \delta}^m(\Omega' \times \mathbb{R}^p)$ such that the following statement holds:

$$ \begin{align} & \forall K(K \in \Omega') \exists N(N \in \mathbb{N}) \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^{n'}) \exists \eta(\eta \in (0.1]) \nonumber \\ & \exists c(c > 0) \forall x(x \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0,1]) \tag{1.4.1} \\ & \qquad |\partial_{\xi}^{\alpha} \partial_{x}^{\beta} a_{\varepsilon}(x, \xi)| \le c(\xi)^{m-\rho|\alpha|+\delta|\beta|} \varepsilon^{-N} (\omega(\varepsilon))^{-(|\beta|-\mu)_+}, \nonumber \end{align} $$
$\{\Omega'_{j,\epsilon}\}_{j \in \mathbb{N}\setminus\{0\}}$ be an open covering of $\Omega'_\epsilon$ such that $\Omega'_{j,\epsilon}$ is relatively compact and $\Omega'_{j,\epsilon} \subset \bar{\Omega}'_{j,\epsilon} \subset \Omega'_{j+1,\epsilon}$ for all $j$. Choosing now cut-off functions $\{\psi_{j,\epsilon}\}_{j \in \mathbb{N}\setminus\{0\}}$ such that $\psi_{j,\epsilon} \in C^\infty(\Omega'_\epsilon)$ and $\psi_{j,\epsilon} \equiv 1$ in a neighborhood of $\bar{\Omega}'_{j,\epsilon}$, we observe that $((1 - \psi_{j,\epsilon}(x))a_\epsilon(x, \xi))_\epsilon$ is a representative of a identically equal to 0 on $\Omega'_{j,\epsilon}$ for all $\epsilon \in (0, 1]$. Hence we see that

$$W_0(x) = \left( \int_{\mathbb{R}^p} \exp[i\varphi_\epsilon(x, \xi)] a_\epsilon(x, \xi) d\xi \right)_\epsilon + \mathbf{N}(R_\varphi),$$

$$W_j(x) = \left( \int_{\mathbb{R}^p} (1 - \psi_{j,\epsilon}(x)) \exp[i\varphi_\epsilon(x, \xi)] a_\epsilon(x, \xi) d\xi \Big|_{\Omega'_{j,\epsilon}} \right)_\epsilon + \mathbf{N}(\Omega'_{j,\epsilon}), \quad (1.3.23)$$

$$j \ge 1$$

is a coherent family of generalized functions which defines $W(x) \in G(\Omega)$ such that $W_0(x)$ and $W_j(x)$ are its restrictions to $R_\varphi$ and $\cup_{\epsilon \in (0,1]} \Omega'_{j,\epsilon}$ respectively.

## 1.4. Pseudodifferential operators with generalized symbol.

**Definition 1.4.1.** Let $m, \mu, \delta$, be real numbers, $\rho, \delta \in [0, 1]$. Let $\omega(\tau)$ be a real valued function $\omega : (0, 1] \to \mathbb{R}, \omega(\tau) > 0$ on the interval $(0, 1]$, such that for some $r$ in $\mathbb{R}$, for some $C > 0$ and for all $\epsilon \in (0, 1]$, $\omega(\epsilon) > C\epsilon^r$. We denote by $S_{\rho, \delta, \omega}^{m, \mu}(\Omega' \times \mathbb{R}^p)$, where $\Omega'$ an open subset of $\mathbb{R}^{n'}$, the set of all $(a_\epsilon)_\epsilon \in S_{\rho, \delta}^m(\Omega' \times \mathbb{R}^p)$ such that the following statement holds:

$$\forall K(K \in \Omega') \exists N(N \in \mathbb{N}) \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^{n'}) \exists \eta(\eta \in (0.1])$$

$$\exists c(c > 0) \forall x(x \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \epsilon(\epsilon \in (0,1]) \quad (1.4.1)$$

$$|\partial_{\xi}^{\alpha} \partial_{x}^{\beta} a_{\epsilon}(x, \xi)| \leq c \langle \xi \rangle^{m-\rho|\alpha|+\delta|\beta|} \epsilon^{-N} (\omega(\epsilon))^{-(|\beta|-\mu)_+},$$
where the exponent $-(|\beta| - \mu)_+ = -\max\{0, |\beta| - \mu\}$ reflects differentiability up to order $\mu$ in the case when $(a_\epsilon)_\epsilon$ is obtained from a non-smooth, classical symbol by convolution with a mollifier with scale $\omega(\epsilon)$.

**Definition 1.4.2.** An element of $N_{\rho, \delta, \omega}^{m, \mu}(\Omega' \times \mathbb{R}^p)$ is a net in $S_{\rho, \delta}^m[\Omega' \times \mathbb{R}^p]$ such that the following statement holds:

$$ 
\begin{align} 
& \forall K(K \in \Omega') \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^{n'}) \forall q(q \in \mathbb{N}) \exists \eta(\eta \in (0,1]) \nonumber \\ 
& \exists c(c > 0) \forall x(x \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0.1]) \tag{1.4.2} \\ 
& \qquad \qquad \qquad \qquad \qquad \qquad \left| \partial_{\xi}^{\alpha} \partial_{x}^{\beta} a_{\varepsilon}(x, \xi) \right| \le c \langle \xi \rangle^{m-\rho|\alpha|+\delta|\beta|} \varepsilon^q (\omega(\varepsilon))^{-(|\beta|-\mu)_+}. \nonumber 
\end{align} 
$$

Nets with this property are called negligible.

**Definition 1.4.3.** The factor space $S_{\rho, \delta, \omega}^{m, \mu}(\Omega' \times \mathbb{R}^p)/N_{\rho, \delta, \omega}^{m, \mu}(\Omega' \times \mathbb{R}^p)$ will be denoted by

$$ \tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega' \times \mathbb{R}^p). \tag{1.4.3} $$

**Remark 1.4.1.** The following mapping properties hold

$$ 
\begin{align} 
\partial_{\xi}^{\alpha} \partial_{x}^{\beta} : \tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m, \mu}(\Omega' \times \mathbb{R}^p) &\to \tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m-\rho|\alpha|+\delta|\beta|, \mu-|\beta|}(\Omega' \times \mathbb{R}^p), \tag{1.4.4} \\ 
(\cdot + \cdot) : \tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m_1, \mu}(\Omega' \times \mathbb{R}^p) \times \tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m_2, \mu}(\Omega' \times \mathbb{R}^p) &\to \tilde{\mathbf{S}}_{\rho, \delta, \omega}^{\max(m_1, m_2), \mu}(\Omega' \times \mathbb{R}^p). \nonumber 
\end{align} 
$$

**Definition 1.4.4.** We denote by $S_{\text{rg}}^{-\infty}(\Omega' \times \mathbb{R}^p)$ the set of all $(a_\epsilon)_\epsilon \in S^{-\infty}[\Omega' \times \mathbb{R}^p]$ such that
where the exponent $-(|\beta| - \mu)_+ = -\max\{0, |\beta| - \mu\}$ reflects differentiability up to order $\mu$ in the case when $(a_\epsilon)_\epsilon$ is obtained from a non-smooth, classical symbol by convolution with a mollifier with scale $\omega(\epsilon)$.

**Definition 1.4.2.** An element of $N_{\rho, \delta, \omega}^{m, \mu}(\Omega' \times \mathbb{R}^p)$ is a net in $S_{\rho, \delta}^m[\Omega' \times \mathbb{R}^p]$ such that the following statement holds:

$$ 
\begin{align} 
& \forall K(K \in \Omega') \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^{n'}) \forall q(q \in \mathbb{N}) \exists \eta(\eta \in (0,1]) \nonumber \\ 
& \exists c(c > 0) \forall x(x \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0.1]) \tag{1.4.2} \\ 
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left| \partial_{\xi}^{\alpha} \partial_{x}^{\beta} a_{\varepsilon}(x, \xi) \right| \le c \langle \xi \rangle^{m-\rho|\alpha|+\delta|\beta|} \varepsilon^q (\omega(\varepsilon))^{-(|\beta|-\mu)_+}. \nonumber 
\end{align} 
$$

Nets with this property are called negligible.

**Definition 1.4.3.** The factor space $S_{\rho, \delta, \omega}^{m, \mu}(\Omega' \times \mathbb{R}^p)/N_{\rho, \delta, \omega}^{m, \mu}(\Omega' \times \mathbb{R}^p)$ will be denoted by

$$ \tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega' \times \mathbb{R}^p). \tag{1.4.3} $$

**Remark 1.4.1.** The following mapping properties hold

$$ 
\begin{align} 
\partial_{\xi}^{\alpha} \partial_{x}^{\beta} : \tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m, \mu}(\Omega' \times \mathbb{R}^p) &\to \tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m-\rho|\alpha|+\delta|\beta|, \mu-|\beta|}(\Omega' \times \mathbb{R}^p), \tag{1.4.4} \\ 
(\cdot + \cdot) : \tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m_1, \mu}(\Omega' \times \mathbb{R}^p) \times \tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m_2, \mu}(\Omega' \times \mathbb{R}^p) &\to \tilde{\mathbf{S}}_{\rho, \delta, \omega}^{\max(m_1, m_2), \mu}(\Omega' \times \mathbb{R}^p). \nonumber 
\end{align} 
$$

**Definition 1.4.4.** We denote by $S_{\text{rg}}^{-\infty}(\Omega' \times \mathbb{R}^p)$ the set of all $(a_\epsilon)_\epsilon \in S^{-\infty}[\Omega' \times \mathbb{R}^p]$ such that
$$
\begin{equation} \tag{1.4.5}
\begin{aligned}
& \forall K(K \in \Omega') \exists N(N \in \mathbb{N}) \forall m(m \in \mathbb{R}) \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^{n'}) \exists \eta(\eta \in (0.1]) \\
& \phantom{\forall K(K \in \Omega') \exists N(N \in \mathbb{N}) \forall m(m \in \mathbb{R}) \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^{n'}) \exists \eta(\eta \in (0.1])} \\
& \exists c(c > 0) \forall x(x \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0,1])
\end{aligned}
\end{equation}
$$

$$
|\partial_{\xi}^{\alpha} \partial_{x}^{\beta} a_{\varepsilon}(x, \xi)| \leq c \langle \xi \rangle^{m-|\alpha|} \varepsilon^{-N}.
$$

We denote by $\mathbb{N}^{-\infty}(\Omega' \times \mathbb{R}^p)$ the set of all $(a_\varepsilon)_\varepsilon \in \mathbb{S}^{-\infty}[\Omega' \times \mathbb{R}^p]$ such that

$$
\begin{equation} \tag{1.4.6}
\begin{aligned}
& \forall K(K \in \Omega') \exists N(N \in \mathbb{N}) \forall q(q \in N) \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^{n'}) \exists \eta(\eta \in (0.1]) \\
& \quad \exists c(c > 0) \forall x(x \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0.1])
\end{aligned}
\end{equation}
$$

$$
|\partial_{\xi}^{\alpha} \partial_{x}^{\beta} a_{\varepsilon}(x, \xi)| \leq c \langle \xi \rangle^{m-|\alpha|} \varepsilon^{-q}.
$$

The factor space $\mathbf{S}_{\text{rg}}^{-\infty}(\Omega' \times \mathbb{R}^p)/\mathbb{N}^{-\infty}(\Omega' \times \mathbb{R}^p)$ we denoted by $\tilde{\mathbf{S}}_{\text{rg}}^{-\infty}(\Omega' \times \mathbb{R}^p)$.

**Remark 1.4.2.** If in addition to the usual assumptions on $\omega(\varepsilon)$ we assume that $(\omega^{-1}(\varepsilon))_{\varepsilon}$ is a slow scale net, we can uniformly bound the contributions of the derivatives in (1.4.1) by a single negative power of $\varepsilon$, obtaining for a suitable constant $c$, for $x \in K$ and for all $\varepsilon$ small enough, hence

$$
|\partial_{\xi}^{\alpha} \partial_{x}^{\beta} a_{\varepsilon}(x, \xi)| \leq c \langle \xi \rangle^{m-|\rho|a + |\delta|\beta} \varepsilon^{-N-1}.
$$

**Definition 1.4.5.** Let $\Omega$ be an open subset of $\mathbb{R}^n$. The elements of the factor spaces $\tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \mathbb{R}^p)$ and $\tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^p)$ will be called symbols and amplitudes of order $m$ and type $(\rho, \delta, \mu, \omega)$ respectively. The elements of the factor spaces $\tilde{\mathbf{S}}_{\text{rg}}^{-\infty}(\Omega \times \mathbb{R}^p)$ and $\tilde{\mathbf{S}}_{\text{rg}}^{-\infty}(\Omega \times \Omega \times \mathbb{R}^p)$ will be called smoothing symbols and smoothing amplitudes, respectively.

**Theorem 1.4.1.** Let $a \in \tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$. The oscillatory integral
$$
\begin{equation} \tag{1.4.5}
\begin{aligned}
& \forall K(K \in \Omega') \exists N(N \in \mathbb{N}) \forall m(m \in \mathbb{R}) \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^{n'}) \exists \eta(\eta \in (0.1]) \\
& \phantom{\forall K(K \in \Omega') \exists N(N \in \mathbb{N}) \forall m(m \in \mathbb{R}) \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^{n'}) \exists \eta(\eta \in (0.1])} \\
& \exists c(c > 0) \forall x(x \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0,1])
\end{aligned}
\end{equation}
$$

$$
|\partial_{\xi}^{\alpha} \partial_{x}^{\beta} a_{\varepsilon}(x, \xi)| \leq c \langle \xi \rangle^{m-|\alpha|} \varepsilon^{-N}.
$$

We denote by $\mathbb{N}^{-\infty}(\Omega' \times \mathbb{R}^p)$ the set of all $(a_\varepsilon)_\varepsilon \in \mathbb{S}^{-\infty}[\Omega' \times \mathbb{R}^p]$ such that

$$
\begin{equation} \tag{1.4.6}
\begin{aligned}
& \forall K(K \in \Omega') \exists N(N \in \mathbb{N}) \forall q(q \in \mathbb{N}) \forall \alpha(\alpha \in \mathbb{N}^p) \forall \beta(\beta \in \mathbb{N}^{n'}) \exists \eta(\eta \in (0.1]) \\
& \quad \exists c(c > 0) \forall x(x \in K) \forall \xi(\xi \in \mathbb{R}^p) \forall \varepsilon(\varepsilon \in (0,1])
\end{aligned}
\end{equation}
$$

$$
|\partial_{\xi}^{\alpha} \partial_{x}^{\beta} a_{\varepsilon}(x, \xi)| \leq c \langle \xi \rangle^{m-|\alpha|} \varepsilon^{-q}.
$$

The factor space $\mathbf{S}_{\text{rg}}^{-\infty}(\Omega' \times \mathbb{R}^p)/\mathbb{N}^{-\infty}(\Omega' \times \mathbb{R}^p)$ we denoted by $\tilde{\mathbf{S}}_{\text{rg}}^{-\infty}(\Omega' \times \mathbb{R}^p)$.

**Remark 1.4.2.** If in addition to the usual assumptions on $\omega(\varepsilon)$ we assume that $(\omega^{-1}(\varepsilon))_{\varepsilon}$ is a slow scale net, we can uniformly bound the contributions of the derivatives in (1.4.1) by a single negative power of $\varepsilon$, obtaining for a suitable constant $c$, for $x \in K$ and for all $\varepsilon$ small enough, hence

$$
|\partial_{\xi}^{\alpha} \partial_{x}^{\beta} a_{\varepsilon}(x, \xi)| \leq c \langle \xi \rangle^{m-|\rho|a + |\delta|\beta} \varepsilon^{-N-1}.
$$

**Definition 1.4.5.** Let $\Omega$ be an open subset of $\mathbb{R}^n$. The elements of the factor spaces $\tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \mathbb{R}^p)$ and $\tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^p)$ will be called symbols and amplitudes of order $m$ and type $(\rho, \delta, \mu, \omega)$ respectively. The elements of the factor spaces $\tilde{\mathbf{S}}_{\text{rg}}^{-\infty}(\Omega \times \mathbb{R}^p)$ and $\tilde{\mathbf{S}}_{\text{rg}}^{-\infty}(\Omega \times \Omega \times \mathbb{R}^p)$ will be called smoothing symbols and smoothing amplitudes, respectively.

**Theorem 1.4.1.** Let $a \in \tilde{\mathbf{S}}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$. The oscillatory integral
$$Au = \int_{\Omega \times \mathbb{R}^n} \exp[i(x-y)\xi]a(x,y,\xi)u(y)dyd\xi = \\ (A_{\varepsilon}u_{\varepsilon}(x))_{\varepsilon} + N(\Omega), \qquad (1.4.7)$$

$$A_{\varepsilon}u_{\varepsilon}(x) = \int_{\Omega \times \mathbb{R}^n} \exp[i(x-y)\xi]a_{\varepsilon}(x,y,\xi)u_{\varepsilon}(y)dyd\xi$$

defines:

(1) a $\tilde{\mathbb{C}}$-linear continuous map from $G_c(\Omega)$ into $G(\Omega)$,

(2) a $\tilde{\mathbb{C}}$-linear continuous map from $G_c^\infty(\Omega)$ into $G^\infty(\Omega)$ if $(\omega^{-1}(\varepsilon))_\varepsilon$ is a slow scale net,

(3) a $\tilde{\mathbb{C}}$-linear continuous map from $G_c(\Omega)$ into $G^\infty(\Omega)$ if $a$ belongs to $S_{\text{rg}}^{-\infty}(\Omega \times \Omega \times \mathbb{R}^n)$.

**Definition 1.4.6.** (1) Let $a \in \tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$. The linear map $A : G_c(\Omega) \to G(\Omega)$ defined via formula

$$u \to (Au)(x) = \int_{\Omega \times \mathbb{R}^n} \exp[i(x-y)\xi]a(x,y,\xi)u(y)dyd\xi \qquad (1.4.8)$$

will be called a (generalized) pseudodifferential operator with amplitude $a$.

(2) The formal transpose of the operator $A$ is the pseudodifferential operator ${}^t A : G_c(\Omega) \to G(\Omega)$ defined via formula

$$u \to ({}^t Au)(y) = \int_{\Omega \times \mathbb{R}^n} \exp[i(x-y)\xi]a(x,y,\xi)u(y)dxd\xi \qquad (1.4.9)$$

**Definition 1.4.7.** Let $A$ be a pseudodifferential operator. The map $k_A \in L(G_c(\Omega \times \Omega), \tilde{\mathbb{C}})$ defined by
$$Au = \int_{\Omega \times \mathbb{R}^n} \exp[i(x-y)\xi]a(x,y,\xi)u(y)dyd\xi = \\ (A_\varepsilon u_\varepsilon(x))_\varepsilon + \mathbf{N}(\Omega), \qquad (1.4.7)$$

$$A_\varepsilon u_\varepsilon(x) = \int_{\Omega \times \mathbb{R}^n} \exp[i(x-y)\xi]a_\varepsilon(x,y,\xi)u_\varepsilon(y)dyd\xi$$

defines:

(1) a $\tilde{\mathbb{C}}$-linear continuous map from $G_c(\Omega)$ into $G(\Omega)$,

(2) a $\tilde{\mathbb{C}}$-linear continuous map from $G_c^\infty(\Omega)$ into $G^\infty(\Omega)$ if $(\omega^{-1}(\varepsilon))_\varepsilon$ is a slow scale net,

(3) a $\tilde{\mathbb{C}}$-linear continuous map from $G_c(\Omega)$ into $G^\infty(\Omega)$ if $a$ belongs to $S_{\text{rg}}^{-\infty}(\Omega \times \Omega \times \mathbb{R}^n)$.

**Definition 1.4.6.** (1) Let $a \in \tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$. The linear map $A : G_c(\Omega) \to G(\Omega)$ defined via formula

$$u \to (Au)(x) = \int_{\Omega \times \mathbb{R}^n} \exp[i(x-y)\xi]a(x,y,\xi)u(y)dyd\xi \qquad (1.4.8)$$

will be called a (generalized) pseudodifferential operator with amplitude $a$.

(2) The formal transpose of the operator $A$ is the pseudodifferential operator ${}^t A : G_c(\Omega) \to G(\Omega)$ defined via formula

$$u \to ({}^t Au)(y) = \int_{\Omega \times \mathbb{R}^n} \exp[i(x-y)\xi]a(x,y,\xi)u(y)dxd\xi \qquad (1.4.9)$$

**Definition 1.4.7.** Let $A$ be a pseudodifferential operator. The map $k_A \in L(G_c(\Omega \times \Omega), \tilde{\mathbb{C}})$ defined by
$$k_A(u) = \int (Au(x, \cdot))(x)dx \qquad (1.4.10)$$

is called the kernel of A.

**Theorem 1.4.2.** Let $a \in \tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$ and $A$ be the corresponding pseudo-differential operator.

(1) For all $u_1 \in G_c(\Omega)$ and $u_2 \in G_c(\Omega)$

$$k_A(u_2 \otimes u_1) = \int_{\Omega} (Au_1)(x)u_2(x)dx = \int_{\Omega} u_1(x)(Au_2)(x)dx \qquad (1.4.11)$$

where $u_2 \otimes u_1 = (u_{2,\epsilon}(x)u_{1,\epsilon}(y))_{\epsilon} + N_c(\Omega \times \Omega)$.

(2) $k_A \in G(\Omega \times \Omega \setminus \Delta)$ where is the diagonal of $\Omega \times \Omega$. Moreover, for open subsets $W$ and $W'$ of with $W \times W' \subseteq \Omega \times \Omega \setminus \Delta$, and for all $u \in G_c(W')$

$$A[u(x)|_W] = \int_{\Omega} k_A(x,y)u(y)dy. \qquad (1.4.12)$$

(3) If $\text{supp}_{x,y}(a) \in \Omega \times \Omega \setminus \Omega'$, where $\Omega'$ is an open neighborhood of $\Delta$, then $k_A \in G(\Omega \times \Omega)$.

(4) If $(\omega^{-1}(\epsilon))_{\epsilon}$ is a slow scale net then (2) and (3) are valid with $G^{\infty}(\Omega \times \Omega \setminus \Delta)$ and $G^{\infty}(\Omega \times \Omega)$ in place of $G(\Omega \times \Omega \setminus \Delta)$ and $G(\Omega \times \Omega)$ respectively.

(5) If $a \in \tilde{S}_{\text{rg}}^{-\infty}(\Omega \times \Omega \times \mathbb{R}^p)$ then $k_A \in G^{\infty}(\Omega \times \Omega)$.

**Theorem 1.4.3.** Let $a \in \tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$ and $(\omega^{-1}(\epsilon))_{\epsilon}$ a slow scale net. Let $A$ be the corresponding pseudodifferential operator. Then for all $u \in G_c(\Omega)$

$$\text{singsupp}_g(Au) \subseteq \text{singsupp}_g(u). \qquad (1.4.13)$$

**Theorem 1.4.4.** Let us consider a linear operator $A : G_c(\Omega) \to G(\Omega)$ of the form
$$k_A(u) = \int (Au(x, \cdot))(x)dx \qquad (1.4.10)$$

is called the kernel of A.

**Theorem 1.4.2.** Let $a \in \tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$ and $A$ be the corresponding pseudo-differential operator.

(1) For all $u_1 \in G_c(\Omega)$ and $u_2 \in G_c(\Omega)$

$$k_A(u_2 \otimes u_1) = \int_{\Omega} (Au_1)(x)u_2(x)dx = \int_{\Omega} u_1(x)(Au_2)(x)dx \qquad (1.4.11)$$

where $u_2 \otimes u_1 = (u_{2,\epsilon}(x)u_{1,\epsilon}(y))_{\epsilon} + N_c(\Omega \times \Omega)$.

(2) $k_A \in G(\Omega \times \Omega \setminus \Delta)$ where is the diagonal of $\Omega \times \Omega$. Moreover, for open subsets $W$ and $W'$ of with $W \times W' \subseteq \Omega \times \Omega \setminus \Delta$, and for all $u \in G_c(W')$

$$A[u(x)|_W] = \int_{\Omega} k_A(x,y)u(y)dy. \qquad (1.4.12)$$

(3) If $\text{supp}_{x,y}(a) \in \Omega \times \Omega \setminus \Omega'$, where $\Omega'$ is an open neighborhood of $\Delta$, then $k_A \in G(\Omega \times \Omega)$.

(4) If $(\omega^{-1}(\epsilon))_{\epsilon}$ is a slow scale net then (2) and (3) are valid with $G^{\infty}(\Omega \times \Omega \setminus \Delta)$ and $G^{\infty}(\Omega \times \Omega)$ in place of $G(\Omega \times \Omega \setminus \Delta)$ and $G(\Omega \times \Omega)$ respectively.

(5) If $a \in \tilde{S}_{\text{rg}}^{-\infty}(\Omega \times \Omega \times \mathbb{R}^p)$ then $k_A \in G^{\infty}(\Omega \times \Omega)$.

**Theorem 1.4.3.** Let $a \in \tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$ and $(\omega^{-1}(\epsilon))_{\epsilon}$ a slow scale net. Let $A$ be the corresponding pseudodifferential operator. Then for all $u \in G_c(\Omega)$

$$\text{singsupp}_g(Au) \subseteq \text{singsupp}_g(u). \qquad (1.4.13)$$

**Theorem 1.4.4.** Let us consider a linear operator $A : G_c(\Omega) \to G(\Omega)$ of the form
$$A[u(x)] = \int_{\Omega} k_{A}(x,y)u(y)dy. \qquad (1.4.14)$$

where $k_A \in G^\infty(\Omega \times \Omega)$. A is an operator with regular generalized kernel if and only if it is a pseudodifferential operator with smoothing amplitude in $\tilde{S}_{\text{rg}}^{-\infty}(\Omega \times \Omega \times \mathbb{R}^p)$.

**Definition 1.4.8.** A pseudodifferential operator A is properly supported if and only if $\text{supp}_{x,y}(k_A)$ is a proper set. An amplitude $a \in \tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$ is called properly supported if and only if $\text{supp}_{x,y}(a)$ is a proper set.
Note that if A is properly supported then $^tA$ is properly supported.

**Theorem 1.4.4.** Let A be a pseudodifferential operator with amplitude $a \in \tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$. (1) If $(\omega(\varepsilon))_\varepsilon$ is bounded then A is properly supported if and only if it can be written with a properly supported amplitude in $\tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$. (2) If $(\omega^{-1}(\varepsilon))_\varepsilon$ is a slow scale net then A is properly supported if and only if it can be written with a properly supported amplitude in $\tilde{S}_{\rho, \text{rg}}^m(\Omega \times \Omega \times \mathbb{R}^n)$.

## I.6. Oscillatory integrals with generalized phase function on $\mathbb{R}^n$.

In this section we argue the most important properties of the integral, depending on a real parameter $\varepsilon$, of the type

$$\left( \int_{\mathbb{R}^n} a_{\varepsilon}(y) \exp[i\varphi_{\varepsilon}(y)] dy \right)_{\varepsilon}. \qquad (1.6.1)$$

We assume that $\varphi_\varepsilon \in C^\infty(\mathbb{R}^n \setminus \{0\})$, $\varepsilon \in (0,1]$ is a phase function of order $k > 0$, $\forall \varepsilon(\varepsilon \in (0,1])$ [$\varphi_\varepsilon \in \Phi^k(\mathbb{R}^n)$] for short, if it is real valued, positively homogeneous of order k, i.e. $\varphi_\varepsilon(tx) = t^k \varphi_\varepsilon(x)$ for $t > 0, \varepsilon \in (0,1]$ and $\nabla \varphi_\varepsilon \neq 0$ for $x \neq 0$.

**Definition 1.6.1.** A function $a \in C^\infty(\mathbb{R}^n)$ is an element of $A_l^m(\mathbb{R}^n)$, $l, m \in \mathbb{R}$, iff
$$A[u(x)] = \int_{\Omega} k_{A}(x,y)u(y)dy. \qquad (1.4.14)$$

where $k_A \in G^\infty(\Omega \times \Omega)$. A is an operator with regular generalized kernel if and only if it is a pseudodifferential operator with smoothing amplitude in $\tilde{S}_{\text{rg}}^{-\infty}(\Omega \times \Omega \times \mathbb{R}^p)$.

**Definition 1.4.8.** A pseudodifferential operator A is properly supported if and only if $\text{supp}_{x,y}(k_A)$ is a proper set. An amplitude $a \in \tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$ is called properly supported if and only if $\text{supp}_{x,y}(a)$ is a proper set.
Note that if A is properly supported then $^tA$ is properly supported.

**Theorem 1.4.4.** Let A be a pseudodifferential operator with amplitude $a \in \tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$. (1) If $(\omega(\varepsilon))_\varepsilon$ is bounded then A is properly supported if and only if it can be written with a properly supported amplitude in $\tilde{S}_{\rho, \delta, \omega}^{m, \mu}(\Omega \times \Omega \times \mathbb{R}^n)$. (2) If $(\omega^{-1}(\varepsilon))_\varepsilon$ is a slow scale net then A is properly supported if and only if it can be written with a properly supported amplitude in $\tilde{S}_{\rho, \text{rg}}^m(\Omega \times \Omega \times \mathbb{R}^n)$.

## I.6. Oscillatory integrals with generalized phase function on $\mathbb{R}^n$.

In this section we argue the most important properties of the integral, depending on a real parameter $\varepsilon$, of the type

$$\left( \int_{\mathbb{R}^n} a_{\varepsilon}(y) \exp[i\varphi_{\varepsilon}(y)] dy \right)_{\varepsilon}. \qquad (1.6.1)$$

We assume that $\varphi_\varepsilon \in C^\infty(\mathbb{R}^n \setminus \{0\})$, $\varepsilon \in (0,1]$ is a phase function of order $k > 0$, $\forall \varepsilon(\varepsilon \in (0,1])$ [$\varphi_\varepsilon \in \Phi^k(\mathbb{R}^n)$] for short, if it is real valued, positively homogeneous of order $k$, i.e. $\varphi_\varepsilon(tx) = t^k \varphi_\varepsilon(x)$ for $t > 0, \varepsilon \in (0,1]$ and $\nabla \varphi_\varepsilon \neq 0$ for $x \neq 0$.

**Definition 1.6.1.** A function $a \in C^\infty(\mathbb{R}^n)$ is an element of $A_l^m(\mathbb{R}^n)$, $l, m \in \mathbb{R}$, iff
$$ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha a(y)| < +\infty. \qquad (1.6.2) $$

We denote the set of nets $(a_\epsilon)_\epsilon \in \mathbf{A}_l^m(\mathbb{R}^n)^{(0,1]}$ by $\mathbf{A}_l^m[\mathbb{R}^n]$. Let $\mathbf{A}_{l,\mathbf{M}}^m(\mathbb{R}^n)$ be the set of all nets $(a_\epsilon)_\epsilon \in \mathbf{A}_l^m[\mathbb{R}^n]$ such that

$$ \forall \alpha(\alpha \in \mathbb{N}^n) \exists N(N \in \mathbb{N}) \left[ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha a_\epsilon(y)| = O(\epsilon^{-N}) \text{ as } \epsilon \to 0 \right] \qquad (1.6.3) $$

and $\mathbb{N}_l^m(\mathbb{R}^n)$ be the set of all nets $(a_\epsilon)_\epsilon \in \mathbf{A}_l^m[\mathbb{R}^n]$ such that

$$ \forall \alpha(\alpha \in \mathbb{N}^n) \forall q(q \in \mathbb{N}) \left[ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha a_\epsilon(y)| = O(\epsilon^q) \text{ as } \epsilon \to 0 \right]. \qquad (1.6.4) $$

The classes $a = \text{cl}[(a_\epsilon)_\epsilon]$ of the factor space $\tilde{\mathbf{A}}_l^m(\mathbb{R}^n) = \mathbf{A}_{l,\mathbf{M}}^m(\mathbb{R}^n)/\mathbb{N}_l^m(\mathbb{R}^n)$ are called generalized amplitudes on $\mathbb{R}^n$.

**Definition 1.6.2.** A function $\varphi \in C^\infty(\mathbb{R}^n\setminus\{0\})$ is an element of $\mathbf{A}_{l,\text{ph}}^m(\mathbb{R}^n\setminus\{0\})$, $l, m \in \mathbb{R}$, iff

$$ \nabla\varphi \neq 0 \text{ for } x \neq 0, \qquad (1.6.5) $$

$$ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha \varphi(y)| < +\infty. $$

We denote the set of nets $(\varphi_\epsilon)_\epsilon \in \mathbf{A}_{l,\text{ph}}^m(\mathbb{R}^n\setminus\{0\})^{(0,1]}$ by $\mathbf{A}_{l,\text{ph}}^m[\mathbb{R}^n]$. Let $\mathbf{A}_{l,\text{ph},\mathbf{M}}^m(\mathbb{R}^n)$ be the set of all nets $(\varphi_\epsilon)_\epsilon \in \mathbf{A}_{l,\text{ph}}^m[\mathbb{R}^n]$ such that
$$ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha a(y)| < +\infty. \qquad (1.6.2) $$

We denote the set of nets $(a_\epsilon)_\epsilon \in A_l^m(\mathbb{R}^n)^{(0,1]}$ by $A_l^m[\mathbb{R}^n]$. Let $A_{l,M}^m(\mathbb{R}^n)$ be the set of all nets $(a_\epsilon)_\epsilon \in A_l^m[\mathbb{R}^n]$ such that

$$ \forall \alpha(\alpha \in \mathbb{N}^n) \exists N(N \in \mathbb{N}) \left[ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha a_\epsilon(y)| = O(\epsilon^{-N}) \text{ as } \epsilon \to 0 \right] \quad (1.6.3) $$

and $N_l^m(\mathbb{R}^n)$ be the set of all nets $(a_\epsilon)_\epsilon \in A_l^m[\mathbb{R}^n]$ such that

$$ \forall \alpha(\alpha \in \mathbb{N}^n) \forall q(q \in \mathbb{N}) \left[ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha a_\epsilon(y)| = O(\epsilon^q) \text{ as } \epsilon \to 0 \right]. \quad (1.6.4) $$

The classes $a = \text{cl}[(a_\epsilon)_\epsilon]$ of the factor space $\tilde{A}_l^m(\mathbb{R}^n) = A_{l,M}^m(\mathbb{R}^n)/N_l^m(\mathbb{R}^n)$ are called generalized amplitudes on $\mathbb{R}^n$.

**Definition 1.6.2.** A function $\varphi \in C^\infty(\mathbb{R}^n\setminus\{0\})$ is an element of $A_{l,ph}^m(\mathbb{R}^n\setminus\{0\})$, $l, m \in \mathbb{R}$, iff

$$ \begin{gathered} \nabla\varphi \neq 0 \text{ for } x \neq 0, \\ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha \varphi(y)| < +\infty. \end{gathered} \qquad (1.6.5) $$

We denote the set of nets $(\varphi_\epsilon)_\epsilon \in A_{l,ph}^m(\mathbb{R}^n\setminus\{0\})^{(0,1]}$ by $A_{l,ph}^m[\mathbb{R}^n]$. Let $A_{l,ph,M}^m(\mathbb{R}^n)$ be the set of all nets $(\varphi_\epsilon)_\epsilon \in A_{l,ph}[\mathbb{R}^n]$ such that
$$ \forall \alpha(\alpha \in \mathbb{N}^n) \exists N(N \in \mathbb{N}) \left[ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha \varphi_\varepsilon(y)| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right]. \quad (1.6.6) $$

The classes $\varphi = \text{cl}[(\varphi_\varepsilon)_\varepsilon]$ of the factor space $\tilde{\mathcal{A}}_{l,ph}^m(\mathbb{R}^n\setminus\{0\}) = \mathcal{A}_{l,ph,M}^m(\mathbb{R}^n\setminus\{0\})/\mathcal{N}_l^m(\mathbb{R}^n\setminus\{0\})$ are called generalized phase function on $\mathbb{R}^n$.

**Definition 1.6.3.** An element $a \in \tilde{\mathcal{A}}_l^m(\mathbb{R}^n)$ is called regular if it has a representative $(a_\varepsilon)_\varepsilon$ fulfilling the following condition:

$$ \exists N(N \in \mathbb{N}) \forall \alpha(\alpha \in \mathbb{N}^n) \left[ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha a_\varepsilon(y)| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right]. \quad (1.6.7) $$

We denote the set of all regular generalized amplitudes on $\mathbb{R}^n$ by $\tilde{\mathcal{A}}_{l,rg}^m(\mathbb{R}^n)$

**Definition 1.6.4.** An element $\varphi \in \tilde{\mathcal{A}}_{l,ph}^m(\mathbb{R}^n\setminus\{0\})$ is called regular if it has a representative $(\varphi_\varepsilon)_\varepsilon$ fulfilling the following condition:

$$ \exists N(N \in \mathbb{N}) \forall \alpha(\alpha \in \mathbb{N}^n) \left[ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha \varphi_\varepsilon(y)| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right]. \quad (1.6.8) $$

We denote the set of all regular generalized phase functions on $\mathbb{R}^n\setminus\{0\}$ by $\tilde{\mathcal{A}}_{l,ph,rg}^m(\mathbb{R}^n\setminus\{0\})$.

**Remark. 1.6.1.** Let $A_{l,N}^m(\mathbb{R}^n)$ be the set of nets in $A_l^m[\mathbb{R}^n]$ defined by (1.6.7) by fixing $N$, and $A_{l,rg}^m(\mathbb{R}^n) = \cup_{N \in \mathbb{N}} A_{l,N}^m(\mathbb{R}^n)$. Since (1.6.7) holds for all representatives of $a$ once it holds for one, we can introduce $\tilde{\mathcal{A}}_{l,rg}^m(\mathbb{R}^n)$ as the quotient $\tilde{\mathcal{A}}_{l,rg}^m(\mathbb{R}^n) = A_{l,rg}^m(\mathbb{R}^n)/\mathcal{N}_l^m(\mathbb{R}^n)$.

**Theorem 1.6.1.** Let $(a_\varepsilon)_\varepsilon \in A_l^m[\mathbb{R}^n]$ and $\varphi_\varepsilon \in \Phi^k(\mathbb{R}^n)$ with $\varepsilon \in (0,1]$ $1-k<l \le 1$. Let $\psi$ be an arbitrary function of $\mathcal{L}(\mathbb{R}^n)$ with $\psi(0) = 1$ and $\phi$ any function in $G_c^\infty(\mathbb{R}^n)$

**Definition 1.6.5.** We define the generalized oscillatory integral of the generalized phase function $\varphi = \text{cl}[(\varphi_\varepsilon)_\varepsilon]$ and amplitude $a = \text{cl}[(a_\varepsilon)_\varepsilon]$ as the class in $\tilde{\mathbb{C}}$:
$$ \forall \alpha(\alpha \in \mathbb{N}^n) \exists N(N \in \mathbb{N}) \left[ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha \varphi_\varepsilon(y)| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right]. \quad (1.6.6) $$

The classes $\varphi = \text{cl}[(\varphi_\varepsilon)_\varepsilon]$ of the factor space $\tilde{\mathcal{A}}_{l,ph}^m(\mathbb{R}^n\setminus\{0\}) = \mathcal{A}_{l,ph,M}^m(\mathbb{R}^n\setminus\{0\})/\mathcal{N}_l^m(\mathbb{R}^n\setminus\{0\})$ are called generalized phase function on $\mathbb{R}^n$.

**Definition 1.6.3.** An element $a \in \tilde{\mathcal{A}}_l^m(\mathbb{R}^n)$ is called regular if it has a representative $(a_\varepsilon)_\varepsilon$ fulfilling the following condition:

$$ \exists N(N \in \mathbb{N}) \forall \alpha(\alpha \in \mathbb{N}^n) \left[ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha a_\varepsilon(y)| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right]. \quad (1.6.7) $$

We denote the set of all regular generalized amplitudes on $\mathbb{R}^n$ by $\tilde{\mathcal{A}}_{l,rg}^m(\mathbb{R}^n)$

**Definition 1.6.4.** An element $\varphi \in \tilde{\mathcal{A}}_{l,ph}^m(\mathbb{R}^n\setminus\{0\})$ is called regular if it has a representative $(\varphi_\varepsilon)_\varepsilon$ fulfilling the following condition:

$$ \exists N(N \in \mathbb{N}) \forall \alpha(\alpha \in \mathbb{N}^n) \left[ \sup_{y \in \mathbb{R}^n} \langle y \rangle^{|\alpha|-m} |\partial^\alpha \varphi_\varepsilon(y)| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right]. \quad (1.6.8) $$

We denote the set of all regular generalized phase functions on $\mathbb{R}^n\setminus\{0\}$ by $\tilde{\mathcal{A}}_{l,ph,rg}^m(\mathbb{R}^n\setminus\{0\})$.

**Remark. 1.6.1.** Let $A_{l,N}^m(\mathbb{R}^n)$ be the set of nets in $A_l^m[\mathbb{R}^n]$ defined by (1.6.7) by fixing $N$, and $A_{l,rg}^m(\mathbb{R}^n) = \cup_{N \in \mathbb{N}} A_{l,N}^m(\mathbb{R}^n)$. Since (1.6.7) holds for all representatives of $a$ once it holds for one, we can introduce $\tilde{\mathcal{A}}_{l,rg}^m(\mathbb{R}^n)$ as the quotient $\tilde{\mathcal{A}}_{l,rg}^m(\mathbb{R}^n) = A_{l,rg}^m(\mathbb{R}^n)/\mathcal{N}_l^m(\mathbb{R}^n)$.

**Theorem 1.6.1.** Let $(a_\varepsilon)_\varepsilon \in A_l^m[\mathbb{R}^n]$ and $\varphi_\varepsilon \in \Phi^k(\mathbb{R}^n)$ with $\varepsilon \in (0,1]$ $1-k<l \le 1$. Let $\psi$ be an arbitrary function of $\mathcal{L}(\mathbb{R}^n)$ with $\psi(0) = 1$ and $\phi$ any function in $G_c^\infty(\mathbb{R}^n)$

**Definition 1.6.5.** We define the generalized oscillatory integral of the generalized phase function $\varphi = \text{cl}[(\varphi_\varepsilon)_\varepsilon]$ and amplitude $a = \text{cl}[(a_\varepsilon)_\varepsilon]$ as the class in $\tilde{\mathbb{C}}$:
$$I_{\phi}[a] = \int_{\mathbb{R}^n} a(y) \exp[i\phi(y)] dy = \mathbf{cl} \left[ \left( \int_{\mathbb{R}^n} a_{\varepsilon}(y) \exp[i\phi_{\varepsilon}(y)] dy \right)_{\varepsilon} \right]. \quad (1.6.9)$$

## I.7. Generalized symbols and amplitudes of weighted type.

**Definition 1.7.1.** In the following for real functions $f$ and $g$ on $\mathbb{R}^n$, we write $f(z) \leq g(z)$ on $A \subseteq \mathbb{R}^n$, if there exists a positive constant $c > 0$ such that $f(z) \leq cg(z)$, for all $z \in A$.

If $f(z) \leq g(z)$ and $g(z) \leq f(z)$, we write $f(z) \sim g(z)$.

**Definition 1.7.2.** A continuous real function $\Lambda(z)$ on $\mathbb{R}^{2n}$ is a weight function if and only if:

(1) there exists $\lambda > 0$ such that $\langle z \rangle^{\lambda} \leq \Lambda(z) \leq \langle z \rangle$ on $\mathbb{R}^{2n}$.

(2) $\Lambda(z) \sim \Lambda(\zeta)$ on $A_{z,\zeta} = \{(z,\zeta) \mid \|z - \zeta\| \leq \lambda\Lambda(z)\}$.

(3) $\forall t(t \to \mathbb{R}^{2n}) : \Lambda(t \cdot z) \leq \Lambda(z)$ on $\mathbb{R}^{2n}$, where $t \cdot z = (t_1z_1, \dots, t_{2n}z_{2n})$.

**Definition 1.7.3.** Let $m \in \mathbb{R}$, $\rho \in (0,1)$, $\Lambda(z)$ be a weight function and $z = (x, \xi) \in \mathbb{R}^{2n}$. We

denote by $S_{\Lambda, \rho, M}^m(\mathbb{R}^{2n})$ or $S_{\Lambda, \rho, M}^m$, the set of all nets $(a_{\varepsilon})_{\varepsilon} \in E[\mathbb{R}^{2n}]$ with the property that

$$\forall \alpha \in \mathbb{N}^{2n} \exists N \in \mathbb{N} \exists c (c > 0) \forall \varepsilon (\varepsilon \in (0,1]) \forall z \in \mathbb{R}^{2n} \left[ |\partial^\alpha a_\varepsilon(z)| \leq c (\Lambda(z))^{m-\rho|\alpha|} \varepsilon^{-N} \right] \quad (1.7.1)$$

We denote by $N_{\Lambda, \rho}^m(\mathbb{R}^{2n})$ or $N_{\Lambda, \rho}^m$, the set of all nets $(a_{\varepsilon})_{\varepsilon} \in E[\mathbb{R}^{2n}]$ with the property

$$\forall \alpha \in \mathbb{N}^{2n} \forall q \in \mathbb{N} \exists c (c > 0) \forall \varepsilon (\varepsilon \in (0,1]) \forall z \in \mathbb{R}^{2n} \left[ |\partial^\alpha a_\varepsilon(z)| \leq c (\Lambda(z))^{m-\rho|\alpha|} \varepsilon^q \right] \quad (1.7.2)$$
$$I_{\phi}[a] = \int_{\mathbb{R}^n} a(y) \exp[i\phi(y)] dy = \mathbf{cl} \left[ \left( \int_{\mathbb{R}^n} a_{\varepsilon}(y) \exp[i\phi_{\varepsilon}(y)] dy \right)_{\varepsilon} \right]. \quad (1.6.9)$$

## I.7. Generalized symbols and amplitudes of weighted type.

**Definition 1.7.1.** In the following for real functions $f$ and $g$ on $\mathbb{R}^n$, we write $f(z) \leq g(z)$ on $A \subseteq \mathbb{R}^n$, if there exists a positive constant $c > 0$ such that $f(z) \leq cg(z)$, for all $z \in A$.

If $f(z) \leq g(z)$ and $g(z) \leq f(z)$, we write $f(z) \sim g(z)$.

**Definition 1.7.2.** A continuous real function $\Lambda(z)$ on $\mathbb{R}^{2n}$ is a weight function if and only if:

(1) there exists $\lambda > 0$ such that $\langle z \rangle^{\lambda} \leq \Lambda(z) \leq \langle z \rangle$ on $\mathbb{R}^{2n}$.

(2) $\Lambda(z) \sim \Lambda(\zeta)$ on $A_{z, \zeta} = \{(z, \zeta) \mid \|z - \zeta\| \leq \lambda \Lambda(z)\}$.

(3) $\forall t(t \to \mathbb{R}^{2n}) : \Lambda(t \cdot z) \leq \Lambda(z)$ on $\mathbb{R}^{2n}$, where $t \cdot z = (t_1 z_1, \dots, t_{2n} z_{2n})$.

**Definition 1.7.3.** Let $m \in \mathbb{R}$, $\rho \in (0, 1)$, $\Lambda(z)$ be a weight function and $z = (x, \xi) \in \mathbb{R}^{2n}$. We

denote by $S_{\Lambda, \rho, M}^m(\mathbb{R}^{2n})$ or $S_{\Lambda, \rho, M}^m$, the set of all nets $(a_{\varepsilon})_{\varepsilon} \in E[\mathbb{R}^{2n}]$ with the property that

$$\forall \alpha \in \mathbb{N}^{2n} \exists N \in \mathbb{N} \exists c (c > 0) \forall \varepsilon (\varepsilon \in (0,1]) \forall z \in \mathbb{R}^{2n} [ |\partial^\alpha a_\varepsilon(z)| \leq c (\Lambda(z))^{m-\rho} |\alpha|^{\varepsilon} N ] \quad (1.7.1)$$

We denote by $N_{\Lambda, \rho}^m(\mathbb{R}^{2n})$ or $N_{\Lambda, \rho}^m$, the set of all nets $(a_{\varepsilon})_{\varepsilon} \in E[\mathbb{R}^{2n}]$ with the property

$$\forall \alpha \in \mathbb{N}^{2n} \forall q \in \mathbb{N} \exists c (c > 0) \forall \varepsilon (\varepsilon \in (0,1]) \forall z \in \mathbb{R}^{2n} [ |\partial^\alpha a_\varepsilon(z)| \leq c (\Lambda(z))^{m-\rho} |\alpha|^{q} ] \quad (1.7.2)$$
**Definition 1.7.4.** The elements of the factor space

$$ \tilde{S}_{\Lambda, \rho}^{m} = S_{\Lambda, \rho, \mathbf{M}}^{m} / N_{\Lambda, \rho}^{m} \qquad (1.7.3) $$

are called generalized symbols of order $m$ and type $(\Lambda, \rho)$.

We say that a generalized symbol $a \in \tilde{S}_{\Lambda, \rho}^{m}$ is regular if it has a representative $(a_{\varepsilon})_{\varepsilon}$ satisfying the following condition

$$ \exists N \in \mathbb{N} \forall \alpha \in \mathbb{N}^{2n} \exists c(c > 0) \forall \varepsilon (\varepsilon \in (0,1]) \forall z \in \mathbb{R}^{2n} [ |\partial^{\alpha} a_{\varepsilon}(z)| \le c(\Lambda(z))^{m-\rho}|\alpha|^{\varepsilon-N} ]. \quad (1.7.4) $$

The set of all regular symbols in $\tilde{S}_{\Lambda, \rho}^{m}$ will be denoted by $\tilde{S}_{\Lambda, \rho, \text{reg}}^{m}$.

**1.8. Pseudodifferential operators acting on $G_{\tau, \mathcal{L}}(\mathbb{R}^n)$**

In the classical theory an integral

$$ A[u(x)] = \int_{\mathbb{R}^{2n}} (\exp[i(x-y)])a(x,y,\xi)u(y)d\xi, \qquad (1.8.1) $$

where $a \in \tilde{S}_{\Lambda, \rho}^{m}$ and $u \in \mathcal{L}(\mathbb{R}^n)$, defines a continuous map $\mathcal{L}(\mathbb{R}^n) \to \mathcal{L}(\mathbb{R}^n)$, which can be extended as a continuous map $\mathcal{L}'(\mathbb{R}^n) \to \mathcal{L}'(\mathbb{R}^n)$.

**Theorem 1.8.1.** Let $a \in \tilde{S}_{\Lambda, \rho}^{m}$, $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and $\varphi$ be a mollifier in $\mathcal{L}(\mathbb{R}^n)$. Then for every $x \in \mathbb{R}^n$

$$ (\exp(ix\xi))a(x,y,\xi)u_{\hat{\phi}}(y) \triangleq \mathbf{cl} \left[ \left( (\exp(ix\xi))a_{\varepsilon}(x,y,\xi)u_{\varepsilon}(y)\hat{\phi}_{\varepsilon} \right)_{\varepsilon} \right] \quad (1.8.2) $$

is a well-defined element of $\tilde{\mathcal{A}}_{0}^{\nu}(\mathbb{R}_{y}^{n} \times \mathbb{R}_{\xi}^{n})$ with $\nu = m_{+} + m_{+}'$.

Therefore one can define the oscillatory integral
**Definition 1.7.4.** The elements of the factor space

$$ \tilde{S}_{\Lambda, \rho}^{m} = S_{\Lambda, \rho, \mathbf{M}}^{m} / N_{\Lambda, \rho}^{m} \qquad (1.7.3) $$

are called generalized symbols of order $m$ and type $(\Lambda, \rho)$.

We say that a generalized symbol $a \in \tilde{S}_{\Lambda, \rho}^{m}$ is regular if it has a representative $(a_{\varepsilon})_{\varepsilon}$ satisfying the following condition

$$ \exists N \in \mathbb{N} \forall \alpha \in \mathbb{N}^{2n} \exists c(c > 0) \forall \varepsilon (\varepsilon \in (0,1]) \forall z \in \mathbb{R}^{2n} [ |\partial^{\alpha} a_{\varepsilon}(z)| \le c(\Lambda(z))^{m-\rho}|\alpha|^{\varepsilon-N} ]. \quad (1.7.4) $$

The set of all regular symbols in $\tilde{S}_{\Lambda, \rho}^{m}$ will be denoted by $\tilde{S}_{\Lambda, \rho, \text{reg}}^{m}$.

**1.8. Pseudodifferential operators acting on $G_{\tau, \mathcal{L}}(\mathbb{R}^n)$**

In the classical theory an integral

$$ A[u(x)] = \int_{\mathbb{R}^{2n}} (\exp[i(x-y)])a(x,y,\xi)u(y)d\xi, \qquad (1.8.1) $$

where $a \in \tilde{S}_{\Lambda, \rho}^{m}$ and $u \in \mathcal{L}(\mathbb{R}^n)$, defines a continuous map $\mathcal{L}(\mathbb{R}^n) \to \mathcal{L}(\mathbb{R}^n)$, which can be extended as a continuous map $\mathcal{L}'(\mathbb{R}^n) \to \mathcal{L}'(\mathbb{R}^n)$.

**Theorem 1.8.1.** Let $a \in \tilde{S}_{\Lambda, \rho}^{m}$, $u \in G_{\tau, \mathcal{L}}(\mathbb{R}^n)$ and $\varphi$ be a mollifier in $\mathcal{L}(\mathbb{R}^n)$. Then for every $x \in \mathbb{R}^n$

$$ (\exp(ix\xi))a(x,y,\xi)u_{\hat{\phi}}(y) \triangleq \text{cl} \left[ \left( (\exp(ix\xi))a_{\varepsilon}(x,y,\xi)u_{\varepsilon}(y)\hat{\phi}_{\varepsilon} \right)_{\varepsilon} \right] \quad (1.8.2) $$

is a well-defined element of $\tilde{\mathcal{A}}_0^v(\mathbb{R}_y^n \times \mathbb{R}_\xi^n)$ with $v = m_+ + m'_+$.

Therefore one can define the oscillatory integral
$$ \int_{\mathbb{R}^{2n}} (\exp[i(x-y)])a(x,y,\xi)u_{\hat{\phi}}(y)dyd\xi = \\ \int_{\mathbb{R}^{2n}} \exp(-iy\xi)(a(x,y,\xi)u_{\hat{\phi}}(y)\exp(ix\xi))dyd\xi. \qquad (1.8.3) $$

where $-y\xi$ is the phase function of order 2 and $a(x,y,\xi)u_{\hat{\phi}}(y)\exp(ix\xi)$ the generalized amplitude in $\tilde{A}_0^{y}(\mathbb{R}_y^n \times \mathbb{R}_\xi^n)$.

## II.1.1. Feynman-Colombeau path integral.

Let us consider $L_2$ transition probability amplitude $K(\vec{x}, t|\vec{x}_0, 0)$

$$ K(\vec{x}, t|\vec{x}_0, 0) = \left\langle \psi_1^*, \exp\left(-\frac{it\hat{H}}{\epsilon}\right) \psi_2 \right\rangle = \\ \int_{\mathbb{R}^n} \left( \psi_1^*(\vec{x}) \left[ \exp\left(-\frac{it\hat{H}}{\epsilon}\right) \psi_2(\vec{x}) \right] \right) d\vec{x}, \qquad (2.1.1.1) $$

where $\psi_1, \psi_2 \in L_2$,

$$ H = -\frac{\epsilon^2}{2m}\Delta + V(\vec{x}, t) \qquad (2.1.1.2) $$

and $H$ is essentially self-adjoint, $\hat{H}$ is the closure of $H$. Propagator (2.1.1.1) is formally given via formula:
$$ \int_{\mathbb{R}^{2n}} (\exp[i(x-y)])a(x,y,\xi)u_{\hat{\phi}}(y)dyd\xi = \\ \int_{\mathbb{R}^{2n}} \exp(-iy\xi)(a(x,y,\xi)u_{\hat{\phi}}(y)\exp(ix\xi))dyd\xi. \qquad (1.8.3) $$

where $-y\xi$ is the phase function of order 2 and $a(x,y,\xi)u_{\hat{\phi}}(y)\exp(ix\xi)$ the generalized amplitude in $\tilde{A}_0^{y}(\mathbb{R}_y^n \times \mathbb{R}_\xi^n)$.

## II.1.1. Feynman-Colombeau path integral.

Let us consider $L_2$ transition probability amplitude $K(\vec{x}, t|\vec{x}_0, 0)$

$$ K(\vec{x}, t|\vec{x}_0, 0) = \left\langle \psi_1^*, \exp\left(-\frac{it\hat{H}}{\epsilon}\right) \psi_2 \right\rangle = \\ \int_{\mathbb{R}^n} \left( \psi_1^*(\vec{x}) \left[ \exp\left(-\frac{it\hat{H}}{\epsilon}\right) \psi_2(\vec{x}) \right] \right) d\vec{x}, \qquad (2.1.1.1) $$

where $\psi_1, \psi_2 \in L_2$,

$$ H = -\frac{\epsilon^2}{2m}\Delta + V(\vec{x}, t) \qquad (2.1.1.2) $$

and $H$ is essentially self-adjoint, $\hat{H}$ is the closure of $H$. Propagator (2.1.1.1) is formally given via formula:
$$ \mathbf{K}(\vec{x}, t|\vec{x}_0, 0) = \lim_{k \to \infty} \Omega_k(n) \int_{\mathbb{R}^{(k-1)n}} \exp\left[\frac{i\Delta\tau}{\epsilon} S_k(\vec{x}, \dots, \vec{x}_0)\right] d\vec{x}_1 \dots d\vec{x}_{k-1}, \quad (2.1.1.3) $$

where

$$ \Omega_{n,k} = \left( \frac{m}{2i\pi\epsilon\Delta\tau} \right)^{\frac{n_k}{2}}, \Delta\tau = \frac{t}{k}, $$

(2.1.1.4)

$$ S_k(\vec{x}, \dots, \vec{x}_0) = \sum_{j=1}^{k} \left[ \frac{m}{2} \left( \frac{\vec{x}_j - \vec{x}_{j-1}}{\Delta\tau} \right) - V(\vec{x}_j) \right], \vec{x} = \vec{x}_k $$

Let $\mathbf{H}$ be a complex separable infinite dimensional Hilbert space, with inner product $\langle \cdot, \cdot \rangle$ and norm $|\cdot|$. Let us consider Schrödinger equation:

$$ -i \frac{\partial \Psi(t)}{\partial t} + \hat{H}(t) \Psi(t) = 0, \quad (2.1.1.5) $$

$$ \Psi(0) = \Psi_0(\vec{x}), $$

where operator $\hat{H}(t)$ given via formula (2.1.2), $\Psi(t) : \mathbb{R} \to \mathbf{H}$ and $\hat{H}(t) : \mathbb{R} \times \mathbf{H} \to \mathbf{H}$.

**Theorem 2.1.1.** [20]. We assume that:

(1) $\Psi_0(\vec{x}) \in L_2(\mathbb{R}^n)$,

(2) $V(\vec{x})$ is continuous and $\sup_{\vec{x} \in \mathbb{R}^n} (|V(\vec{x})|) < +\infty$.

Then corresponding solution of the Schrödinger equation (2.1.5) exist and can be represented via formulae (2.1.1.3)-(2.1.1.4).

Let $\mathbf{H}$ be a complex separable infinite dimensional Hilbert space, with inner product
$$ \mathbf{K}(\vec{x}, t|\vec{x}_0, 0) = \lim_{k \to \infty} \Omega_k(n) \int_{\mathbb{R}^{(k-1)n}} \exp\left[\frac{i\Delta\tau}{\epsilon} S_k(\vec{x}, \dots, \vec{x}_0)\right] d\vec{x}_1 \dots d\vec{x}_{k-1}, \quad (2.1.1.3) $$

where

$$ \Omega_{n,k} = \left( \frac{m}{2i\pi\epsilon\Delta\tau} \right)^{\frac{nk}{2}}, \Delta\tau = \frac{t}{k}, $$

(2.1.1.4)

$$ S_k(\vec{x}, \dots, \vec{x}_0) = \sum_{j=1}^{k} \left[ \frac{m}{2} \left( \frac{\vec{x}_j - \vec{x}_{j-1}}{\Delta\tau} \right) - V(\vec{x}_j) \right], \vec{x} = \vec{x}_k $$

Let $\mathbf{H}$ be a complex separable infinite dimensional Hilbert space, with inner product $\langle \cdot, \cdot \rangle$ and norm $|\cdot|$. Let us consider Schrödinger equation:

$$ -i \frac{\partial \Psi(t)}{\partial t} + \hat{H}(t) \Psi(t) = 0, \quad (2.1.1.5) $$

$$ \Psi(0) = \Psi_0(\vec{x}), $$

where operator $\hat{H}(t)$ given via formula (2.1.2), $\Psi(t) : \mathbb{R} \to \mathbf{H}$ and $\hat{H}(t) : \mathbb{R} \times \mathbf{H} \to \mathbf{H}$.

**Theorem 2.1.1.** [20]. We assume that:

(1) $\Psi_0(\vec{x}) \in L_2(\mathbb{R}^n)$,

(2) $V(\vec{x})$ is continuous and $\sup_{\vec{x} \in \mathbb{R}^n} (|V(\vec{x})|) < +\infty$.

Then corresponding solution of the Schrödinger equation (2.1.5) exist and can be represented via formulae (2.1.1.3)-(2.1.1.4).

Let $\mathbf{H}$ be a complex separable infinite dimensional Hilbert space, with inner product
⟨,⟩ and norm |·|. Let us consider Colombeau-Schrödinger type equation

$$ -i \left( \frac{\partial \Psi_{\varepsilon}(t)}{\partial t} \right)_{\varepsilon \in (0,1]} + \left( \hat{H}_{\varepsilon}(t) \Psi_{\varepsilon}(t) \right)_{\varepsilon \in (0,1]} = 0, \quad (2.1.1.6) $$

where: (1) $\forall \varepsilon(\varepsilon \neq 0)[\Psi_{\varepsilon}(t) : \mathbb{R} \to \mathbf{H}]$, (2) $\hat{H}_0(t) = \hat{H}(t)$ and operator $\hat{H}(t)$ is given via Eq.(2.1.1.2) and $\sup_{\vec{x} \in \mathbb{R}^n} (|V(\vec{x})|) = \infty$,

$$ (3) \quad \forall \varepsilon(\varepsilon \neq 0) \left[ \hat{H}_{\varepsilon}(t) : \mathbb{R} \times \mathbf{H} \to \mathbf{H} \right], (4) \quad \mathbf{cl}\left(\left(\hat{H}_{\varepsilon}(t)\right)_{\varepsilon}\right) : \tilde{\mathbb{R}} \times G_{\mathbf{H}} \to G_{\mathbf{H}}. $$

From Eq.(2.1.1.6) one obtain formal "infinitesimal" solution

$$ (\Psi_{\varepsilon}(t + dt))_{\varepsilon \in (0,1]} = \left( \exp\left\{-idt \left[\hat{H}_{\varepsilon}(t)\right]\right\} \Psi_{\varepsilon}(t) \right)_{\varepsilon \in (0,1]} \quad (2.1.1.7) $$

Using simple formal calculation one obtain formal solution of the Eq.(2.1.1.6) via formula :

$$ (\Psi_{\varepsilon}(t))_{\varepsilon \in (0,1]} = \left( \lim_{\delta_N \to 0} \left[ \exp\{-i\Delta t_{N-1} \hat{H}_{\varepsilon}(t_{N-1})\} \dots \exp\{-i\Delta t_0 \hat{H}_{\varepsilon}(t_0)\} \Psi_{\varepsilon}(t_0) \right] \right)_{\varepsilon \in (0,1]}, \quad (2.1.1.8) $$

where $t_0 < t_1 < \dots < t_N = t$, $\Delta t_j = t_{j+1} - t_j$, $\delta_N = \max_{0 \le j \le N-1} \Delta t_j$

If $\forall \varepsilon[\varepsilon \in (0,1]]$ limit $\Psi_{\varepsilon}(t) = \lim_{\delta_N \to 0}[\circ]$ in the formula (2.1.1.8) exists we obtain Colombeau generalized function $(\Psi_{\varepsilon}(t))_{\varepsilon \in (0,1]}$ We denote RH of Eq.(2.1.1.8) as

$$ (\Psi_{\varepsilon}(t))_{\varepsilon \in (0,1]} = \left( \left[ \prod_{\tau=t_0}^{t} \exp\{-id\tau \hat{H}_{\varepsilon}(\tau)\} \right] \Psi_{\varepsilon}(t_0) \right)_{\varepsilon \in (0,1]} \quad (2.1.1.9) $$

and call it Maslov-Colombeau chronological exponential or Maslov-Colombeau T-exponent.

Let $\mathbf{B}$ be a complex separable infinite dimensional Banach space with the norm
⟨,⟩ and norm |·|. Let us consider Colombeau-Schrödinger type equation

$$ -i \left( \frac{\partial \Psi_{\varepsilon}(t)}{\partial t} \right)_{\varepsilon \in (0,1]} + \left( \hat{H}_{\varepsilon}(t) \Psi_{\varepsilon}(t) \right)_{\varepsilon \in (0,1]} = 0, \qquad (2.1.1.6) $$

where: (1) $\forall \varepsilon(\varepsilon \neq 0)[\Psi_{\varepsilon}(t) : \mathbb{R} \to \mathbf{H}]$, (2) $\hat{H}_0(t) = \hat{H}(t)$ and operator $\hat{H}(t)$ is given via Eq.(2.1.1.2) and $\sup_{\vec{x} \in \mathbb{R}^n} (|V(\vec{x})|) = \infty$,

$$ (3) \quad \forall \varepsilon(\varepsilon \neq 0) \left[ \hat{H}_{\varepsilon}(t) : \mathbb{R} \times \mathbf{H} \to \mathbf{H} \right], (4) \quad \mathbf{cl}\left(\left(\hat{H}_{\varepsilon}(t)\right)_{\varepsilon}\right) : \tilde{\mathbb{R}} \times G_{\mathbf{H}} \to G_{\mathbf{H}}. $$

From Eq.(2.1.1.6) one obtain formal "infinitesimal" solution

$$ (\Psi_{\varepsilon}(t+dt))_{\varepsilon \in (0,1]} = \left( \exp\left\{-idt \left[\hat{H}_{\varepsilon}(t)\right]\right\} \Psi_{\varepsilon}(t) \right)_{\varepsilon \in (0,1]} \qquad (2.1.1.7) $$

Using simple formal calculation one obtain formal solution of the Eq.(2.1.1.6) via formula :

$$ (\Psi_{\varepsilon}(t))_{\varepsilon \in (0,1]} = \left( \lim_{\delta_N \to 0} \left[ \exp\{-i\Delta t_{N-1} \hat{H}_{\varepsilon}(t_{N-1})\} \dots \exp\{-i\Delta t_0 \hat{H}_{\varepsilon}(t_0)\} \Psi_{\varepsilon}(t_0) \right] \right)_{\varepsilon \in (0,1]}, \qquad (2.1.1.8) $$

where $t_0 < t_1 < \dots < t_N = t$, $\Delta t_j = t_{j+1} - t_j$, $\delta_N = \max_{0 \le j \le N-1} \Delta t_j$

If $\forall \varepsilon[\varepsilon \in (0,1]]$ limit $\Psi_{\varepsilon}(t) = \lim_{\delta_N \to 0}[\circ]$ in the formula (2.1.1.8) exists we obtain Colombeau generalized function $(\Psi_{\varepsilon}(t))_{\varepsilon \in (0,1]}$ We denote RH of Eq.(2.1.1.8) as

$$ (\Psi_{\varepsilon}(t))_{\varepsilon \in (0,1]} = \left( \left[ \prod_{\tau=t_0}^{t} \exp\{-id\tau \hat{H}_{\varepsilon}(\tau)\} \right] \Psi_{\varepsilon}(t_0) \right)_{\varepsilon \in (0,1]} \qquad (2.1.1.9) $$

and call it Maslov-Colombeau chronological exponential or Maslov-Colombeau T-exponent.

Let $\mathbf{B}$ be a complex separable infinite dimensional Banach space with the norm
||◦||. The Banach space C'_B = C([0, t], B) consist of continuous functions
ψ(t), ψ : [0, t] → B, with the norm

$$
\|\psi\|_{C_B^t} = \sup_{0 \le \tau \le t} \|\psi(\tau)\|_B. \qquad (2.1.1.10)
$$

If $t' < t$ we denote $\psi'(t) \triangleq \psi(t) \upharpoonright [0, t']$, thus $\psi'(t) \in C([0, t'], \mathbf{B})$.
Suppose that:

$$
(1) \varepsilon_0 > 0, T_0 > 0,
$$

(2) $\forall t, \tilde{\varepsilon}$ such that $0 \le t \le T, (-t) \le \tilde{\varepsilon} \le \varepsilon_0$ there exists a function

$$
S_{t, \tilde{\varepsilon}}[\circ] : \mathbf{C}([0, t'], \mathbf{B}) \times [-t, \varepsilon_0] \to \mathbf{B} \quad (2.1.1.11)
$$

such that $\forall v \in C([0, t'], \mathbf{B})$, $\forall t \in [0, T_0]$ function $q(t, \tilde{\varepsilon}) = \|S_{t, \tilde{\varepsilon}}[v]\|_{\mathbf{B}}$ is continuous on variable $\tilde{\varepsilon}$,

(3) if $\tilde{\varepsilon} \le 0$, then $S_{t,\tilde{\varepsilon}}[v] = S_t[v] \triangleq v(t)$.

Let's consider now finite splitting Δ<sub>N</sub> of the closed interval [0, T<sub>0</sub>] :

$$
\Delta_N = \{0 = t_0 < t_1 < \dots < t_N = T_0\}. \qquad (2.1.1.12)
$$

Note that for every $v_0 \in \mathbf{B}$ there exists an sequence $v_j(t)$, $j = 0, 1, \dots, N-1$ such that $v_{j+1}(t) = S_{t_j, t-t_j}[v_j]$, $0 \le t \le t_{j+1}$.

**Definition 2.1.1.1.** [20].

$$
v_N(t) = S_{t_N, \Delta t_{N-1}} \circ \dots \circ S_{t_j, \Delta t_j} \circ \dots \circ S_{t_0, \Delta t_0}[v_0], \qquad (2.1.1.13)
$$

$$
\Delta t_j = t_{j+1} - t_j, j = 0, 1, \dots, N-1.
$$

**Definition 2.1.1.2.** [20] Suppose that:

(1) $v_0 \in D \subsetneq B$ and

(2) s-lim$_{\delta_{N} \to 0} v_N(t)$ there exists.