Setting $g_k = (f_1)_k + (f_2)_k + (f_3)_k$, we know that $\Gamma g_k$ and $\Gamma((\partial_t + v \cdot \nabla_x)g_k) = \partial_t \Gamma g_k$ belong to $L_{t,x,v}^2$. We can thus write for all $s < t$,

$$ \begin{aligned} \|g_k(t)\|_{L_{x,v}^2}^2 - \|g_k(s)\|_{L_{x,v}^2}^2 &= \iint_{\mathbb{R}^{2d}} |(\Gamma g_k)(t, x, v)|^2 dx \, dv - \iint_{\mathbb{R}^{2d}} |(\Gamma g_k)(s, x, v)|^2 dx \, dv \\ &= 2 \operatorname{Re} \iint_{\mathbb{R}^{2d}} \int_s^t \partial_t (\Gamma g_k)(\tau, x, v) \overline{(\Gamma g_k)(\tau, x, v)} d\tau \, dx \, dv \\ &= 2 \operatorname{Re} \int_s^t \iint_{\mathbb{R}^{2d}} \partial_t (\Gamma g_k)(\tau, x, v) \overline{(\Gamma g_k)(\tau, x, v)} dx \, dv \, d\tau \\ &= 2 \operatorname{Re} \int_s^t \iint_{\mathbb{R}^{2d}} ((\partial_t + v \cdot \nabla_x)g_k)(\tau, x, v) \overline{g_k(\tau, x, v)} dx \, dv \, d\tau \end{aligned} $$

where we used Fubini's theorem, and change of variables $\Gamma(t), \Gamma(s)$ and $\Gamma(\tau)$ in the $(x, v)$-integrals. The left hand-side converges to $\|f(t)\|_{L_{x,v}^2}^2 - \|f(s)\|_{L_{x,v}^2}^2$. For the right-hand side, we split the integral using the decomposition of $(\partial_t + v \cdot \nabla_x)g_k$ into three terms and argue differently for each. Realising the $v$-integral as the duality bracket between $\dot{H}_v^{-\beta}$ and $\dot{H}_v^{\beta}$, we have, using the dominated convergence theorem for the $(\tau, x)$-integral,

$$ \begin{aligned} & \int_s^t \iint_{\mathbb{R}^{2d}} ((S_1 + (-\Delta_v)^\beta f)_k - (-\Delta_v)^\beta g_k) \overline{g_k} \, dx \, dv \, d\tau \\ &= \int_s^t \int_{\mathbb{R}^d} \langle (S_1 + (-\Delta_v)^\beta f)_k - (-\Delta_v)^\beta g_k, g_k \rangle \, dx \, d\tau \rightarrow \int_s^t \int_{\mathbb{R}^d} \langle S_1, f \rangle \, dx \, d\tau. \end{aligned} $$

Similarly, realizing the $x$-integral as the duality bracket between $\dot{H}_x^{-\frac{\beta}{2\beta+1}}$ and $\dot{H}_x^{\frac{\beta}{2\beta+1}}$ and using dominated convergence in the $(t, v)$-integral

$$ \int_s^t \iint_{\mathbb{R}^{2d}} (S_2)_k \overline{g_k} \, dx \, dv \, d\tau = \int_s^t \int_{\mathbb{R}^d} \langle (S_2)_k, g_k \rangle \, dv \, d\tau \rightarrow \int_s^t \int_{\mathbb{R}^d} \langle S_2, f \rangle \, dv \, d\tau. $$

Eventually, using merely dominated convergence for the $(t, x, v)$-integral

$$ \int_s^t \iint_{\mathbb{R}^{2d}} (S_3)_k \overline{g_k} \, dx \, dv \, d\tau \rightarrow \int_s^t \iint_{\mathbb{R}^d} S_3 \overline{f} \, dx \, dv \, d\tau. \quad \square $$

The following result is a generalisation of Theorem 1.16. The $d/2$ constraint is likely inessential with an appropriate definition of the spaces $\dot{\mathcal{L}}_\beta^\gamma$, which we have chosen not to develop here.

**Theorem 3.9** (Kinetic embedding and transfer of regularity - polarized version). Let $-\beta \le \varepsilon \le \beta$ such that $\beta \pm \varepsilon < d/2$. Let $f, \tilde{f} \in \mathcal{D}'(\Omega)$ be such that $f \in \dot{\mathcal{L}}_\beta^{\beta+\varepsilon}$ and $\tilde{f} \in \dot{\mathcal{L}}_\beta^{\beta-\varepsilon}$. Then

$$ \begin{gathered} \text{(i)} \quad f \in L_{t,v}^2 \dot{H}_x^{\frac{\beta+\varepsilon}{2\beta+1}} \cap C_0(\mathbb{R}_t; \dot{X}_\beta^\varepsilon) \text{ with} \\ \|D_x^{\frac{\beta+\varepsilon}{2\beta+1}} f\|_{L_{t,x,v}^2} + \sup_{t \in \mathbb{R}} \|f(t)\|_{\dot{X}_\beta^\varepsilon} \lesssim_{d,\beta,\varepsilon} \|f\|_{\dot{\mathcal{L}}_\beta^{\beta+\varepsilon}} \end{gathered} $$
**Remark 5.4.** Following the same strategy, one can also solve the backward kinetic Cauchy problem for $-(\partial_t + v \cdot \nabla_x) + A^*$ on intervals $(-\infty, 0]$. Step 2 uses instead uniqueness for distributional solutions of the forward Kolmogorov operator on $(-\infty, 0) \times \mathbb{R}^{2d}$. As integration is on this set, the change of sign for the kinetic field gives the condition (H2) in Step 6. The other steps are rigorously the same.

**5.2. Inhomogeneous kinetic spaces.** The theory of homogeneous kinetic spaces adapts to inhomogeneous kinetic spaces using the inhomogeneous Sobolev spaces $H_v^\gamma$ and $H_x^{\frac{2\beta+1}{2\beta+1}}$ for all $\beta > 0$ and $\gamma \in \mathbb{R}$ (no restriction $\gamma < d/2$ here). It amounts to adding the $L_{t,x,v}^2$ norm to the homogeneous ones. This leads us to define on $\Omega$ the inhomogeneous spaces $X_\beta^\gamma$ as in (2.2) and (2.3), $\mathcal{F}_\beta^\gamma$ as in (2.4), $\mathcal{G}_\beta^\gamma$ as in (2.6), and $\mathcal{L}_\beta^\gamma$ as in (2.7) for which the analog of Theorem 2.2 holds.

**Theorem 5.5 (Kinetic embeddings and transfer of regularity: inhomogeneous case).**
Assume $\gamma \in [0, 2\beta]$.

(i) $\mathcal{L}_\beta^\gamma \hookrightarrow L_{t,v}^2 H_x^{\frac{\gamma}{2\beta+1}}$ with $\|f\|_{L_{t,v}^2 H_x^{\frac{2\beta+1}{2\beta+1}}} \lesssim_{d,\beta,\gamma} \|f\|_{\mathcal{L}_\beta^\gamma}$

(ii) $\mathcal{L}_\beta^\gamma \hookrightarrow C_0(\mathbb{R}_t; X_\beta^{\gamma-\beta})$ with $\sup_{t \in \mathbb{R}} \|f(t)\|_{X_\beta^{\gamma-\beta}} \lesssim_{d,\beta,\gamma} \|f\|_{\mathcal{L}_\beta^\gamma}$.

(iii) $S(\Omega)$ is densely contained in all $\mathcal{F}_\beta^\gamma, \mathcal{G}_\beta^\gamma$ and $\mathcal{L}_\beta^\gamma$.

(iv) The families of spaces $(\mathcal{F}_\beta^\gamma)$ and $(\mathcal{G}_\beta^\gamma)$ have the complex interpolation property.

**Remark 5.6.** To the knowledge of the authors, the first estimate (i) for $\mathcal{G}_\beta^\gamma$ in the special case $\gamma = \beta = 1$ appears first in [8].

This is proved using the inhomogeneous Kolmogorov operators $K_{1,\beta}^+$ corresponding to $(\pm (\partial_t + v \cdot \nabla_x) + (-\Delta_v)^\beta + 1)^{-1}$ as in Section 2.3 with the analogs of Lemma 2.10 and Proposition 2.11 without any restriction on $\gamma$. The proofs are similar and we skip the details. Theorem 1.1 and its generalization Theorem 3.6 are also true with inhomogeneous assumptions and conclusions.

**5.3. Kinetic Cauchy problems on strips.** We now come to equations with relaxed ellipticity conditions, still imposing measurability. We assume that there exist $0 < \Lambda < \infty$ and $0 \le c^0 < \infty$, such that uniformly for $(t, x) \in \mathbb{R} \times \mathbb{R}^d$,

$$ (5.3) \quad \forall f, g \in L_{t,x}^2 H_v^\beta, \quad |a_{t,x}(f,g)| \le \Lambda (\|f\|_{\dot{H}_v^\beta} + c^0 \|f\|_{L_v^2}) (\|g\|_{\dot{H}_v^\beta} + c^0 \|g\|_{L_v^2}) $$

and that there exists $\lambda > 0, c_0 \ge 0$ such that for all $f \in H_v^\beta$ and uniformly for $(t, x) \in \mathbb{R} \times \mathbb{R}^d$,

$$ (5.4) \qquad \mathrm{Re} \, a_{t,x}(f,f) \ge \lambda \|f\|_{\dot{H}_v^\beta}^2 - c_0 \|f\|_{L_v^2}^2. $$

Note that this allows the possibility to include lower-order operators acting in velocity with suitable coefficients.

The inhomogeneous version of existence and uniqueness of weak solutions defined in $F^\beta = L_{t,x}^2 H_v^\beta$ on $\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d$ of $(\partial_t + v \cdot \nabla_x)f + Af + cf = S$ is completely similar with $c > c_0$. That is, the source and the solution belong to the respective corresponding inhomogeneous spaces. The inhomogeneous kinetic operator $K_{c,A} = (\partial_t + v \cdot \nabla_x + A + c)^{-1}$ has the inhomogeneous boundedness corresponding to (4.4).
(ii) $\tilde{f} \in L_{t,v}^2 \dot{H}_x^{\frac{\beta-\varepsilon}{2\beta+1}} \cap C_0(\mathbb{R}_t; \dot{X}_\beta^{-\varepsilon})$ with

$$ \|D_x^{\frac{\beta-\varepsilon}{2\beta+1}} \tilde{f}\|_{L_{t,x,v}^2} + \sup_{t \in \mathbb{R}} \|\tilde{f}(t)\|_{\dot{X}_\beta^{-\varepsilon}} \lesssim_{d,\beta,\varepsilon} \|\tilde{f}\|_{L_\beta^{\beta-\varepsilon}}. $$

(iii) The map $t \mapsto \langle f(t), \tilde{f}(t) \rangle$, where the bracket is for the duality between $\dot{X}_\beta^\varepsilon, \dot{X}_\beta^{-\varepsilon}$ is absolutely continuous on $\mathbb{R}$ and its almost everywhere derivative can be computed as follows. Write

$$ (\partial_t + v \cdot \nabla_x) f = S_1 + S_2 + S_3 $$

$$ \text{with } S_1 \in L_{t,x}^2 \dot{H}_v^{-\beta+\varepsilon}, S_2 \in L_{t,v}^2 \dot{H}_x^{\frac{-\beta+\varepsilon}{2\beta+1}} \text{ and } S_3 \in L_t^1 \dot{X}_\beta^{-\beta+\varepsilon}, \text{ and} $$

$$ (\partial_t + v \cdot \nabla_x) \tilde{f} = \tilde{S}_1 + \tilde{S}_2 + \tilde{S}_3 $$

$$ \text{with } \tilde{S}_1 \in L_{t,x}^2 \dot{H}_v^{-\beta-\varepsilon}, \tilde{S}_2 \in L_{t,v}^2 \dot{H}_x^{\frac{-\beta-\varepsilon}{2\beta+1}} \text{ and } \tilde{S}_3 \in L_t^1 \dot{X}_\beta^{-\beta-\varepsilon}. \text{ Then for a.e. } t \in \mathbb{R} $$

$$ \frac{d}{dt} \langle f(t), \tilde{f}(t) \rangle = \int_{\mathbb{R}^d} (\langle S_1, \tilde{f} \rangle + \langle f, \tilde{S}_1 \rangle) dx + \int_{\mathbb{R}^d} (\langle S_2, \tilde{f} \rangle + \langle f, \tilde{S}_2 \rangle) dv + \langle S_3, \tilde{f} \rangle + \langle f, \tilde{S}_3 \rangle, $$

where each bracket denotes a different sesquilinear duality extending the $L^2$ inner product by tracing the spaces involved. Putting absolute values around each bracket gives an integrable function in the missing variables.

*Proof.* Items (i) and (ii) are already in Theorem 2.2. Item (iii) is the polarised version (1.13) of Theorem 1.16 where the exponents are symmetric with respect to $\beta$ for duality reasons. Details are left to the reader. $\square$

## 4. CONSTRUCTION OF WEAK SOLUTIONS TO ROUGH EQUATIONS WHEN $\beta < d/2$

This section is devoted to the construction of weak solutions when $\beta < d/2$. It relies on a theorem due to J.-L. Lions [28, Chap. II, Thm. 1.1], which can be used in this range for $\beta$. Since this article is written in French, we recall the statement before proving Theorem 1.5.

### 4.1. Existence and uniqueness of weak solutions.
The proof of the existence of weak solutions to Kolmogorov-Fokker-Planck equations with rough coefficients uses the following theorem.

**Theorem 4.1** (Lions – [28]). Let $F$ be a Hilbert space equipped with a scalar product $\langle \cdot, \cdot \rangle_F$ and the associated norm $\| \cdot \|_F$. Let $\mathcal{H} \subset F$ be a prehibertian space equipped with a scalar product $\langle \cdot, \cdot \rangle_{\mathcal{H}}$ and the associated norm $\| \cdot \|_{\mathcal{H}}$. Assume that there exists $c_1 > 0$ such that for all $f \in \mathcal{H}$, $\|f\|_F \le c_1 \|f\|_{\mathcal{H}}$. Let $E$ be a sesquilinear form on $F \times \mathcal{H}$ such that

(H1) for all $h \in \mathcal{H}$, the linear form $f \mapsto E(f, h)$ is continuous on $F$,

(H2) there exists $\alpha > 0$ such that $\mathrm{Re} E(h, h) \ge \alpha \|h\|_{\mathcal{H}}^2$ for all $h \in \mathcal{H}$.

Then for any semi-linear continuous form $L$ on $\mathcal{H}$, there exists $f \in F$ such that $E(f, h) = L(h)$ for all $h \in \mathcal{H}$.
For the kinetic Cauchy problem, we may work on a finite or infinite time interval. Define all the inhomogeneous kinetic spaces as before on $\Omega_I = I \times \mathbb{R}^d \times \mathbb{R}^d$ where $I$ is an open interval. The embeddings are proved as follows.

For $I$ infinite, say $(0, \infty)$, use the extension to the inhomogeneous backward Kolmogorov operator of estimates in Proposition 2.11 and uniqueness in Lemma 3.1 on $\Omega_+$. For $(-\infty, T)$ use the same arguments for the forward Kolmogorov operator.

For $I$ finite, say $I = (0, T)$, use cut-off in time: if $\chi$ is Lipschitz and supported in $[0, T)$ and is 1 near 0, and $f \in \mathcal{L}_{\beta}^{\gamma}$ on $\Omega_{(0,T)}$, then $\chi f \in \mathcal{L}_{\beta}^{\gamma}$ on $\Omega_+$ so that we have continuity of $f$ in $X_{\beta}^{\gamma-\beta}$ in any subinterval $[0, T']$ of $[0, T)$. Also doing the same thing at the other endpoint of $(0, T)$, we conclude that $f \in C([0, T]; X_{\beta}^{\gamma-\beta})$. If $\gamma = \beta$, we also obtain the absolute continuity of $t \mapsto \|f(t)\|_{L_{x,v}^2}$ on $[0, T]$ and its extension to duality brackets $\langle f(t), \tilde{f}(t) \rangle$ of pairs $(f, \tilde{f}) \in \mathcal{L}_{\beta}^{\beta+\epsilon} \times \mathcal{L}_{\beta}^{\beta-\epsilon}$, $-\beta \le \epsilon \le \beta$.

Moreover, $L_t^2(I; S(\mathbb{R}^{2d}))$ is dense in all of them and one can also interpolate the $\mathcal{F}$ and $\mathcal{G}$ scales.

The inhomogeneous version of Theorem 5.2 is the following.

**Theorem 5.7** (Weak solutions for the kinetic Cauchy problem in inhomogeneous spaces). Let $0 < T < \infty$. Assume that the sesquilinear form a satisfies (5.3) and (5.4) on $(0, T) \times \mathbb{R}^d \times \mathbb{R}^d$. Let $S \in L_{t,x}^2 H_v^{-\beta} + L_{t,v}^2 H_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ and $\psi \in L_{x,v}^2$. There exists a unique $f \in L_{t,x}^2 H_v^{\beta}$ such that

$$ (5.5) \qquad \begin{cases} (\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S, \\ f(0) = \psi \end{cases} $$

in the sense of Definition 5.1 for test functions in $\mathcal{D}([0, T) \times \mathbb{R}^d \times \mathbb{R}^d)$.

Moreover, $f \in L_{t,v}^2 H_x^{\frac{\beta}{2\beta+1}} \cap C([0, T]; L_{x,v}^2)$, $f(t)$ converges to $\psi$ in $L_{x,v}^2$ as $t \to 0$, and

$$ \sup_{t \in [0,T]} \|f(t)\|_{L_{x,v}^2} + \|D_v^{\beta} f\|_{L_{t,x,v}^2} + \|D_x^{\frac{\beta}{2\beta+1}} f\|_{L_{t,x,v}^2} + \|f\|_{L_{t,x,v}^2} \\ \lesssim \|S\|_{L_{t,x}^2 H_v^{-\beta} + L_{t,v}^2 H_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2} + \|\psi\|_{L_{x,v}^2} $$

for an implicit constant depending on $d, \beta, \lambda, \Lambda, c_0, T$. Furthermore, $f$ satisfies the energy equality (4.1) for all times $0 \le s < t \le T$.

*Proof.* For existence, extend $a_{t,x}$ by the form associated with $\lambda(-\Delta_v)^\beta$. We may call it the canonical extension, and note that it preserves the ellipticity constants $\lambda, \Lambda$. Extend $S$ by $0$ for $t \ge T$. In the case $c_0 = 0 = c^0$, we have nothing else to do than restrict the construction on $[0, \infty)$ to $[0, T]$. If $c_0 > 0$, use the exponential change of variable trick $f \to e^{-ct}f$ to go back to the kinetic Cauchy problem $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f + cf = e^{-ct}S$ with initial data $\psi$. Solve this problem with $c > c_0$ on $[0, \infty)$ exactly as in Theorem 5.2 but with inhomogeneous spaces, then restrict to $[0, T]$ and undo the change of variable mentioned above.

For uniqueness, recall that any $f \in \mathcal{L}_{\beta}^{\beta}$ on $\Omega_{(0,T)}$ belongs to $C([0, T]; L_{x,v}^2)$ with the absolute continuity of $t \mapsto \|f(t)\|_{L_{x,v}^2}^2$. If $f \in F^{\beta}$ and $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = 0$ on $\Omega_{(0,T)}$ with $f(0) = 0$, then $f \in \mathcal{F}_{\beta}^{\beta} \subset \mathcal{L}_{\beta}^{\beta}$ on $\Omega_{(0,T)}$. Thus $f$ satisfies the energy equality
The following lemma is a consequence of the transfer of regularity obtained in Theorem 1.16 and is valid for all $\beta > 0$.

**Lemma 4.2 (Energy equality).** Let $\beta > 0$ and $S = S_1 + S_2 + S_3$ with $S_1 \in L_{t,x}^2 \dot{H}_v^{-\beta}$ and $S_2 \in L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}}$ and $S_3 \in L_t^1 L_{x,v}^2$. Any weak solution to $(\partial_t + v \cdot \nabla_x)f + Af = S$ in the sense of Definition 1.2 satisfies the energy equality: for any $s, t \in \mathbb{R}$ with $s < t$,

$$ \|f(t)\|_{L_{x,v}^2}^2 + 2 \operatorname{Re} \int_s^t \int_{\mathbb{R}^d} a_{\tau,x}(f, f) \, dx \, d\tau = \\ (4.1) \quad \|f(s)\|_{L_{x,v}^2}^2 + 2 \operatorname{Re} \int_s^t \left( \int_{\mathbb{R}^d} \langle S_1, f \rangle \, dx + \int_{\mathbb{R}^d} \langle S_2, f \rangle \, dv + \iint_{\mathbb{R}^{2d}} S_3 \bar{f} \, dx \, dv \right) \, d\tau. $$

*Proof.* As $(\partial_t + v \cdot \nabla_x)f = S - Af$, we know from Theorem 1.16 that $t \mapsto \|f(t)\|_{L_{x,v}^2}^2$ is absolutely continuous, with limit 0 at $\pm\infty$ and, for $s < t$, $\|f(t)\|_{L_{x,v}^2}^2 - \|f(s)\|_{L_{x,v}^2}^2$ is equal to

$$ 2 \operatorname{Re} \int_s^t \left( \int_{\mathbb{R}^d} \langle S_1 - Af, f \rangle \, dx + \int_{\mathbb{R}^d} \langle S_2, f \rangle \, dv + \iint_{\mathbb{R}^{2d}} S_3 \bar{f} \, dx \, dv \right) \, d\tau. $$

The bracket inside the first integral denotes the one expressing the $\dot{H}_v^{-\beta}$, $\dot{H}_v^{\beta}$ duality as a function of $(\tau, x)$, so by definition and writing the variables for clarity

$$ \langle Af, f \rangle(\tau, x) = a_{\tau,x}(f(\tau, x), f(\tau, x)) = a_{\tau,x}(f, f) $$

by convention in the last equality.

<figure><img src="image_6.png" /></figure>

*Proof of Theorem 1.5 when $\beta < d/2$.* Lemma 4.2 already proves that any weak solution satisfies the energy equality.

In the second step, we prove uniqueness. Assume that $f \in \dot{F}^{\beta}$ with $(\partial_t + v \cdot \nabla_x)f + Af = 0$ (in the sense of Definition 1.2). The energy equality reads when $s < t$,

$$ \|f(t)\|_{L_{x,v}^2}^2 - \|f(s)\|_{L_{x,v}^2}^2 = -2 \operatorname{Re} \int_s^t \int_{\mathbb{R}^d} a_{\tau,x}(f, f) \, dx \, d\tau. $$

Letting now $s \to -\infty$ and $t \to +\infty$ and using that limits of $L_{x,v}^2$ norms are zero, we obtain $\operatorname{Re} \int_{-\infty}^{+\infty} \int_{\mathbb{R}^d} a_{\tau,x}(f, f) \, dx \, d\tau = 0$. Ellipticity implies $f = 0$.

In a third step, we prove the existence in the case where $S \in L_{t,x}^2 \dot{H}_v^{-\beta} = (\dot{F}^{\beta})'$ by use of Theorem 4.1. This is the only place where we need the assumption $\beta < d/2$ in this proof. Indeed, we recall that when $\beta < d/2$, $\dot{H}_v^{\beta}$ is a Hilbert space, so is $\dot{F}^{\beta} = L_{t,x}^2 \dot{H}_v^{\beta}$.

To verify the hypotheses there, we let $\mathcal{H} = \mathcal{D}(\Omega)$ equipped with the prehilbertian norm $\|h\|_{\mathcal{H}} = \|h\|_{\dot{F}^{\beta}}$. In particular, the inclusion $\mathcal{H} \subset \dot{F}^{\beta}$ is continuous.

Next, consider the sesquilinear form $E$ on $\dot{F}^{\beta} \times \mathcal{H}$ defined by

$$ E(f, h) = - \iiint_{\Omega} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv + a(f, h), $$

where $a$ is defined in (1.6).
(4.1): for any $t \le T$, $\|f(t)\|_{L_{x,v}^2}^2 + 2 \operatorname{Re} \int_0^t \int_{\mathbb{R}^d} a_{\tau,x}(f, f) \,dx\,d\tau = \|f(0)\|_{L_{x,v}^2}^2 = 0$, and it follows that $f = 0$. $\square$

**Remark 5.8.** The same arguments show that one can also solve the backward kinetic Cauchy problem for $-(\partial_t + v \cdot \nabla_x) + \mathcal{A}^*$ on intervals $[0, T]$ with final value at $T$, proceeding with Theorem 5.2 on $(-\infty, T]$) for operators $-(\partial_t + v \cdot \nabla_x) + \mathcal{A}^* + c$.

## 6. THE SPECIAL CASE OF LOCAL DIFFUSION OPERATORS

In this section, we focus on $\beta = 1$ and give different definitions of the homogeneous spaces that are valid in any dimension $d \ge 1$.

Define $\dot{W}^{1,2}(\mathbb{R}^d) = \{f \in \mathcal{D}'(\mathbb{R}^d); \nabla f \in L^2(\mathbb{R}^d)\}$, equipped with the semi-norm $\|\nabla f\|_{L^2}$. Remark that $\dot{H}^1(\mathbb{R}^d) \subset \dot{W}^{1,2}(\mathbb{R}^d)$ when $d \ge 3$. The inhomogeneous version is $W^{1,2}(\mathbb{R}^d) = \{f \in L^2(\mathbb{R}^d); \nabla f \in L^2(\mathbb{R}^d)\}$ equipped with the norm $(\|f\|_{L^2}^2 + \|\nabla f\|_{L^2}^2)^{1/2}$. It agrees with $H^1(\mathbb{R}^d)$ when $d \ge 1$. Hence, our inhomogeneous theory is unchanged and we only have to address the homogeneous one.

**Lemma 6.1.** Any element of $\dot{W}^{1,2}(\mathbb{R}^d)$ identifies to an element in $L_{\text{loc}}^2(\mathbb{R}^d) \cap \mathcal{S}'(\mathbb{R}^d)$, and $\mathcal{D}(\mathbb{R}^d)$ is dense in $\dot{W}^{1,2}(\mathbb{R}^d)$ for the semi-norm.

*Proof.* This is known. Here is a proof for the comfort of the reader. Let $f \in \dot{W}^{1,2}(\mathbb{R}^d)$. As $f$ is a distribution, we may regularize it to $f_\varepsilon$ in this space by mollification, and $f_\varepsilon \in C^\infty$. Use Poincaré inequalities for $f_\varepsilon$ and limiting arguments to deduce that $f \in L_{\text{loc}}^2(\mathbb{R}^d)$. Poincaré inequalities again give a polynomial growth in $R$ of the $L^2$ norms of $f$ on balls $B(0, R)$. Thus $f$ is a tempered distribution as well. Lastly, for the density, we claim that we can approximate $f$ by compactly supported functions in the $\|\nabla f\|_2$ semi-norm. Pick $f_k(x) = (f(x) - c_k)\varphi(2^{-k}x)$, $k \ge 1$, with $\varphi$ a smooth function supported on the ball $B(0, 2)$ and that is 1 on the unit ball and where $c_k$ is the mean of $f$ on the ball $B(0, 2^{k+1})$. It is easy to see that $\nabla f_k$ converges weakly to $\nabla f$ in $L^2$ and by Mazur's lemma, there exist convex combinations of subsequences of $\nabla f_k$ having strong convergence to $\nabla f$ in $L^2$. From there, use mollification to obtain an approximation in $\mathcal{D}(\mathbb{R}^d)$. $\square$

We introduce the kinetic space

$$ (6.1) \qquad \dot{\mathcal{L}} = \{f \in \mathcal{D}'(\Omega); f \in L_{t,x}^2 \dot{W}_v^{1,2} \ \& \ (\partial_t + v \cdot \nabla_x)f \in L_t^2 \dot{X}_1^{-1} + L_t^1 L_{x,v}^2\} $$

with semi-norm $\|f\|_{\dot{\mathcal{L}}}$ defined by

$$ \|f\|_{\dot{\mathcal{L}}} = \|\nabla_v f\|_{L_{t,x,v}^2} + \|(\partial_t + v \cdot \nabla_x)f\|_{L_t^2 \dot{X}_1^{-1} + L_t^1 L_{x,v}^2}. $$

The spaces with negative indices are defined as before and $\dot{H}_v^{-1}$ identifies to the dual of $\dot{W}_v^{1,2}$, so that we also write $\dot{H}_v^{-1} = \dot{W}_v^{-1,2}$ and $\dot{X}_1^{-1} = L_x^2 \dot{W}_v^{-1,2} + L_v^2 \dot{H}_x^{-1/3}$. Note that constants belong to $\dot{\mathcal{L}}$ (contrarily to $\dot{\mathcal{L}}_1^1$ to which we naturally compare $\dot{\mathcal{L}}$).

Specializing the results obtained for the Fokker-Planck operators $K^\pm := K_1^\pm$ in Proposition 2.11 (recall that the limitation on $\gamma$ there was only due to definition), we can state the following boundedness result.

**Proposition 6.2** (Boundedness of the Fokker-Planck operators). We have the bounded map

$$ K^\pm: L_{t,x}^2 \dot{W}_v^{-1,2} + L_{t,v}^2 \dot{H}_x^{-1/3} + L_t^1 L_{x,v}^2 \to L_{t,x}^2 \dot{W}_v^{1,2} \cap L_{t,v}^2 \dot{H}_x^{1/3} \cap C_0(\mathbb{R}_t; L_{x,v}^2). $$
We need to check (H1), that is: for all $h \in \mathcal{H}$, $f \mapsto E(f, h)$ is continuous on $\dot{\mathbb{F}}^{\beta}$.
To this end, we show that $|E(f, h)| \lesssim \|f\|_{\dot{\mathbb{F}}^{\beta}} \|h\|_{\dot{\mathcal{F}}^{\beta}}$, where the $\dot{\mathcal{F}}^{\beta}$ norm is defined in (2.5). This implies (H1) as $\|h\|_{\dot{\mathcal{F}}^{\beta}} < \infty$ for $h \in \mathcal{H}$. Let $h \in \mathcal{H}$ and $f \in \dot{\mathbb{F}}^{\beta}$. Then, by Hölder inequality and Sobolev embedding, we have

$$ \left| \iiint_{\Omega} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv \right| \le \|f\|_{L_{t,x}^2 L_v^q} \|(\partial_t + v \cdot \nabla_x)h\|_{L_{t,x}^2 L_v^{q'}} \\ \lesssim \|f\|_{L_{t,x}^2 \dot{H}_v^{\beta}} \|(\partial_t + v \cdot \nabla_x)h\|_{L_{t,x}^2 \dot{H}_v^{-\beta}}. $$

For the other term, we use the assumption on $a$ in order to get,

$$ |a(f, h)| \le \Lambda \|f\|_{\dot{\mathbb{F}}^{\beta}} \|h\|_{\dot{\mathbb{F}}^{\beta}}. $$

Gathering the two terms using the definition of the $\dot{\mathcal{F}}^{\beta}$ norm in (2.5) gives us the conclusion.

We next check (H2), namely that $\mathrm{Re} E(h, h) \ge \lambda \|h\|_{\mathcal{H}}^2$ for all $h \in \mathcal{H}$. Indeed, by integrating by parts,

$$ (4.2) \qquad - \mathrm{Re} \iiint_{\Omega} h(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv = 0. $$

$$ \text{Thus, } \mathrm{Re} E(h, h) = \mathrm{Re} a(h, h) \ge \lambda \|h\|_{\dot{\mathbb{F}}^{\beta}}^2 = \lambda \|h\|_{\mathcal{H}}^2. $$

As $S \in (\dot{\mathbb{F}}^{\beta})'$, the map $h \mapsto \iint_{\mathbb{R} \times \mathbb{R}^d} \langle S, h \rangle \, dx \, dt$, where the brackets inside the integral are the $\dot{H}_v^{-\beta}$, $\dot{H}_v^{\beta}$ duality as usual, is a continuous semi-linear functional on $\mathcal{H}$. Applying the Theorem 4.1, there exists $f \in \dot{\mathbb{F}}^{\beta}$ such that

$$ E(f, h) = \iint_{\mathbb{R} \times \mathbb{R}^d} \langle S, h \rangle \, dx \, dt \quad \text{for all } h \in \mathcal{H}. $$

In our language, this means that $f$ is a weak solution to $(\partial_t + v \cdot \nabla_x) f + \mathcal{A} f = S$ in the sense of Definition 1.2. And since $(\partial_t + v \cdot \nabla_x) f = S - \mathcal{A} f \in (\dot{\mathbb{F}}^{\beta})'$, we have that $f \in \dot{\mathcal{F}}^{\beta}$.

The second and third steps show the isomorphism property of $(\partial_t + v \cdot \nabla_x) + \mathcal{A}$ from $\dot{\mathcal{F}}^{\beta}$ onto $(\dot{\mathbb{F}}^{\beta})'$.

In the fourth step, we assume $S \in L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$. We prove the existence of a weak solution by a duality scheme and the use of kinetic embedding and transfer of regularity in Theorem 1.16. The same argument as above applies to $((\partial_t + v \cdot \nabla_x) + \mathcal{A})^* = - (\partial_t + v \cdot \nabla_x) + \mathcal{A}^*$, because (4.2) is not changed in changing the sign of the field and $\mathcal{A}^*$ is also elliptic. It is also an isomorphism from $\dot{\mathcal{F}}^{\beta}$ onto $(\dot{\mathbb{F}}^{\beta})'$. In particular, Theorem 1.16 implies that

$$ (-(\partial_t + v \cdot \nabla_x) + \mathcal{A}^*)^{-1} : (\dot{\mathbb{F}}^{\beta})' \to \dot{\mathcal{F}}^{\beta} \hookrightarrow \dot{\mathcal{L}}^{\beta} \hookrightarrow L_{t,v}^2 \dot{H}_x^{\frac{\beta}{2\beta+1}} \cap C_0(\mathbb{R}_t; L_{x,v}^2). $$

Hence, the operator $\mathcal{T} : L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2 \to \dot{\mathbb{F}}^{\beta}$ defined by

$$ \langle \mathcal{T} S, h \rangle = \langle S, -((\partial_t + v \cdot \nabla_x) + \mathcal{A}^*)^{-1} h \rangle, \quad \text{for } h \in (\dot{\mathbb{F}}^{\beta})' \text{ and } S \in L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2 $$
is continuous.
**Corollary 6.3** (Uniqueness up to constant). Let $f \in D'(\Omega)$ for which $\nabla_v f \in L_{t,x,v}^2$. Then $\pm(\partial_t + v \cdot \nabla_x)f - \Delta_v f = 0$ in the sense of distributions implies that $f$ is constant.

*Proof.* We follow the strategy of Lemma 3.1 and notation there. By Lemma 6.1 $f \in S'(\Omega)$ but $\hat{f}$ may not be locally integrable on $\Omega$. Still, one can solve the ordinary differential equation in a distributional sense and obtain that $G = K(t, 0)G_0$ with $G_0 \in D'(\mathbb{R}^{2d})$. Next the condition $(\xi - t\varphi)G \in L_{t,\varphi,\xi}^2$ implies that the restriction of $G$ to the complement of $\{\xi - t\varphi = 0\}$ is locally square integrable. This forces $G_0$ to be supported at the origin $(\varphi, \xi) = (0, 0)$. Indeed, pick a compact set $C$ in $\mathbb{R}^{2d} \setminus \{(0, 0)\}$. Then one can find a compact interval $I \subset \mathbb{R}$ such that $\min|\xi - t\varphi|^2 > 0$ on $I \times C$. Thus $G \in L^2(I \times C)$ and, using the equation, we see that $G \in C(I; L^2(C))$. This implies that $G_0 \in L^2(C)$. Using again $(\xi - t\varphi)G \in L_{t,\varphi,\xi}^2$, this gives us

$$ \iiint_{\mathbb{R} \times C} |\xi - t\varphi|^2 \exp \left( -2 \int_0^t |\xi - \tau\varphi|^2 d\tau \right) |G_0(\varphi, \xi)|^2 dt d\varphi d\xi < \infty, $$

which yields $G_0 = 0$ on $C$. Hence $G$ is supported in $\mathbb{R} \times \{(0, 0)\}$. Hence $(\xi - t\varphi)G = 0$ as a distribution (because it is locally integrable and supported on a null Lebesgue set). Thus $\partial_t G = 0$ from the equation. Differentiating $\xi G = t\varphi G$ with respect to $t$ leads to $\varphi G = 0$, and consequently $\xi G = 0$. This implies that $G = T_0 \otimes \delta$ where $\delta$ is the Dirac mass at the origin in $\mathbb{R}^{2d}$ and $T_0 \in D'(\mathbb{R})$. One uses again $\partial_t G = 0$ to conclude that $T_0$ is constant. Going back to $f$, we conclude that $f$ is constant. $\square$

**Proposition 6.4.** The Kolmogorov operators $K^\pm$ are isomorphisms from $L_{t,x}^2 \dot{W}_v^{-1,2}$ onto $\dot{F}/\mathbb{C}$, from $L_t^2 \dot{X}_1^{-1}$ onto $\dot{G}/\mathbb{C}$ and from $L_t^2 \dot{X}_1^{-1} + L_t^1 L_{x,v}^2$ onto $\dot{L}/\mathbb{C}$, using quotient norms, where the spaces $\dot{F}$ and $\dot{G}$ are defined similarly to $\dot{L}$ replacing $L_{t,x}^2 \dot{H}_v^1$ by $L_{t,x}^2 \dot{W}_v^{1,2}$.

*Proof.* This follows from the estimates and uniqueness, as in Lemma 3.3. $\square$

**Corollary 6.5.** Let $f \in \dot{L}$. Then there exists a constant $c \in \mathbb{C}$ such that $f - c \in C_0(\mathbb{R}_t; L_{x,v}^2) \cap L_{t,v}^2 \dot{H}_x^{\frac{1}{3}}$ with estimate

$$ \|f - c\|_{L_{t,v}^2 \dot{H}_x^{\frac{1}{3}}} + \sup_{t \in \mathbb{R}} \|f(t) - c\|_{L_{x,v}^2} \lesssim_d \|f\|_{\dot{L}}. $$

The space in (iii) of Theorem 2.2 is dense in $\dot{L}$. Moreover, for $f \in \dot{L}$, the absolute continuity stated in Theorem 1.1 (and its generalization Theorem 3.6) holds with $f-c$.

*Proof.* Follow the strategy of proof of Theorem 2.2. Details are left to the reader. $\square$

**Remark 6.6.** One can now see that $\dot{L} = \dot{L}_1^1 + \mathbb{C}$ when $d \ge 3$.

Let us turn to the construction of weak solutions. The difficulty with this homogeneous space here is that $L_{t,x}^2 \dot{W}_v^{1,2}$ is not a Hilbert space so that we cannot apply the Lions theorem directly. We use an approximation procedure starting from the inhomogeneous case. We let $A = A(t, x, v)$ be a bounded and coercive matrix with measurable coefficients (real and symmetric is not necessary) in such a way that

$$ a_{t,x}(f,g) = \int_{\mathbb{R}^d} \langle A(t,x,v) \nabla_v f(v), \nabla_v g(v) \rangle dv, \quad f,g \in \dot{W}_v^{1,2}, $$

satisfies (1.7) and (1.8) on $\dot{W}_v^{1,2}$. We define $\mathcal{A}$ as in (1.4) with the form $a$ defined by (1.9) on $L_{t,x}^2 \dot{W}_v^{1,2}$. Here we assume either $t \in \mathbb{R}$ or $t \in (0, \infty)$.
If $S \in L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ satisfies the extra assumption $S \in (\dot{F}^\beta)'$, then $\mathcal{T}S$ agrees with $((\partial_t + v \cdot \nabla_x) + \mathcal{A})^{-1}S$ as previously defined, so the function $f = \mathcal{T}S$ satisfies $E(f, h) = \langle S, h \rangle$ for all $h \in \mathcal{H} = \mathcal{D}(\Omega)$.

We now remove this extra assumption. Approximate $S \in L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ by $S_j \in \mathcal{S}_K$ as in Lemma 2.12, and thus, letting $f_j = \mathcal{T}S_j$, we have $E(f_j, h) = \langle S_j, h \rangle$ when $h \in \mathcal{D}(\Omega)$. For fixed $h$, since $S_j \to S$ in $L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$, using the proper meaning of $\langle S, h \rangle$ explained in Theorem 1.16, $\langle S_j, h \rangle \to \langle S, h \rangle$. As $f_j \to \mathcal{T}S$ in $\dot{F}^\beta$, $E(f_j, h) \to E(\mathcal{T}S, h)$ using the estimate proved for (H1) above. We have shown that $\mathcal{T}S$ is a weak solution to (1.1). Eventually, the fact that $\mathcal{T}S \in L_{t,v}^2 \dot{H}_x^{\frac{\beta}{2\beta+1}} \cap C_0(\mathbb{R}_t; L_{x,v}^2)$ follows from Theorem 1.16. $\square$

## 5. KINETIC CAUCHY PROBLEMS

This section is devoted to the construction of weak solutions to kinetic Cauchy problems on strips for time intervals of the form $[0, \infty)$ or $[0, T]$ with $T$ finite. In order to construct solutions on strips, we will introduce inhomogeneous kinetic spaces and prove an embedding theorem. We begin by working on infinite intervals and obtain homogeneous estimates.

### 5.1. The kinetic Cauchy problem in homogeneous spaces.
We turn our attention to the kinetic Cauchy problem

$$ (5.1) \qquad \begin{cases} (\partial_t + v \cdot \nabla_x) f + \mathcal{A} f = S, \\ f(0) = \psi, \end{cases} $$

For the kinetic Cauchy problem on an infinite time interval, we replace $\Omega$ by $\Omega_+ = (0, \infty) \times \mathbb{R}^d \times \mathbb{R}^d$ and one can again use the homogeneous spaces on $\Omega_+$ with the same notation, see Section 2.2.2. We use the same notation as well for the operator $\mathcal{A}$.

**Definition 5.1** (Weak solutions to the kinetic Cauchy problem). Let $\beta > 0$. Let $S \in \mathcal{D}'(\Omega_+)$ and $\psi \in L_{x,v}^2$. The distribution $S$ is assumed to extend to a continuous linear functional on $\mathcal{D}(\Omega_+)$. A distribution $f \in \mathcal{D}'(\Omega_+)$ is said to be a weak solution to (5.1) if $f \in \dot{U}^\beta$ and for all $h \in \mathcal{D}(\Omega_+)$,

$$ -\iiint_{\Omega_+} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv + a(f, h) = \langle S, h \rangle + \iint_{\mathbb{R}^{2d}} \psi(x, v) \bar{h}(0, x, v) \, dx \, dv. $$

Again, $f \in \dot{U}^\beta$ implies $f \in L_{t,x}^2 S'_v \cap L_{t,x}^2 L_{\text{loc},v}^2$ with our definition. When $h$ is smooth, the expression $(\partial_t + v \cdot \nabla_x) \bar{h}$ is nothing but the pointwise partial derivative calculation. Thus, the left-hand side is well-defined; in the right-hand side, $\langle S, h \rangle$ expresses the ad-hoc duality for the extension of $S$ (it will be clear in the following). The function $\psi$ could also be a distribution in $\mathbb{R}^{2d}$, in which case the corresponding term should be a sesquilinear duality bracket $\langle \psi, h(0) \rangle$. When $h \in \mathcal{D}(\Omega_+)$, the last term disappears and this means in particular that $f$ is a weak solution to the first equation $(\partial_t + v \cdot \nabla_x) f + \mathcal{A} f = S$ in the sense of Definition 1.2 on $\Omega_+$. The second equation in (5.1) is encoded by allowing test functions that do not vanish at $t = 0$.
**Theorem 6.7.** (i) On $\mathbb{R}$, if $S \in L_t^2 \dot{X}_1^{-1} + L_t^1 L_{x,v}^2$, then there is a weak solution $f \in L_{t,x}^2 \dot{W}_v^{1,2}$, unique up to a constant, to $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$. For the appropriate constant, $f - c \in C_0(\mathbb{R}_t; L_{x,v}^2) \cap L_{t,v}^2 \dot{H}_x^{\frac{1}{3}}$ and satisfies the energy equality.

(ii) On $[0, \infty)$, if $S \in L_t^2 \dot{X}_1^{-1} + L_t^1 L_{x,v}^2$ and $\psi \in L_{x,v}^2$, then there is a unique weak solution in $L_{t,x}^2 \dot{W}_v^{1,2}$ to the kinetic Cauchy problem $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$ with $f(0) = \psi$. We have $f \in C_0([0, \infty); L_{x,v}^2) \cap L_{t,v}^2 \dot{H}_x^{\frac{1}{3}}$ and satisfies the energy equality.

*Proof.* We start with uniqueness. In case (i), a weak solution $f \in L_{t,x}^2 \dot{W}_v^{1,2}$ to $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$ belongs to $\dot{\mathcal{L}}$. Thus, there is a constant $c$ such that $\tilde{f} = f - c \in C_0(\mathbb{R}_t; L_{x,v}^2)$ and satisfies the energy equality. If we assume $S = 0$, then for all $s < t$,

$$ \|\tilde{f}(t)\|_{L_{x,v}^2}^2 - \|\tilde{f}(s)\|_{L_{x,v}^2}^2 = -2 \operatorname{Re} \int_s^t \int_{\mathbb{R}^d} a_{\tau,x}(\tilde{f}, \tilde{f}) \, dx \, d\tau. $$

In particular, this yields after taking limits at $\pm\infty$, $\operatorname{Re} \int_{-\infty}^{+\infty} \int_{\mathbb{R}^d} a_{\tau,x}(\tilde{f}, \tilde{f}) \, dx \, d\tau = 0$, hence $\operatorname{Re} a_{\tau,x}(\tilde{f}, \tilde{f}) = 0$ for almost all $(\tau, x)$ (because it is non negative). This yields $\|\tilde{f}(t)\|_{L_{x,v}^2}^2 - \|\tilde{f}(s)\|_{L_{x,v}^2}^2 = 0$ for all $s < t$, that is $t \mapsto \|\tilde{f}(t)\|_{L_{x,v}^2}^2$ is constant. As it has zero limits at $\pm\infty$, $\tilde{f} = 0$ and $f$ is constant.

In case (ii), we can produce the same argument following Steps 1 and 2 in the proof of Theorem 5.2 to show that on $\Omega_+$, for all $f \in \dot{\mathcal{L}}$, there is a constant $c$ such that

$$ \|f - c\|_{C_0([0, \infty); L_{x,v}^2)} \lesssim_d \|f\|_{\dot{\mathcal{L}}}. $$

Now consider a weak solution $f$ with $S = 0$ and $\psi = 0$. As we assume that $f(0) = 0$ at time 0, $f \in C_0([0, \infty); L_{x,v}^2)$ and satisfies the energy equality. Therefore, for $t > 0$, $\|f(t)\|_{L_{x,v}^2}^2 = -2 \operatorname{Re} \int_0^t \int_{\mathbb{R}^d} a_{\tau,x}(f, f) \, dx \, d\tau \le 0$, thus $f(t) = 0$.

Now we turn to existence. We begin with (ii) in the case where $S \in L_{t,x}^2 \dot{W}_v^{-1,2}$ and $\psi \in L_{x,v}^2$. We prove existence on any finite strip $\Omega_{(0,T)} = (0, T) \times \mathbb{R}^d \times \mathbb{R}^d$ of a weak solution $f^T$ with $\nabla_v f^T \in L^2(\Omega_{(0,T)})$ and $f^T \in C([0, T]; L_{x,v}^2)$. Assume this is done. Such solutions also verify the energy equality. If $T < T'$ and $g = f^{T'} - f^T$, then $g$ is a weak solution in the same class on $[0, T]$ with zero source and zero initial data. The energy equality yields when $t \in [0, T]$, $\|g(t)\|_{L_{x,v}^2}^2 = -2 \operatorname{Re} \int_0^t \int_{\mathbb{R}^d} a_{\tau,x}(g, g) \, dx \, d\tau \le 0$, thus $g(t) = 0$. Since $f^T$ and $f^{T'}$ agree on the smaller strip, this allows us to define $f$ on $\Omega_+$ and $f$ is a weak solution to the kinetic Cauchy problem in (ii) with the desired properties.

To construct $f^T$, we argue using the inhomogeneous setup on $[0, T]$ that is now fixed and we work on this interval. For $\varepsilon > 0$, using Theorem 5.7 with $\mathcal{A} + \varepsilon$, we solve in $L_{t,x}^2 \dot{W}_v^{1,2}$ the kinetic Cauchy problem $(\partial_t + v \cdot \nabla_x)f_\varepsilon + \mathcal{A}f_\varepsilon + \varepsilon f_\varepsilon = e^{-\varepsilon t}S$ with $f_\varepsilon(0) = \psi$. Note that $(\partial_t + v \cdot \nabla_x)(e^{\varepsilon t}f_\varepsilon) + \mathcal{A}(e^{\varepsilon t}f_\varepsilon) = S$ and $(e^{\varepsilon t}f_\varepsilon)(0) = \psi$. Using the energy equality for $e^{\varepsilon t}f_\varepsilon$ (absolute continuity follows from that of $f_\varepsilon$) and the fact that $S \in L_x^2 \dot{W}_v^{-1,2}$, we can see that $\|\nabla_v(e^{\varepsilon t}f_\varepsilon)\|_{L_{t,x,v}^2}$ and $\sup_{t \in [0, T]} \|e^{\varepsilon t}f_\varepsilon(t)\|_{L_{x,v}^2}$ are uniformly bounded with respect to $\varepsilon$. Hence, a classical weak*-limit argument furnishes a weak
**Theorem 5.2** (Existence and uniqueness of weak solutions to the kinetic Cauchy problem). Let $\beta > 0$. Let $S \in L_{t,x}^2 \dot{H}_v^{-\beta} + L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ and $\psi \in L_{x,v}^2$. There exists a unique weak solution $f \in \dot{U}^\beta$ to the kinetic Cauchy problem in the sense of Definition 5.1.

Moreover, $f \in L_{t,v}^2 \dot{H}_x^{\frac{\beta}{2\beta+1}} \cap C_0([0, \infty); L_{x,v}^2)$, hence there is $L_{x,v}^2$ convergence of $f(t)$ to $\psi$ as $t \to 0$, and

$$ \|D_v^\beta f\|_{L_{t,x,v}^2} + \|D_x^{\frac{\beta}{2\beta+1}} f\|_{L_{t,x,v}^2} + \sup_{t \in [0,\infty)} \|f(t)\|_{L_{x,v}^2} \\ \lesssim \|S\|_{L_{t,x}^2 \dot{H}_v^{-\beta} + L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2} + \|\psi\|_{L_{x,v}^2} $$

for an implicit constant depending only on $d, \beta, \lambda, \Lambda$. In addition, $f$ satisfies the energy equality (4.1) for non-negative times.

*Proof of Theorem 5.2 in the case $\beta < d/2$. The strategy has several steps.*

Step 1: The estimates of the backward Kolmogorov operator $K_\beta^-$ in Corollary 2.18 hold on $\Omega_+$, by restriction to the time interval $(0, \infty)$ with $C_0(\mathbb{R}_t; L_{x,v}^2)$ replaced by $C_0([0, \infty); L_{x,v}^2)$.

Step 2: The uniqueness statement in Lemma 3.1 holds for the backward Kolmogorov operator on $\Omega_+$ when $\gamma = \beta$.

Step 3: Combining Steps 1 and 2, the space $\dot{L}_\beta^\beta$ on $\Omega_+$ embeds into $C_0([0, \infty); L_{x,v}^2)$. In particular, for all $f \in \dot{L}_\beta^\beta$,

$$ \|f\|_{C_0([0,\infty); L_{x,v}^2)} \lesssim_{d,\beta} \|f\|_{\dot{L}_\beta^\beta}. $$

Step 4: More generally, the statement of Theorem 1.16 holds on $\Omega_+$ with $C_0(\mathbb{R}_t; L_{x,v}^2)$ replaced by $C_0([0, \infty); L_{x,v}^2)$. The absolute continuity on $[0, \infty)$ is proved similarly using the backward Kolmogorov operator to construct approximations for the proof. In particular, any weak solution to the kinetic Cauchy problem as in the statement satisfies the energy equality (4.1) of Lemma 4.2 for non-negative times and converges in the sense of $L_{x,v}^2$ limit at $t = 0$.

Step 5: The uniqueness in the kinetic Cauchy problem proceeds with the energy equality as in the proof of Theorem 1.5.

Step 6: We use $\beta < d/2$ here. Existence when $S \in L_{t,x}^2 \dot{H}_v^{-\beta} = (\dot{F}^\beta)'$, and $\psi \in L_{x,v}^2$ also proceeds as in the proof of Theorem 1.5 with modifications as follows. We change here $\mathcal{H}$ to $\mathcal{H}_+ = \mathcal{D}(\bar{\Omega}_+)$ equipped with prehilbertian norm $\|h\|_{\mathcal{H}_+}^2 = \|h\|_{\dot{F}^\beta}^2 + \|h(0)\|_{L_{x,v}^2}^2$.

Clearly $\mathcal{H}_+$ is continuously included in the Hilbert space $\dot{F}^\beta$ on $\Omega_+$.

Setting

$$ E_+(f, h) := - \iiint_{\Omega_+} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv + a(f, h), $$

the proof of condition (H1) with $|E_+(f, h)| \lesssim \|f\|_{\dot{F}^\beta} \|h\|_{\dot{F}_\beta^\beta}$ for $f \in \dot{F}^\beta$ and $h \in \mathcal{H}_+$ is unchanged. Now, for condition (H2), if $h \in \mathcal{H}_+$ then integration by parts in $t$ and $x$
solution $f^T \in L_t^\infty L_{x,v}^2 \cap L_{t,x}^2 W_v^{1,2}$. As $f^T$ belongs to the inhomogeneous version $\mathcal{L}$ of $\dot{\mathcal{L}}$ on $\Omega_{(0,T)}$, continuity in time follows.

Now, we turn to existence in (i) when $S \in L_{t,x}^2 \dot{W}_v^{-1,2}$ on $\Omega$ this time. By applying (ii) with time interval $[-n, \infty)$, $n \in \mathbb{N}$, instead on $[0, \infty)$, consider the weak solution $f_n$ on $[-n, \infty)$ to the kinetic Cauchy problem $(\partial_t + v \cdot \nabla_x) f_n + \mathcal{A} f_n = S$ and $f_n(-n) = 0$. The extension of this solution by $0$ when $t \le -n$ belongs to $L_{t,x}^2 \dot{W}_v^{1,2} \cap C_0(\mathbb{R}_t; L_{x,v}^2) \cap L_{t,v}^2 \dot{H}_x^{\frac{1}{3}}$ on $\Omega$, uniformly in $n$. Extract a weak*-limit in $L_{t,x}^2 \dot{W}_v^{1,2} \cap L_t^\infty L_{x,v}^2 \cap L_{t,v}^2 \dot{H}_x^{\frac{1}{3}}$. This weak*-limit is a weak solution on $\mathbb{R}$ and as it is already a bounded function of time valued in $L_{x,v}^2$, it further belongs to $C_0(\mathbb{R}_t; L_{x,v}^2)$ using the embedding in Corollary 6.5.

It remains to solve the problems for source terms in $L_{t,v}^2 \dot{H}_x^{-\frac{1}{3}} \cap L_t^1 L_{x,v}^2$. For this, we use the duality scheme in Step 4 of the proof of Theorem 1.12 first on $\mathbb{R}$. Next, on $(0, \infty)$ with zero initial data, we use this previous case and the causality principle as in Step 7 of the proof of Theorem 5.2. $\square$

**Remark 6.8.** When $d \ge 3$, as $L_{t,x}^2 \dot{H}_v^1 \subset L_{t,x}^2 \dot{W}_v^{1,2}$, uniqueness implies that the weak solutions in Theorem 6.7, (ii), and in Theorem 5.2 when $\beta = 1$ are the same. Similarly, the weak solution $f-c$ in Theorem 6.7, (i), is the same as the one in Theorem 1.12.

Let us illustrate our results by a consequence for optimal regularity, to be compared with [8, Theorem 1.5] where $f$ and $\nabla_v f$ were both supposed to belong to $L_{t,x,v}^2$ for the conclusion $\Delta_v f \in L_{t,x,v}^2$.

**Corollary 6.9 (Optimal regularity).** Let $f \in \mathcal{D}'(\Omega)$ with $\sup_{\sigma \in \mathbb{R}} \|\nabla_v f\|_{L^2(\Omega_{(\sigma,\sigma+1)})} < \infty$ (in particular, if $\nabla_v f \in L_{t,x,v}^2$) be such that $S = (\partial_t + v \cdot \nabla_x) f - \Delta_v f \in L_{t,x,v}^2$. Then, $(\partial_t + v \cdot \nabla_x) f, \Delta_v f \in L_{t,x,v}^2$ and $\nabla_v f \in C_0(\mathbb{R}_t; L_{x,v}^2)$.

*Proof.* Again the estimates obtained using Fourier methods in Proposition 2.13 imply in particular the boundedness

$$ \mathcal{K}^{\pm}: L_{t,x,v}^2 \to L_{t,x}^2 \dot{W}_v^{2,2} \cap C_0(\mathbb{R}_t; L_x^2 \dot{W}_v^{1,2}) $$

where $\dot{W}^{2,2}(\mathbb{R}^d) = \{f \in \mathcal{D}'(\mathbb{R}^d); \nabla^2 f \in L^2(\mathbb{R}^d)\}$. Thus, $\mathcal{K}^+ S$ is a tempered distribution solution of the same equation as $f$. We show that $f - \mathcal{K}^+ S$ is constant, which concludes the proof from the properties of $\mathcal{K}^+ S$.

Remark that from the above bounds $\nabla_v \mathcal{K}^+ S$ satisfies the same condition as $\nabla_v f$ in the hypothesis. Applying the method of proof of Corollary 6.3 to $f - \mathcal{K}^+ S$ and calling $G$ the Fourier transform of $\Gamma(f - \mathcal{K}^+ S)$, the only thing that changes is the condition $(\xi - t\varphi)G \in L_{t,\varphi,\xi}^2$ replaced by $(\xi - t\varphi)G \in L^2(\sigma, \sigma + 1; L_{\varphi,\xi}^2)$ uniformly in $\sigma \in \mathbb{R}$. This is enough to conclude that the restriction of $G_0$ to any compact set $C \subset \mathbb{R}^{2d} \setminus \{(0,0)\}$ is square integrable and that

$$ \sup_{\sigma \in \mathbb{R}} \iiint_{(\sigma, \sigma+1) \times C} |\xi - t\varphi|^2 \exp \left( -2 \int_0^t |\xi - \tau\varphi|^2 d\tau \right) |G_0(\varphi, \xi)|^2 dt d\varphi d\xi < \infty. $$

By taking $\sigma \to -\infty$, this is only possible if $G_0 = 0$ almost everywhere on $C$. The rest of the proof is verbatim the one of Corollary 6.3. $\square$

**Remark 6.10.** At the expense of more technicalities, one can make the same discussion with the higher-order homogeneous space $\dot{W}^{m,2}(\mathbb{R}^d) = \{f \in \mathcal{D}'(\mathbb{R}^d); \nabla^m f \in$
shows that

$$ -\mathrm{Re} \iiint_{\Omega_+} h(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv = \frac{1}{2} \|h(0)\|_{L_{x,v}^2}^2. $$

Thus,

$$ \mathrm{Re} E_+(h, h) = \frac{1}{2} \|h(0)\|_{L_{x,v}^2}^2 + \mathrm{Re} a(h, h) \geq \min \left( \lambda, \frac{1}{2} \right) \|h\|_{\mathcal{H}_+}^2. $$

Finally, the map $\mathcal{H}_+ \ni h \mapsto L(h) = \iint_{(0,\infty) \times \mathbb{R}^d} \langle S, h \rangle \, dx \, dt + \langle \psi, h(0) \rangle$, where the brackets inside the integral are the $\dot{H}_v^{-\beta}$, $\dot{H}_v^{\beta}$ duality as usual, is a continuous semi-linear functional on $\mathcal{H}_+$. For the first term, this is as before, and for the second, this is simply the Cauchy-Schwarz inequality,

$$ |\langle \psi, h(0) \rangle| \le \|\psi\|_{L_{x,v}^2} \|h(0)\|_{L_{x,v}^2} \le \|\psi\|_{L_{x,v}^2} \|h\|_{\mathcal{H}_+}. $$

Applying Theorem 4.1 again, there exists $f \in \dot{\mathbb{F}}^{\beta}$ such that for all $h \in \mathcal{D}(\overline{\Omega}_+)$, $E_+(f, h) = L(h)$, that is explicitly

$$ (5.2) \quad - \iiint_{\Omega_+} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv + \iint_{(0,\infty) \times \mathbb{R}^d} a_{t,x}(f, h) \, dx \, dt \\ = \iint_{(0,\infty) \times \mathbb{R}^d} \langle S, h \rangle \, dx \, dt + \langle \psi, h(0) \rangle. $$

In particular, $f$ is a weak solution on $\Omega_+$ to $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$. It remains to show $f(0) = \psi$ as we already know that $t \mapsto f(t)$ is continuous on $[0, \infty)$ in $L_{x,v}^2$. We remark again that $t \mapsto \langle f(t), h(t) \rangle$ is absolutely continuous and

$$ \langle f(t), h(t) \rangle - \langle f(0), h(0) \rangle = - \int_s^t \int_{\mathbb{R}^d} a_{\tau,x}(f, h) \, dx \, d\tau \\ + \int_0^t \int_{\mathbb{R}^d} \langle S, h \rangle \, dx \, d\tau + \int_0^t \iint_{\mathbb{R}^{2d}} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dx \, dv \, d\tau. $$

Letting $t \to +\infty$, the right hand side tends to $-\langle \psi, h(0) \rangle$ by (5.2) and the left hand side to $-\langle f(0), h(0) \rangle$ from Step 3. As $h(0)$ can be arbitrary in $\mathcal{D}(\mathbb{R}^{2d})$, we obtain $u(0) = \psi$.

Step 7: It remains to prove existence when $S \in L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ and $\psi = 0$. In order to do so, we extend $S$ by $0$ for $t < 0$ and take the weak solution $f$ to $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$ on $\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d$ of Theorem 1.5. Using the energy equality, we have when $s < t \le 0$,

$$ \|f(t)\|_{L_{x,v}^2}^2 - \|f(s)\|_{L_{x,v}^2}^2 = -2 \mathrm{Re} \int_s^t \int_{\mathbb{R}^d} a_{t,x}(f, f) \, dx \, dt, $$

and letting $s \to -\infty$, we obtain $\|f(t)\|_{L_{x,v}^2}^2 \le 0$. Therefore, $f$ restricted to $\Omega_+$ is a weak solution on $\Omega_+$ with $f(0) = 0$. $\square$

**Corollary 5.3 (Causality principle).** Let $\beta > 0$. Let $S \in L_{t,x}^2 \dot{H}_v^{-\beta} + L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ on $\Omega_+$. If $S_0$ is the zero extension of $S$ for $t < 0$, then $K_\mathcal{A}^+ S_0$, with $K_\mathcal{A}^+$ as in (1.9), vanishes for $t \le 0$ and its restriction to $\Omega_+$ is the solution to the kinetic Cauchy problem (5.1) with zero initial value.
$L^2(\mathbb{R}^d)$, $m$ integer, which contains polynomials of degree less than $m$, applying this to higher order Kolmogorov operators $K_m^\pm$. For weak solutions, ellipticity (1.8) should be in the sense of a Gårding inequality on $\dot{W}_v^{m,2}$. Embedding (on $\mathbb{R}$ or $[0, \infty)$) and uniqueness of weak solutions on $\mathbb{R}$ are obtained modulo polynomials of degree less than $m$.

## 7. EXTENSIONS AND LINK WITH OTHER KINETIC EMBEDDINGS

This section is devoted to extensions of our results in two directions: in the case where $\gamma > 2\beta$ and in the case where $\beta > d/2$.

### 7.1. Beyond the limitation $\gamma \le 2\beta$
As mentioned in the introduction there is a notion of kinetic spaces developed in [28] where embeddings similar to ours are proved without restriction on $\gamma$, while we are limited with $0 \le \gamma \le 2\beta$. It is worth explaining the links between the two works. For this, we need to introduce more spaces and make some extensions (mostly without proofs, which are adaptations of the previous arguments).

#### 7.1.1. Homogeneous source spaces revisited.
We first extend the range of parameter $\gamma$ in our spaces $\dot{X}_\beta^\gamma$. We just mention at this stage that we did not do it in the article because it would not have helped to prove the existence of weak solutions in Theorem 1.12, where we only impose ellipticity in the $v$-variable.

In the proof of Lemma 2.9, we established the following isomorphism via the Fourier transform in $\mathbb{R}^{2d}$

$$ \dot{X}_\beta^\gamma \to L_{\varphi, \xi}^2(w^{2\gamma} d\varphi d\xi), $$

where $w(\varphi, \xi) = \sup(|\xi|, |\varphi|^{1/(2\beta)})$, provided $\gamma < d/2$. When $\gamma > 0$, the only reason for the restriction $\gamma < d/2$ is our definition of $\dot{X}_\beta^\gamma$ as the intersection of two spaces, separating the variables in some sense. We remark that a simple calculation shows for $\gamma \in \mathbb{R}$,

$$ w^{-\gamma} \in L_{\text{loc}}^2(\mathbb{R}^{2d}) \iff \gamma < (\beta + 1)d. $$

This allows us to define $\dot{X}_\beta^\gamma$ for $\gamma < (\beta + 1)d$ in the spirit of homogeneous Sobolev spaces by

$$ \dot{X}_\beta^\gamma = \{f \in S'(\mathbb{R}^{2d}) ; \exists g \in L^2(\mathbb{R}^{2d}), \hat{f} = w^{-\gamma} \hat{g}\} $$

with norm $\|f\|_{\dot{X}_\beta^\gamma} = \|g\|_{L_{\varphi, \xi}^2}$. In this range, it is a well-defined Hilbert space, contained in $S'(\mathbb{R}^{2d})$ with elements having Fourier transforms in $L_{\text{loc}}^1(\mathbb{R}^{2d})$. When $\gamma < d/2$, this space is equal, as a set and with equivalent norm, to the one defined in (2.2) and (2.3).

With this definition, our proofs on the Fourier transform side (where there is no restriction on exponents) show that the statements of Lemma 2.9, Proposition 2.11 hold for $\gamma < (\beta + 1)d$. Also the uniqueness as in Lemma 3.1 applies with $f \in L_t^2 \dot{X}_\beta^\gamma$ and $\gamma < (\beta + 1)d$, thanks to Remark 3.2.
Proof in the case $\beta < d/2$. This is the argument in Step 7. $\square$

<figure><img src="image_7.png" /></figure>

**Remark 5.4.** Following the same strategy, one can also solve the backward kinetic Cauchy problem for $-(\partial_t + v \cdot \nabla_x) + A^*$ on intervals $(-\infty, 0]$. Step 2 uses uniqueness for distributional solutions of the forward Kolmogorov operator on $(-\infty, 0) \times \mathbb{R}^{2d}$. As integration is on this set, the change of sign for the kinetic field gives the condition (H2) in Step 6. The other steps are rigorously the same.

## 6. STATEMENTS ON INHOMOGENEOUS SPACES

It should be clear that inhomogeneous situations yield similar results and come with slight modifications in statements and proofs. We explain this in this section with a few details when necessary.

**6.1. Embeddings for inhomogeneous kinetic spaces.** The theory of homogeneous kinetic spaces adapts to inhomogeneous kinetic spaces using the inhomogeneous Sobolev spaces $H_v^\gamma$ and $H_x^{\frac{2\beta+1}{\gamma}}$ for all $\beta > 0$ and $\gamma \in \mathbb{R}$ (no restriction $\gamma < d/2$ here). It amounts to adding the $L_{t,x,v}^2$ norm to the homogeneous ones. This leads us to define on $\Omega$ the inhomogeneous spaces $X_\beta^\gamma$ as in (2.2) and (2.3), $\mathcal{F}_\beta^\gamma$ as in (2.4), $\mathcal{G}_\beta^\gamma$ as in (2.6), and $\mathcal{L}_\beta^\gamma$ as in (2.7) for which the analog of Theorem 2.2 holds.

**Theorem 6.1 (Kinetic embeddings and transfer of regularity: inhomogeneous case).**
Assume $\gamma \in [0, 2\beta]$.

(i) $\mathcal{L}_\beta^\gamma \hookrightarrow L_{t,v}^2 H_x^{\frac{2\beta+1}{\gamma}}$ with $\|f\|_{L_{t,v}^2 H_x^{\frac{2\beta+1}{\gamma}}} \lesssim_{d,\beta,\gamma} \|f\|_{\mathcal{L}_\beta^\gamma}$

(ii) $\mathcal{L}_\beta^\gamma \hookrightarrow C_0(\mathbb{R}_t; X_\beta^{\gamma-\beta})$ with $\sup_{t \in \mathbb{R}} \|f(t)\|_{X_\beta^{\gamma-\beta}} \lesssim_{d,\beta,\gamma} \|f\|_{\mathcal{L}_\beta^\gamma}$.

(iii) There is a subspace of $L_{t,x,v}^2$ densely contained in all $\mathcal{F}_\beta^\gamma$, $\mathcal{G}_\beta^\gamma$ and $\mathcal{L}_\beta^\gamma$.

(iv) The families of spaces $(\mathcal{F}_\beta^\gamma)_\gamma$ and $(\mathcal{G}_\beta^\gamma)_\gamma$ have the complex interpolation property.

**Remark 6.2.** To the knowledge of the authors, the first estimate (i) for $\mathcal{G}_\beta^\gamma$ in the special case $\gamma = \beta = 1$ appears first in [10].

This is proved using the inhomogeneous Kolmogorov operators $\mathcal{K}_{1,\beta}^+$ corresponding to $(\pm (\partial_t + v \cdot \nabla_x) + (-\Delta_v)^\beta + 1)^{-1}$ as in Section 2.3 and where the density with the analogs of Lemma 2.12 and Proposition 2.16 without any restriction on $\gamma$. The proofs using weighted estimates with weight $\sup(1, |\xi - t\varphi|, |\varphi|^{\frac{1}{2\beta+1}})$ are similar and we skip the details. Theorem 1.16 and its generalisation Theorem 3.9 are also true with inhomogeneous assumptions and conclusions without the condition $\beta \pm \varepsilon < d/2$.

**6.2. Kinetic Cauchy problems on strips.** We now come to equations with relaxed ellipticity conditions, still imposing measurability. We assume that there exist $0 < \Lambda < \infty$ and $0 \le c^0 < \infty$, such that uniformly for $(t, x) \in \mathbb{R} \times \mathbb{R}^d$,

$$ (6.1) \quad \forall f, g \in L_{t,x}^2 H_v^\beta, \quad |a_{t,x}(f,g)| \le \Lambda (\|f\|_{\dot{H}_v^\beta} + c^0 \|f\|_{L_v^2}) (\|g\|_{\dot{H}_v^\beta} + c^0 \|g\|_{L_v^2}) $$

and that there exists $\lambda > 0, c_0 \ge 0$ such that for all $f \in H_v^\beta$ and uniformly for $(t, x) \in \mathbb{R} \times \mathbb{R}^d$,

$$ (6.2) \quad \operatorname{Re} a_{t,x}(f,f) \ge \lambda \|f\|_{\dot{H}_v^\beta}^2 - c_0 \|f\|_{L_v^2}^2. $$
### 7.1.2. More homogeneous kinetic spaces.
We remark that even with this new definition of $\dot{X}_{\beta}^{\gamma}$, the upper bound on $\gamma$ in the definitions of our kinetic spaces $\dot{\mathcal{F}}_{\beta}^{\gamma}$, $\dot{\mathcal{G}}_{\beta}^{\gamma}$, $\dot{\mathcal{L}}_{\beta}^{\gamma}$ is $d/2$ because we impose them to be subspaces of $L_{t,x}^2 \dot{H}_v^{\gamma}$. Nevertheless, changing this condition to $f \in L_t^2 \dot{X}_{\beta}^{\gamma}$ allows to introduce when $\gamma < (\beta + 1)d$,

$$ (7.1) \qquad \dot{T}_{\beta}^{\gamma} = \{f \in D'(\Omega); f \in L_t^2 \dot{X}_{\beta}^{\gamma} \ \& \ (\partial_t + v \cdot \nabla_x) f \in L_t^2 \dot{X}_{\beta}^{\gamma-2\beta}\} $$

with Hilbertian norm $\|f\|_{\dot{T}_{\beta}^{\gamma}}$ defined by

$$ \|f\|_{\dot{T}_{\beta}^{\gamma}}^{2} = \|f\|_{L_{t}^{2} \dot{X}_{\beta}^{\gamma}}^{2} + \|(\partial_{t} + v \cdot \nabla_{x})f\|_{L_{t}^{2} \dot{X}_{\beta}^{\gamma-2\beta}}^{2}, $$

and the larger space

$$ (7.2) \qquad \dot{U}_{\beta}^{\gamma} = \{f \in D'(\Omega); f \in L_t^2 \dot{X}_{\beta}^{\gamma} \ \& \ (\partial_t + v \cdot \nabla_x) f \in L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{X}_{\beta}^{\gamma-\beta}\} $$

with norm $\|f\|_{\dot{U}_{\beta}^{\gamma}}$ defined by

$$ \|f\|_{\dot{U}_{\beta}^{\gamma}} = \|f\|_{L_t^2 \dot{X}_{\beta}^{\gamma}} + \|(\partial_t + v \cdot \nabla_x) f\|_{L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{X}_{\beta}^{\gamma-\beta}} $$

Note that the $x$-regularity is already included in the definition of these spaces, hence $\dot{T}_{\beta}^{\gamma} \subset \dot{G}_{\beta}^{\gamma}$ and $\dot{U}_{\beta}^{\gamma} \subset \dot{\mathcal{L}}_{\beta}^{\gamma}$ when $0 \le \gamma < d/2$. When $\gamma \le 0$, the opposite inclusions hold.

There is an immediate observation that $(-\Delta_v)^\beta$ maps $L_t^2 \dot{X}_\beta^\gamma$ into $L_t^2 \dot{X}_\beta^{\gamma-2\beta}$ when $\gamma < (\beta + 1)d$. Indeed, writing $\hat{f} = w^{-\gamma}\hat{g}$ with $\hat{g} \in L_{t,\varphi,\xi}^2$, we have $|\xi|^{2\beta}\hat{f} = w^{-\gamma+2\beta}(|\xi|^{2\beta}w^{-2\beta}\hat{g})$. As $|\xi|^{2\beta}w^{-2\beta}\hat{g}$ is in $L_{t,\varphi,\xi}^2$ and $\gamma - 2\beta < \gamma$, we have that $|\xi|^{2\beta}\hat{f}$ is a tempered distribution with inverse (partial) Fourier transform in $L_t^2 \dot{X}_\beta^{\gamma-2\beta}$.

### 7.1.3. Another homogeneous kinetic embedding.
In this scale, the kinetic embedding statement becomes the following one.

**Theorem 7.1 (Homogeneous kinetic embedding: extension).** Assume $\gamma < (\beta + 1)d$.

(i) $\dot{T}_{\beta}^{\gamma} \subset \dot{U}_{\beta}^{\gamma} \to C_0(\mathbb{R}_t; \dot{X}_{\beta}^{\gamma-\beta})$ with $\sup_{t \in \mathbb{R}} \|f(t)\|_{\dot{X}_{\beta}^{\gamma-\beta}} \lesssim_{d,\beta,\gamma} \|f\|_{\dot{U}_{\beta}^{\gamma}}$.

(ii) The subspaces $\mathcal{K}_{\beta}^{\pm}(S_K)$ of $S(\Omega)$ in Theorem 2.2 are dense in $\dot{T}_{\beta}^{\gamma}$ and $\dot{U}_{\beta}^{\gamma}$.

(iii) The family of spaces $(\dot{T}_{\beta}^{\gamma})_{\gamma}$ has the complex interpolation property.

Again this is only the definition of spaces that imposes the upper bound on $\gamma$. The inclusion (i) in this statement is weaker than (ii) in Theorem 2.2 but for a larger range.

We also have modifications of the statements of Theorem 1.1 and of Theorem 3.6 assuming $f \in \dot{U}_{\beta}^{\beta+\epsilon}$ and $\tilde{f} \in \dot{U}_{\beta}^{\beta-\epsilon}$, with only constraint $\beta \pm \epsilon < (\beta + 1)d$. In the latter statement, as there is no transfer of regularity, (i) and (ii) only contain the sup norm estimates. Next, (iii) has to be modified slightly with assumptions on the sums $S_1 + S_2, \tilde{S}_1 + \tilde{S}_2$ rather than individual assumptions. The dualities in the formula for the derivative also have to be adapted. This is left to the reader.

The isomorphism statement Lemma 3.3 is modified as follows: when $\gamma < (\beta + 1)d$, $\mathcal{K}_{\beta}^{\pm}$ are isomorphisms from $L_t^2 \dot{X}_{\beta}^{\gamma-2\beta}$ onto $\dot{T}_{\beta}^{\gamma}$ and from $L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{X}_{\beta}^{\gamma-\beta}$ onto $\dot{U}_{\beta}^{\gamma}$. This is what induces the interpolation property for the scale of spaces $\dot{T}_{\beta}^{\gamma}$.

The above discussion also extends to homogeneous spaces on $\Omega_+$.
Note that this allows the possibility of including lower-order operators acting in velocity with suitable coefficients.

The inhomogeneous version of existence and uniqueness of weak solutions defined in $F^\beta = L_{t,x}^2 H_v^\beta$ on $\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d$ of $(\partial_t + v \cdot \nabla_x) f + \mathcal{A} f + cf = S$ is completely similar with $c > c_0$. That is, the source and the solution belong to the corresponding inhomogeneous spaces. The inhomogeneous kinetic operator $\mathcal{K}_{c,A} = (\partial_t + v \cdot \nabla_x + \mathcal{A} + c)^{-1}$ has the inhomogeneous boundedness corresponding to (1.10).

For the kinetic Cauchy problem, we may work on a finite or infinite time interval. Define all the inhomogeneous kinetic spaces as before on $\Omega_I = I \times \mathbb{R}^d \times \mathbb{R}^d$ where $I$ is an open interval. The embeddings are proved as follows.

For $I$ infinite, say $(0, \infty)$, use the extension to the inhomogeneous backward Kolmogorov operator of estimates in Proposition 2.16 and uniqueness in Lemma 3.1 on $\Omega_+$. For $(-\infty, T)$, use the same arguments for the forward Kolmogorov operator.

For $I$ finite, say $I = (0, T)$, use cut-off in time: if $\chi: \mathbb{R} \to [0, 1]$ is Lipschitz and supported in $[0, T)$ and is 1 near 0, and $f \in \mathcal{L}_\beta^\gamma$ on $\Omega_{(0,T)}$, then $\chi f \in \mathcal{L}_\beta^\gamma$ on $\Omega_+$ so that we have continuity of $f$ in $X_\beta^{\gamma-\beta}$ in any subinterval $[0, T']$ of $[0, T)$. Also doing the same thing at the other endpoint of $(0, T)$ and gluing the two information, we conclude that $f \in C([0, T]; X_\beta^{\gamma-\beta})$. If $\gamma = \beta$, we also obtain the absolute continuity of $t \mapsto \|f(t)\|_{L_{x,v}^2}$ on $[0, T]$ and its extension to duality brackets $\langle f(t), \tilde{f}(t) \rangle$ of pairs $(f, \tilde{f}) \in \mathcal{L}_\beta^{\beta+\varepsilon} \times \mathcal{L}_\beta^{\beta-\varepsilon}$, $-\beta \le \varepsilon \le \beta$.

Moreover, $L_t^2(I; \mathcal{S}(\mathbb{R}^{2d}))$ is dense in all of them, and one can also interpolate the $\mathcal{F}$ and $\mathcal{G}$ scales.

The inhomogeneous version of Theorem 5.2 is the following.

**Theorem 6.3** (Weak solutions for the kinetic Cauchy problem in inhomogeneous spaces). Let $\beta > 0$ and $0 < T < \infty$. Assume that the sesquilinear form $\mathcal{A}$ satisfies (6.1) and (6.2) on $(0, T) \times \mathbb{R}^d \times \mathbb{R}^d$. Let $S \in L_{t,x}^2 H_v^{-\beta} + L_{t,v}^2 H_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ and $\psi \in L_{x,v}^2$. There exists a unique $f \in L_{t,x}^2 H_v^\beta$ such that

$$ (6.3) \qquad \begin{cases} (\partial_t + v \cdot \nabla_x) f + \mathcal{A} f = S, \\ f(0) = \psi \end{cases} $$

in the sense of Definition 5.1 for test functions in $\mathcal{D}([0, T) \times \mathbb{R}^d \times \mathbb{R}^d)$.

Moreover, $f \in L_{t,v}^2 H_x^{\frac{\beta}{2\beta+1}} \cap C([0, T]; L_{x,v}^2)$, $f(t)$ converges to $\psi$ in $L_{x,v}^2$ as $t \to 0$, and

$$ \sup_{t \in [0, T]} \|f(t)\|_{L_{x,v}^2} + \|D_v^\beta f\|_{L_{t,x,v}^2} + \|D_x^{\frac{\beta}{2\beta+1}} f\|_{L_{t,x,v}^2} + \|f\|_{L_{t,x,v}^2} \\ \lesssim \|S\|_{L_{t,x}^2 H_v^{-\beta} + L_{t,v}^2 H_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2} + \|\psi\|_{L_{x,v}^2} $$

for an implicit constant depending on $d, \beta, \lambda, \Lambda, c_0, T$. Furthermore, $f$ satisfies the energy equality (4.1) for all times $0 \le s < t \le T$.

*Proof.* For existence, extend $a_{t,x}$ by the form associated with $\lambda(-\Delta_v)^\beta$. We may call it the canonical extension, and note that it preserves the ellipticity constants $\lambda, \Lambda$. Extend $S$ by $0$ for $t \ge T$. Use the exponential change of variable trick $f \to e^{-ct}f$ to go back to the kinetic Cauchy problem $(\partial_t + v \cdot \nabla_x) f + \mathcal{A} f + cf = e^{-ct}S$ with initial
7.1.4. **Inhomogeneous versions.** Consider the inhomogeneous versions $\mathcal{T}_{\beta}^{\gamma}(I)$, $\mathcal{U}_{\beta}^{\gamma}(I)$ on $\Omega$, $\Omega_{+}$ or on strips $(0, T) \times \mathbb{R}^d \times \mathbb{R}^d$ where $I = \mathbb{R}$, $(0, \infty)$ or $(0, T)$, respectively. With these spaces, there are no conditions on $\gamma \in \mathbb{R}$. The natural generalizations to Theorem 7.1 and absolute continuity hold. In particular, one has

$$ (7.3) \quad \mathcal{T}_{\beta}^{\gamma}(0, T) \hookrightarrow C([0, T]; X_{\beta}^{\gamma-\beta}) \text{ with } \sup_{t \in [0, T]} \|f(t)\|_{X_{\beta}^{\gamma-\beta}} \lesssim_{d, \beta, \gamma, T} \|f\|_{\mathcal{T}_{\beta}^{\gamma}(0, T)}. $$

If $T = \infty$, the homogeneous estimate is the one corresponding to (i) in Theorem 7.1 (with $\gamma < (\beta + 1)d$).

**Proposition 7.2.** If $0 \le \gamma \le 2\beta$, then $\mathcal{G}_{\beta}^{\gamma}(0, T) = \mathcal{T}_{\beta}^{\gamma}(0, T)$. As a consequence, Theorem 5.5(i), and (7.3) are the same statements in this range of $\gamma$.

*Proof.* When $\gamma \ge 0$, the definition yields the inclusion $\mathcal{T}_{\beta}^{\gamma}(0, T) \subset \mathcal{G}_{\beta}^{\gamma}(0, T)$. The inhomogeneous version of the transfer of regularity contained in Theorem 5.5(ii) shows that if $0 \le \gamma \le 2\beta$ then $\mathcal{G}_{\beta}^{\gamma}(0, T) \subset \mathcal{T}_{\beta}^{\gamma}(0, T)$. $\square$

When $\gamma \notin [0, 2\beta]$, this equality of spaces does not seem true, hence the kinetic embedding for $\mathcal{G}_{\beta}^{\gamma}(0, T)$ is not clear as well.

7.1.5. *Comparison with other embeddings.* We can now make a precise link with [28] where the estimate (7.3) was proved (see Theorem 5.13 in this work). More precisely, the spaces $\mathcal{T}_{\beta}^{\gamma}(0, T)$ are introduced (with different notation and in the range $\gamma \ge \beta$: a close examination shows this restriction was not necessary and their arguments work for all $\gamma$) as part of a more general framework (with temporal weights) for solving the kinetic Cauchy problem for the Kolmogorov equation with constant diffusion $(0 < \beta \le 1)$ and (7.3) is a key ingredient in the argument. While the techniques used there relied on Fourier transform as well, the method is different. It is proved by first establishing a weaker embedding in $C([0, T]; X)$ where $X$ is a Sobolev space with a large enough negative exponent. This argument is possible when working with inhomogeneous spaces. Then by an interpolation argument, the exact trace space was characterized as $X_{\beta}^{\gamma-\beta}$. Some estimates were mentioned to hold when $T = \infty$ but not the homogeneous embedding that would correspond to (7.3).

**7.2. Beyond the half-dimension condition.** We describe here a way to remove the constraint $\beta < d/2$ while keeping homogeneous estimates.

7.2.1. *The two main results revisited.* The limitation $\beta < d/2$ in our two main results in the homogeneous situation is mostly a matter of definition in trying to avoid superfluous assumptions. We have seen that this limitation disappears under inhomogeneous assumptions (Theorem 5.7). This is because there is a control of $\|f(t)\|_{L_{x,v}^2}$ in $L_t^2$ sense. But then the price we paid in the inhomogeneous setting is that this control appears in the estimates.

If one wants to keep homogeneous estimates, then it suffices to add a mild qualitative inhomogeneous information that is not used quantitatively. More precisely, instead of working with general distributions, we assume at the beginning a control on $\|f(t)\|_{L_{x,v}^2}$ in $L_{\text{loc},t}^2$ sense, which allows us to apply positive fractional powers of $D_v$ and $D_x$ to $f$.
data $\psi$. Solve this problem with $c > c_0$ on $[0, \infty)$ exactly as in Theorem 5.2 but with inhomogeneous spaces, then restrict to $[0, T]$ and undo the change of variable mentioned above.

For uniqueness, recall that any $f \in \mathcal{L}_{\beta}^{\beta}$ on $\Omega_{(0,T)}$ belongs to $C([0, T]; L_{x,v}^2)$ with the absolute continuity of $t \mapsto \|f(t)\|_{L_{x,v}^2}^2$. If $f \in F^{\beta}$ and $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = 0$ on $\Omega_{(0,T)}$ with $f(0) = 0$, then $f \in \mathcal{F}_{\beta}^{\beta} \subset \mathcal{L}_{\beta}^{\beta}$ on $\Omega_{(0,T)}$. Thus $f$ satisfies the energy equality (4.1): for any $t \le T$, $\|f(t)\|_{L_{x,v}^2}^2 + 2 \operatorname{Re} \int_0^t \int_{\mathbb{R}^d} a_{\tau,x}(f, f) \,dx\,d\tau = \|f(0)\|_{L_{x,v}^2}^2 = 0$, and it follows that $f = 0$. $\square$

**Remark 6.4.** The same arguments show that one can also solve the backward kinetic Cauchy problem for $-(\partial_t + v \cdot \nabla_x) + \mathcal{A}^*$ on intervals $[0, T]$ with final value at $T$, proceeding with Theorem 5.2 on $(-\infty, T]$) for operators $-(\partial_t + v \cdot \nabla_x) + \mathcal{A}^* + c$.

## 7. PROOF OF THEOREM 1.5 AND THEOREM 5.2 WHEN $\beta \ge d/2$.

Basically, the main technical difference is the construction of weak solutions having the desired homogeneous estimates since we work in $\Omega$ or $\Omega_+$. Recall the notation

$$ \dot{Y}^{\beta} = L_{t,x}^{2} \dot{H}_{v}^{\beta} \cap L_{t,v}^{2} \dot{H}_{x}^{\frac{\beta}{2\beta+1}} \cap C_{0}(\mathbb{R}_{t}; L_{x,v}^{2}) $$

and

$$ \dot{Z}^{\beta} = L_{t,x}^{2} \dot{H}_{v}^{-\beta} + L_{t,v}^{2} \dot{H}_{x}^{-\frac{\beta}{2\beta+1}} + L_{t}^{1} L_{x,v}^{2} $$

used when working on $\mathbb{R}$. We shall do the proofs at the same time, assuming that the form associated to $\mathcal{A}$ is defined for $t \in \mathbb{R}$ (we know we can always extend it anyway) and divide the argument into 6 steps.

Step 1: Uniqueness in $\dot{U}^{\beta}$ on $\Omega_+$ and $\Omega$.

Step 2: Existence in $\dot{U}^{\beta}$ for $S \in L_{t,x}^2 \dot{H}_v^{-\beta}$ on $\Omega_+$.

Step 3: Existence in $\dot{U}^{\beta}$ for $S \in L_{t,x}^2 \dot{H}_v^{-\beta}$ on $\Omega$.

Step 4: On $\Omega$, $\mathcal{K}_{\mathcal{A}}^{+}: L_{t,x}^2 \dot{H}_v^{-\beta} \to \dot{Y}^{\beta}$.

Step 5: On $\Omega$, $\mathcal{K}_{\mathcal{A}}^{+}: \dot{Z}^{\beta} \to \dot{Y}^{\beta}$ (end of proof of Theorem 1.5).

Step 6: Causality (end of proof of Theorem 5.2).

Step 1: Uniqueness is a consequence of the energy equality for weak solutions. The one in $\Omega$ is proved in Lemma 4.2. For the one in $\Omega_+$, one follows the first steps of the proof of Theorem 5.2 when $\beta < d/2$. Indeed, all the results used there are proved for $\beta > 0$.

Step 2: Assume $S \in L_{t,x}^2 \dot{H}_v^{-\beta}$ and $\psi \in L_{x,v}^2$ and $0 < T < \infty$. As $L_{t,x}^2 \dot{H}_v^{-\beta} \subset L_{t,x}^2 H_v^{-\beta}$, we may apply Theorem 6.3 with inhomogeneous spaces. Hence, there exists a weak solution $f_{\varepsilon} \in L_{t,x}^2 H_v^{\beta} \cap L_{t,v}^2 H_x^{\frac{\beta}{2\beta+1}} \cap C([0, T]; L_{x,v}^2)$ of the kinetic Cauchy problem on $(0, T)$

$$ \begin{cases} (\partial_t + v \cdot \nabla_x) f_{\varepsilon} + \mathcal{A} f_{\varepsilon} + \varepsilon f_{\varepsilon} = e^{-\varepsilon t} S \\ f_{\varepsilon}(0) = \psi. \end{cases} $$
We now indicate to the reader how things should be adapted. We follow the scheme of our discussion and start with tools allowing us to prove Theorem 1.1 when $\beta \ge d/2$.

To begin with, the mapping properties of the Kolmogorov operators $K_{\beta}^{\pm}$ from Proposition 2.11 read

$$ (7.4) \quad K_{\beta}^{\pm} : L_{t,x}^{2} \dot{H}_{v}^{-\beta} + L_{t,v}^{2} \dot{H}_{x}^{-\frac{\beta}{2\beta+1}} + L_{t}^{1} L_{x,v}^{2} \to L_{t,x}^{2} \dot{H}_{v}^{\beta} \cap L_{t,v}^{2} \dot{H}_{x}^{\frac{\beta}{2\beta+1}} \cap C_{0}(\mathbb{R}_{t}; L_{x,v}^{2}). $$

The norm estimates that go with make sense for all $\beta > 0$ if we take into account that the target space is contained in $L_{\text{loc},t}^{2} L_{x,v}^{2}$.

Next, the proof of uniqueness (Lemma 3.1) holds with $\gamma = \beta$ and assuming $f \in L_{\text{loc},t}^{2} L_{x,v}^{2}$, see Remark 3.2.

The isomorphism properties of Lemma 3.3 hold at $\gamma = \beta \ge d/2$ with modification of the kinetic spaces at $\gamma = \beta$: replace $f \in D'(\Omega)$ by $f \in L_{\text{loc},t}^{2} L_{x,v}^{2}$, but keep the same norms.

The proof of Theorem 1.1 holds when $\beta \ge d/2$ assuming $f \in L_{\text{loc},t}^{2} L_{x,v}^{2}$ and taking into account the above modifications (it also uses the density Lemma 2.10 for exponent $-\beta$ which is unchanged).

Next, we come to weak solutions. We extend their definitions (Definition 1.9 on $\mathbb{R}$, Definition 5.1 on $(0, \infty)$ and its variant on $(0, T)$) when $\beta \ge d/2$ by requiring that a weak solution also belongs to $L_{\text{loc},t}^{2} L_{x,v}^{2}$.

Theorem 1.12 and Theorem 5.2 are valid with uniqueness class $L_{\text{loc},t}^{2} L_{x,v}^{2} \cap L_{t,x}^{2} \dot{H}_{v}^{\beta}$ when $\beta \ge d/2$. The proof of uniqueness uses Lemma 4.1 which is a mere corollary of Theorem 1.1. But now the proof of existence does not apply. Instead, we proceed the other way around and we adapt the method used to prove for existence in Theorem 6.7. That is, we start by solving the kinetic Cauchy problem on $(0, T)$ with inhomogeneous spaces for $(\partial_t + v \cdot \nabla_x) f_{\varepsilon} + A f_{\varepsilon} + \varepsilon f_{\varepsilon} = e^{-\varepsilon t} S$ when $S \in L_{t,x}^{2} \dot{H}_{v}^{-\beta}$ and $\psi \in L_{x,v}^{2}$. We observe that we have uniform bounds for $\|D_{v}^{\beta}(e^{\varepsilon t} f_{\varepsilon})\|_{L^{2}(0,T; L_{x,v}^{2})} + \sup_{t \in [0,T]} \|e^{\varepsilon t} f_{\varepsilon}(t)\|_{L_{x,v}^{2}}$ in $\varepsilon > 0$. Hence, we extract with $\varepsilon \to 0$ a weak*-limit $f^T \in L_{t}^{\infty} L_{x,v}^{2} \cap L_{t,x}^{2} \dot{H}_{v}^{\beta}$ that is a weak solution on $\Omega_{(0,T)}$ of $(\partial_t + v \cdot \nabla_x) f^T + A f^T = S$ with same data. Then, the compatibility for different $T$ implied by uniqueness allows us to take $T = \infty$, giving a proof for Theorem 5.2.

Now, we push backward the initial time to $-\infty$ with zero initial condition to obtain, again with a weak*-limit argument, a weak solution in $\Omega$.

Eventually, we consider sources in $L_{t,v}^{2} \dot{H}_{x}^{-\frac{\beta}{2\beta+1}} \cap L_{t}^{1} L_{x,v}^{2}$ on $\Omega$ first by the duality scheme and next on $\Omega_{+}$ using the causality principle. This finishes the proof for Theorem 1.12 when $\beta \ge d/2$.

The operator $K_A$ makes sense even when $\beta \ge d/2$ with the modification of the definition of weak solutions. Then Corollary 4.3 (boundedness and isomorphism) extends to $\beta \ge d/2$ with the modification of the homogeneous kinetic spaces as described above. Causality in Corollary 5.3 is still valid with this new definition.

**Remark 7.3.** When $\beta = 1$, the weak solutions built in the above paragraph are the same as the ones in Theorem 6.7 when $d \ge 1$.

### 7.2.2. Concerning scales of homogeneous kinetic spaces.
A non-optimal way to address the case $\gamma \ge d/2$ in Theorem 2.2 is to impose in the definition of kinetic spaces
Rewriting the equation as $(\partial_t + v \cdot \nabla_x)(e^{\epsilon t} f_\epsilon) + \mathcal{A}(e^{\epsilon t} f_\epsilon) = S$ and using the energy equality on all subintervals $[0, \tau]$ leads to the bound

$$ \|D_v^\beta (e^{\epsilon t} f_\epsilon)\|_{L^2(0,T; L_{x,v}^2)}^2 + \sup_{t \in [0,T]} \|e^{\epsilon t} f_\epsilon(t)\|_{L_{x,v}^2}^2 \lesssim \|S\|_{L_{t,x}^2 \dot{H}_v^{-\beta}}^2 + \|\psi\|_{L_{x,v}^2}^2 $$

with uniform implicit constant with respect to $\epsilon > 0$ and $T$. Hence, we can extract when $\epsilon \to 0$ a weak*-limit for the sequence $e^{\epsilon t} f_\epsilon \in L_t^\infty L_{x,v}^2 \cap L_{t,x}^2 \dot{H}_v^\beta$ and obtain a weak solution $f^T \in \dot{U}^\beta$ in $\Omega_{(0,T)}$ of $(\partial_t + v \cdot \nabla_x)f^T + \mathcal{A}f^T = S$ with same initial data $\psi$. Observe that this solution satisfies

$$ \|D_v^\beta f^T\|_{L^2(0,T; L_{x,v}^2)}^2 + \sup_{t \in [0,T]} \|f^T(t)\|_{L_{x,v}^2}^2 \lesssim \|S\|_{L_{t,x}^2 \dot{H}_v^{-\beta}}^2 + \|\psi\|_{L_{x,v}^2}^2 $$

with constant uniform with respect to $T$. Then, the compatibility for different $T$ implied by uniqueness allows us to define $f$ in $\Omega_+$ as $f = f^T$ on $\Omega_{(0,T)}$. This $f$ is a weak solution on $\Omega_+$ with initial data $\psi$, and it satisfies the above bound for $T = \infty$ (by a monotone convergence argument as $T \uparrow \infty$). In particular, $f \in \dot{U}^\beta$ in $\Omega_+$.

**Step 3:** Assume $S \in L_{t,x}^2 \dot{H}_v^{-\beta}$ on $\Omega$ and $\psi = 0$. We can apply Step 2 on any interval $[-k, \infty)$, $k \in \mathbb{N}$, with zero initial condition and push backward the initial time $-k$ to $-\infty$ to obtain, again with a weak*-limit argument, using uniform bounds with respect to $k$, a weak solution $f \in \dot{U}^\beta$ in $\Omega$.

**Step 4:** Existence and uniqueness show that one can define $\mathcal{K}_A^+ : L_{t,x}^2 \dot{H}_v^{-\beta} \to \dot{U}^\beta$ as $\mathcal{K}_A^+ S$ is the unique solution $f \in \dot{U}^\beta$ to $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$ on $\Omega$. It follows that $f \in \dot{L}_\beta^\beta$. By Theorem 2.2, $f \in L_{t,x}^2 \dot{H}_v^\beta \cap L_{t,x}^2 \dot{H}_x^{\frac{2\beta+1}{\beta}} \cap C_0(\mathbb{R}_t; L_{x,v}^2)$. This establishes the desired boundedness

$$ \mathcal{K}_A^+ : L_{t,x}^2 \dot{H}_v^{-\beta} \to \dot{Y}^\beta. $$

**Step 5:** We wish to apply the duality scheme as in an earlier argument, but we have to do it with care. The result of Step 4 applies to the backward equation for $\mathcal{A}^*$ and yields

$$ \mathcal{K}_{A^*}^- : L_{t,x}^2 \dot{H}_v^{-\beta} \to \dot{Y}^\beta. $$

Note that if $S \in \mathcal{S}_K$ and $h \in L_{t,x}^2 \dot{H}_v^{-\beta}$, then

$$ |\langle \mathcal{K}_A^+ S, h \rangle| = |\langle S, \mathcal{K}_{A^*}^- h \rangle| \le \|S\|_{\dot{Z}^\beta} \|\mathcal{K}_{A^*}^- h\|_{\dot{Y}^\beta} \lesssim \|S\|_{\dot{Z}^\beta} \|h\|_{L_{t,x}^2 \dot{H}_v^{-\beta}} $$

using Step 4 in the last inequality. Consequently, we can interpret $\mathcal{K}_A^+ S$ as a bounded linear functional on $L_{t,x}^2 \dot{H}_v^{-\beta}$, hence in $(L_{t,x}^2 \dot{H}_v^{-\beta})'$, with norm controlled by $\|S\|_{\dot{Z}^\beta}$. Since $\mathcal{S}_K \subset L_{t,x}^2 \dot{H}_v^{-\beta}$, we know by Step 4 that $\mathcal{K}_A^+ S \in \dot{Y}^\beta \subset \dot{U}^\beta$, the completion of which for the semi-norm $\| \cdot \|_{\dot{U}^\beta} = \| \cdot \|_{L_{t,x}^2 \dot{H}_v^\beta}$ is that dual space. In particular, for any $f \in \dot{U}^\beta$, $\|f\|_{\dot{U}^\beta} = \|f\|_{(L_{t,x}^2 \dot{H}_v^{-\beta})'}$ under identification. It follows that

$$ \|\mathcal{K}_A^+ S\|_{\dot{U}^\beta} \lesssim \|S\|_{\dot{Z}^\beta}. $$

From the equation we deduce that $\mathcal{K}_A^+ S \in \dot{L}_\beta^\beta$ with

$$ \|\mathcal{K}_A^+ S\|_{\dot{L}_\beta^\beta} \lesssim \|S\|_{\dot{Z}^\beta}. $$
the condition $f \in L_{\text{loc},t}^2 L_{x,v}^2$, keeping the norms unchanged. Then, homogeneous in-equalities in (i) and (ii) remain valid. Density in (iii) also holds. But interpolation is probably lost (when $\gamma \ge d/2$) as is completeness. The proof and conclusion of Theorem 3.6 with this modification are also valid.

However, to modify Lemma 3.3 we need a definition ensuring completeness and this is not clear.

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Now, as noted in Corollary 3.6, $\dot{\mathcal{L}}_{\beta}^{\beta}$ is a complete normed space, so arguing by density of $S_K$ in $\dot{Z}^{\beta}$, we conclude that $K_{\mathcal{A}}^{+}$ extends boundedly from $\dot{Z}^{\beta}$ to $\dot{\mathcal{L}}_{\beta}^{\beta}$, which is contained in $\dot{U}^{\beta}$. Eventually, this extension maps $\dot{Z}^{\beta}$ into weak solutions by passing to the limit in the weak formulation of the equation. This completes the proof of existence on $\Omega$ for source terms in $\dot{Z}^{\beta}$.

**Step 6:** The causality principle follows again from the energy equality and allows one to construct weak solutions on $\Omega_{+}$ as in Step 7 of the proof of Theorem 5.2 when $\beta < d/2$. This concludes the argument, and also proves Corollary 5.3 when $\beta \ge d/2$.

**Remark 7.1.** As we have now fully proved Theorem 5.2 when $\beta > 0$, we may go back to Theorem 6.3 in the case $c_0 = c^0 = 0$ in (6.1) and (6.2). Assume $S \in L_{t,x}^2 \dot{H}_v^{-\beta} + L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ on $\Omega_{(0,T)}$ and $\psi \in L_{x,v}^2$. We have nothing else to do than restrict the construction on $[0, \infty)$ to $[0, T]$, having extended $S$ by 0 for $t \ge T$. The weak solution enjoys the homogeneous estimates in $L_{t,x}^2 \dot{H}_v^{\beta} \cap L_{t,v}^2 \dot{H}_x^{\frac{\beta}{2\beta+1}} \cap C_0([0, T]; L_{x,v}^2)$ with bounds independent of $T$ if $S$ is normalized. It also has an $L_{t,x,v}^2$ bound on $\Omega_{(0,T)}$ but this bound blows up with $T$.

## 8. THE SPECIAL CASE OF LOCAL DIFFUSION OPERATORS

In this section, we focus on $\beta = 1$ and give different definitions of the homogeneous spaces that are valid in any dimension $d \ge 1$.

Define $\dot{W}^{1,2}(\mathbb{R}^d) = \{f \in \mathcal{D}'(\mathbb{R}^d); \nabla f \in L^2(\mathbb{R}^d)\}$, equipped with the semi-norm $\|\nabla f\|_{L^2}$. Remark that $\dot{H}^1(\mathbb{R}^d) \subset \dot{W}^{1,2}(\mathbb{R}^d)$ when $d \ge 3$. The inhomogeneous version is $W^{1,2}(\mathbb{R}^d) = \{f \in L^2(\mathbb{R}^d); \nabla f \in L^2(\mathbb{R}^d)\}$ equipped with the norm $(\|f\|_{L^2}^2 + \|\nabla f\|_{L^2}^2)^{1/2}$. It agrees with $H^1(\mathbb{R}^d)$ when $d \ge 1$. Hence, our inhomogeneous theory is unchanged and we only have to address the homogeneous one.

**Lemma 8.1.** Any element of $\dot{W}^{1,2}(\mathbb{R}^d)$ identifies to an element in $L_{\text{loc}}^2(\mathbb{R}^d) \cap \mathcal{S}'(\mathbb{R}^d)$, and $\mathcal{D}(\mathbb{R}^d)$ is dense in $\dot{W}^{1,2}(\mathbb{R}^d)$ for the semi-norm.

*Proof.* This is known. Here is a proof for the comfort of the reader. Let $f \in \dot{W}^{1,2}(\mathbb{R}^d)$. As $f$ is a distribution, we may regularise it to $f_{\varepsilon}$ in this space by mollification, and $f_{\varepsilon} \in C^{\infty}$. Use Poincaré inequalities for $f_{\varepsilon}$ and limiting arguments to deduce that $f \in L_{\text{loc}}^2(\mathbb{R}^d)$. Poincaré inequalities again give a polynomial growth in $R$ of the $L^2$ norms of $f$ on balls $B(0, R)$. Thus $f$ is a tempered distribution as well. Lastly, for the density, we claim that we can approximate $f$ by compactly supported functions in the $\|\nabla f\|_2$ semi-norm. Pick $f_k(x) = (f(x) - c_k)\varphi(2^{-k}x)$, $k \ge 1$, with $\varphi$ a smooth function supported on the ball $B(0, 2)$ and that is 1 on the unit ball and where $c_k$ is the mean of $f$ on the ball $B(0, 2^{k+1})$. It is easy to see that $\nabla f_k$ converges weakly to $\nabla f$ in $L^2$ and by Mazur's lemma, there exist convex combinations of subsequences of $\nabla f_k$ having strong convergence to $\nabla f$ in $L^2$. From there, use mollification to obtain an approximation in $\mathcal{D}(\mathbb{R}^d)$. $\square$

We introduce the kinetic space

$$ (8.1) \qquad \dot{\mathcal{L}} = \{f \in \mathcal{D}'(\Omega); f \in L_{t,x}^2 \dot{W}_v^{1,2} \ \& \ (\partial_t + v \cdot \nabla_x)f \in L_t^2 \dot{X}_1^{-1} + L_t^1 L_{x,v}^2\} $$
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UNIVERSITÉ PARIS-SACLAY, CNRS, LABORATOIRE DE MATHÉMATIQUES D'ORSAY, 91405 ORSAY, FRANCE

Email address: pascal.auscher@universite-paris-saclay.fr

DÉPARTEMENT DE MATHÉMATIQUES ET APPLICATIONS, ÉCOLE NORMALE SUPÉRIEURE, UNIVERSITÉ PSL, CNRS, 75005 PARIS, FRANCE

Email address: cyril.imbert@ens.psl.eu

INSTITUT FÜR ANALYSIS UND NUMERIK, WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER, ORLÉANS-RING 10, 48149 MÜNSTER, GERMANY.

Email address: lukas niebel@uni-muenster.de
with semi-norm $||f||_{\dot{L}}$ defined by

$$ ||f||_{\dot{L}} = ||\nabla_v f||_{L^2_{t,x,v}} + ||(\partial_t + v \cdot \nabla_x)f||_{L^2_t \dot{X}_1^{-1} + L^1_t L^2_{x,v}} $$

The spaces with negative indices are defined as before and $\dot{H}_v^{-1}$ identifies to the dual of $\dot{W}_v^{1,2}$, so that we also write $\dot{H}_v^{-1} = \dot{W}_v^{-1,2}$ and $\dot{X}_1^{-1} = L_x^2 \dot{W}_v^{-1,2} + L_v^2 \dot{H}_x^{-1/2}$. Note that the $f \in L_{t,x}^2 \dot{W}^{1,2}$ condition excludes non-zero constants (in the first version of this article, we wrote the opposite, which would be true if we replace this condition by $\nabla f \in L_{t,x,v}^2$). Hence, all results in this section are modified accordingly.

Redoing for the Kolmogorov operators $K^{\pm} := K_1^{\pm}$ the proof in Corollary 2.18, we can state the following boundedness result.

**Proposition 8.2** (Boundedness of the Kolmogorov operators). *We have the bounded map*

$$ K^{\pm}: L_{t,x}^2 \dot{W}_v^{-1,2} + L_{t,v}^2 \dot{H}_x^{-1/2} + L_t^1 L_{x,v}^2 \to L_{t,x}^2 \dot{W}_v^{1,2} \cap L_{t,v}^2 \dot{H}_x^{1/2} \cap C_0(\mathbb{R}_t; L_{x,v}^2). $$

**Corollary 8.3** (Uniqueness). *Let $f \in D'(\Omega)$ for which $\nabla_v f \in L_{t,x,v}^2$. Then $\pm(\partial_t + v \cdot \nabla_x)f - \Delta_v f = 0$ in the sense of distributions implies that $f$ is constant. If we further impose $f \in L_{t,x}^2 \dot{W}_v^{1,2}$ then $f = 0$.*

*Proof.* Assume $(\partial_t + v \cdot \nabla_x)f - \Delta_v f = 0$ as the other case is similar. We follow the strategy of Lemma 3.1 and the notation there. By Lemma 8.1, $f \in L_{t,x}^2 S'_v \subset S'(\Omega)$ but $\hat{f}$ may not be locally integrable on $\Omega$. Still, one can solve the ordinary differential equation in a distributional sense and obtain that $G = K(t, 0)G_0$ in $D'(\Omega)$ with $G_0 \in D'_{\varphi, \xi}$. Next the condition $(\xi - t\varphi)G \in L_{t,\varphi,\xi}^2$ implies that the restriction of $G$ to the complement of $\{\xi - t\varphi = 0\}$ is locally square integrable. This forces $G_0$ to be supported at the origin $(\varphi, \xi) = (0, 0)$. Indeed, pick a compact set $C$ in $\mathbb{R}_{\varphi, \xi}^{2d} \setminus \{(0, 0)\}$. Then one can find a compact interval $I \subset \mathbb{R}_t$ such that $\min |\xi - t\varphi|^2 > 0$ on $I \times C$. Thus $G \in L^2(I \times C)$ and, using the equation, we see that $G \in C(I; L^2(C))$. This implies that $G_0 \in L^2(C)$. Using again $(\xi - t\varphi)G \in L_{t,\varphi,\xi}^2$, this gives us

$$ \iiint_{\mathbb{R} \times C} |\xi - t\varphi|^2 \exp \left( -2 \int_0^t |\xi - \tau\varphi|^2 d\tau \right) |G_0(\varphi, \xi)|^2 dt d\varphi d\xi < \infty, $$

which yields $G_0 = 0$ on $C$. This proves that $G$ is supported in $\mathbb{R}_t \times \{(0, 0)\}$. This implies $(\xi - t\varphi)G = 0$ as a distribution because it is locally integrable in $\Omega$ and supported in the null Lebesgue set $\mathbb{R}_t \times \{(0, 0)\}$. Thus $\partial_t G = 0$ from the equation. Differentiating $\xi G = t\varphi G$ with respect to $t$ leads to $\varphi G = 0$, and consequently $\xi G = 0$. This implies that $G = T_0 \otimes \delta$ where $\delta$ is the Dirac mass at the origin in $\mathbb{R}_{\varphi, \xi}^{2d}$ and $T_0 \in D'(\mathbb{R}_t)$. One uses again $\partial_t G = 0$ to conclude that $T_0$ is constant. Going back to $f$, we conclude that $f$ is constant. And this constant must be 0 if $f \in L_{t,x}^2 \dot{W}_v^{-1,2}$. $\square$

**Proposition 8.4.** *The Kolmogorov operators $K^{\pm}$ are isomorphisms from $L_{t,x}^2 \dot{W}_v^{-1,2}$ onto $\dot{F}$, from $L_t^2 \dot{X}_1^{-1}$ onto $\dot{G}$ and from $L_t^2 \dot{X}_1^{-1} + L_t^1 L_{x,v}^2$ onto $\dot{L}$, using quotient norms, where the spaces $\dot{F}$ and $\dot{G}$ are defined similarly to $\dot{L}$ replacing $L_{t,x}^2 \dot{H}_v^1$ by $L_{t,x}^2 \dot{W}_v^{1,2}$. In particular, $\dot{F}$, $\dot{G}$ and $\dot{L}$ are complete normed spaces.*

*Proof.* This follows from the estimates and uniqueness, as in Lemma 3.5. $\square$
**Corollary 8.5.** Let $f \in \dot{\mathcal{L}}$. Then $f \in C_0(\mathbb{R}_t; L_{x,v}^2) \cap L_{t,v}^2 \dot{H}_x^{\frac{1}{3}}$ with estimate

$$ \|f\|_{L_{t,v}^2 \dot{H}_x^{\frac{1}{3}}} + \sup_{t \in \mathbb{R}} \|f(t)\|_{L_{x,v}^2} \lesssim_d \|f\|_{\dot{\mathcal{L}}} $$

The space in item (iii) of Theorem 2.2 is dense in $\dot{\mathcal{L}}$. Moreover, for $f \in \dot{\mathcal{L}}$, the absolute continuity stated in Theorem 1.16 (and its generalization Theorem 3.9) holds.

*Proof.* Follow the strategy of proof of Theorem 2.2. Details are left to the reader. □

**Remark 8.6.** One can now see that $\dot{\mathcal{L}} = \dot{\mathcal{L}}_1^1$ when $d \ge 3$. It is only because we have not defined $\dot{H}_v^1$ when $d = 1, 2$ that we do not have a statement for $\dot{\mathcal{L}}_1^1$ in that case, see [5].

Let us turn to the construction of weak solutions. The difficulty with this homogeneous space here is that $L_{t,x}^2 \dot{W}_v^{1,2}$ is not a Hilbert space, so that we cannot apply the Lions theorem directly. We use an approximation procedure starting from the inhomogeneous case. We let $A = A(t, x, v)$ be a bounded and coercive matrix with measurable coefficients (real and symmetric is not necessary) in such a way that

$$ a_{t,x}(f,g) = \int_{\mathbb{R}^d} \langle A(t,x,v) \nabla_v f(v), \nabla_v g(v) \rangle \, dv, \quad f,g \in \dot{W}_v^{1,2}, $$

satisfies (1.4) and (1.5) on $\dot{W}_v^{1,2}$. We define $\mathcal{A}$ as in (1.2) with the form $a$ defined by (1.6) on $L_{t,x}^2 \dot{W}_v^{1,2}$. Here we assume either $t \in \mathbb{R}$ or $t \in (0, \infty)$.

**Theorem 8.7.** (i) On $\mathbb{R}$, if $S \in L_t^2 \dot{X}_1^{-1} + L_t^1 L_{x,v}^2$, then there is a unique weak solution $f \in L_{t,x}^2 \dot{W}_v^{1,2}$ to $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$. We have $f \in C_0(\mathbb{R}_t; L_{x,v}^2) \cap L_{t,v}^2 \dot{H}_x^{\frac{1}{3}}$ and satisfies the energy equality.

(ii) On $[0, \infty)$, if $S \in L_t^2 \dot{X}_1^{-1} + L_t^1 L_{x,v}^2$ and $\psi \in L_{x,v}^2$, then there is a unique weak solution in $L_{t,x}^2 \dot{W}_v^{1,2}$ to the kinetic Cauchy problem $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$ with $f(0) = \psi$. We have $f \in C_0([0, \infty); L_{x,v}^2) \cap L_{t,v}^2 \dot{H}_x^{\frac{1}{3}}$ and satisfies the energy equality.

*Proof.* We start with uniqueness. In case (i), a weak solution $f \in L_{t,x}^2 \dot{W}_v^{1,2}$ to $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$ belongs to $\dot{\mathcal{L}}$. Thus, $f \in C_0(\mathbb{R}_t; L_{x,v}^2)$ and satisfies the energy equality. If we assume $S = 0$, then for all $s < t$,

$$ \|f(t)\|_{L_{x,v}^2}^2 - \|f(s)\|_{L_{x,v}^2}^2 = -2 \operatorname{Re} \int_s^t \int_{\mathbb{R}^d} a_{\tau,x}(f,f) \, dx \, d\tau. $$

In particular, this yields after taking limits at $\pm\infty$, $\operatorname{Re} \int_{-\infty}^{+\infty} \int_{\mathbb{R}^d} a_{\tau,x}(f,f) \, dx \, d\tau = 0$, hence $\operatorname{Re} a_{\tau,x}(f,f) = 0$ for almost all $(\tau, x)$ (because it is non negative). This yields $\|f(t)\|_{L_{x,v}^2}^2 - \|f(s)\|_{L_{x,v}^2}^2 = 0$ for all $s < t$, that is $t \mapsto \|f(t)\|_{L_{x,v}^2}^2$ is constant. As it has zero limits at $\pm\infty$, $f = 0$.

In case (ii), we can produce the same argument following Steps 1 and 2 in the proof of Theorem 5.2 to show that on $\Omega_+$, for all $f \in \dot{\mathcal{L}}$,

$$ \|f\|_{C_0([0,\infty); L_{x,v}^2)} \lesssim_d \|f\|_{\dot{\mathcal{L}}} $$
Now consider a weak solution $f$ with $S = 0$ and $\psi = 0$. As we assume that $f(0) = 0$ at time $0$, the energy equality yields for $t > 0$, $\|f(t)\|_{L_{x,v}^2}^2 = -2 \operatorname{Re} \int_0^t \int_{\mathbb{R}^d} a_{\tau,x}(f, f) dx d\tau \le 0$, thus $f(t) = 0$.

Now we turn to existence. It follows the argument done in Section 7. We give details for completeness. We begin with (ii) in the case where $S \in L_{t,x}^2 \dot{W}_v^{-1,2}$ and $\psi \in L_{x,v}^2$. We prove existence on any finite strip $\Omega_{(0,T)} = (0, T) \times \mathbb{R}^d \times \mathbb{R}^d$ of a weak solution $f^T$ with $\nabla_v f^T \in L^2(\Omega_{(0,T)})$ and $f^T \in C([0, T]; L_{x,v}^2)$. Assume this is done. Such solutions also verify the energy equality. If $T < T'$ and $g = f^{T'} - f^T$, then $g$ is a weak solution in the same class on $[0, T]$ with zero source and zero initial data. The energy equality yields when $t \in [0, T]$, $\|g(t)\|_{L_{x,v}^2}^2 = -2 \operatorname{Re} \int_0^t \int_{\mathbb{R}^d} a_{\tau,x}(g, g) dx d\tau \le 0$, thus $g(t) = 0$. Since $f^T$ and $f^{T'}$ agree on the smaller strip, this allows us to define $f$ on $\Omega_+$ and $f$ is a weak solution to the kinetic Cauchy problem in (ii) with the desired properties.

To construct $f^T$, we argue using the inhomogeneous setup on $[0, T]$ that is now fixed, and we work on this interval. For $\varepsilon > 0$, using Theorem 6.3 with $\mathcal{A} + \varepsilon$, we solve in $L_{t,x}^2 W_v^{1,2}$ the kinetic Cauchy problem $(\partial_t + v \cdot \nabla_x)f_\varepsilon + \mathcal{A}f_\varepsilon + \varepsilon f_\varepsilon = e^{-\varepsilon t}S$ with $f_\varepsilon(0) = \psi$. Note that $(\partial_t + v \cdot \nabla_x)(e^{\varepsilon t}f_\varepsilon) + \mathcal{A}(e^{\varepsilon t}f_\varepsilon) = S$ and $(e^{\varepsilon t}f_\varepsilon)(0) = \psi$. Using the energy equality for $e^{\varepsilon t}f_\varepsilon$ (absolute continuity follows from that of $f_\varepsilon$) and the fact that $S \in L_x^2 \dot{W}_v^{-1,2}$, we can see that $\|\nabla_v(e^{\varepsilon t}f_\varepsilon)\|_{L_{t,x,v}^2}$ and $\sup_{t \in [0,T]} \|e^{\varepsilon t}f_\varepsilon(t)\|_{L_{x,v}^2}$ are uniformly bounded with respect to $\varepsilon$. Hence, a classical weak*-limit argument furnishes a weak solution $f^T \in L_t^\infty L_{x,v}^2 \cap L_{t,x}^2 W_v^{1,2}$. As $f^T$ belongs to the inhomogeneous version $\mathcal{L}$ of $\dot{\mathcal{L}}$ on $\Omega_{(0,T)}$, continuity in time follows.

Now, we turn to existence in (i) when $S \in L_{t,x}^2 \dot{W}_v^{-1,2}$ on $\Omega$ this time. By applying (ii) with time interval $[-n, \infty)$, $n \in \mathbb{N}$, instead on $[0, \infty)$, consider the weak solution $f_n$ on $[-n, \infty)$ to the kinetic Cauchy problem $(\partial_t + v \cdot \nabla_x)f_n + \mathcal{A}f_n = S$ and $f_n(-n) = 0$. The extension of this solution by $0$ when $t \le -n$ belongs to $L_{t,x}^2 \dot{W}_v^{1,2} \cap C_0(\mathbb{R}_t; L_{x,v}^2) \cap L_{t,x}^2 \dot{H}_x^{\frac{1}{3}}$ on $\Omega$, uniformly in $n$. Extract a weak*-limit in $L_{t,x}^2 \dot{W}_v^{1,2} \cap L_t^\infty L_{x,v}^2 \cap L_{t,x}^2 \dot{H}_x^{\frac{1}{3}}$. This weak*-limit is a weak solution on $\mathbb{R}$ and as it is already a bounded function of time valued in $L_{x,v}^2$, it further belongs to $C_0(\mathbb{R}_t; L_{x,v}^2)$ using the embedding in Corollary 8.5.

It remains to solve the problems for source terms in $L_{t,x}^2 \dot{H}_x^{-\frac{1}{3}} \cap L_t^1 L_{x,v}^2$. For this, we use the duality scheme in Section 7 first on $\mathbb{R}$ starting from the operator

$$ \tilde{\mathcal{K}}_{\mathcal{A}^*}^{-} : L_{t,x}^2 \dot{W}_v^{-1,2} \to \dot{Y}^\beta \hookrightarrow \dot{\mathcal{L}} $$

defined as $\tilde{\mathcal{K}}_{\mathcal{A}^*}^{-} S$ is the weak solution for the backward equation, as we just did for the forward equation, and using that $\dot{\mathcal{L}}$ is a complete normed space. Next, on $(0, \infty)$ with zero initial data, we use this previous case and the causality principle as in Step 7 of the proof of Theorem 5.2. $\square$

**Remark 8.8.** When $d \ge 3$, as $L_{t,x}^2 \dot{H}_v^1 \subset L_{t,x}^2 \dot{W}_v^{1,2}$, uniqueness implies that the weak solutions in Theorem 8.7, (ii), and in Theorem 5.2 when $\beta = 1$ are the same. Similarly, the weak solution $f$ in Theorem 8.7 (i) is the same as the one in Theorem 1.5. This holds as well when $d = 1, 2$ provided one defines $\dot{H}_v^1$ appropriately, see [5].
Let us illustrate our results by a consequence for optimal regularity, to be compared with [10, Theorem 1.5] where $f$ and $\nabla_v f$ were both supposed to belong to $L_{t,x,v}^2$ for the conclusion $\Delta_v f \in L_{t,x,v}^2$.

**Corollary 8.9 (Optimal regularity).** Let $f \in D'(\Omega)$ with $\sup_{\sigma \in \mathbb{R}} \|\nabla_v f\|_{L^2(\Omega_{(\sigma, \sigma+1)})} < \infty$ (in particular, if $\nabla_v f \in L_{t,x,v}^2$) be such that $S = (\partial_t + v \cdot \nabla_x)f - \Delta_v f \in L_{t,x,v}^2$. Then, $(\partial_t + v \cdot \nabla_x)f, \Delta_v f \in L_{t,x,v}^2$ and $\nabla_v f \in C_0(\mathbb{R}_t; L_{x,v}^2)$.

*Proof.* Again, the estimates obtained using Fourier methods in Proposition 2.19 imply in particular the boundedness

$$ \mathcal{K}^{\pm}: L_{t,x,v}^2 \to L_{t,x}^2 \dot{W}_v^{2,2} \cap C_0(\mathbb{R}_t; L_x^2 \dot{W}_v^{1,2}) $$

where $\dot{W}^{2,2}(\mathbb{R}^d) = \{f \in D'(\mathbb{R}^d); \nabla^2 f \in L^2(\mathbb{R}^d)\}$. Thus, $\mathcal{K}^+S$ is a tempered distribution solution of the same equation as $f$. We show that $f - \mathcal{K}^+S$ is constant, which concludes the proof from the properties of $\mathcal{K}^+S$.

Remark that from the above bounds $\nabla_v \mathcal{K}^+S$ satisfies the same condition as $\nabla_v f$ in the hypothesis. Applying the method of proof of Corollary 8.3 to $f - \mathcal{K}^+S$ and calling $G$ the Fourier transform of $\Gamma(f - \mathcal{K}^+S)$, the only thing that changes is the condition $(\xi - t\varphi)G \in L_{t,\varphi,\xi}^2$ replaced by $(\xi - t\varphi)G \in L^2(\sigma, \sigma + 1; L_{\varphi,\xi}^2)$ uniformly in $\sigma \in \mathbb{R}$. This is enough to conclude that the restriction of $G_0$ to any compact set $C \subset \mathbb{R}^{2d} \setminus \{(0,0)\}$ is square integrable and that

$$ \sup_{\sigma \in \mathbb{R}} \iiint_{(\sigma, \sigma+1) \times C} |\xi - t\varphi|^2 \exp \left( -2 \int_0^t |\xi - \tau\varphi|^2 d\tau \right) |G_0(\varphi, \xi)|^2 dt d\varphi d\xi < \infty. $$

By taking $\sigma \to -\infty$, this is only possible if $G_0 = 0$ almost everywhere on $C$. The rest of the proof is verbatim the one of Corollary 8.3. $\square$

**Remark 8.10.** At the expense of more technicalities, one can make the same discussion with the higher-order homogeneous space $\dot{W}^{m,2}(\mathbb{R}^d) = \{f \in D'(\mathbb{R}^d); \nabla^m f \in L^2(\mathbb{R}^d)\}$, $m$ integer, which contains polynomials of degree less than $m$, applying this to higher order Kolmogorov operators $\mathcal{K}_m^{\pm}$. For weak solutions, ellipticity (1.5) should be in the sense of a Gårding inequality on $\dot{W}_v^{m,2}$. Embedding (on $\mathbb{R}$ or $[0, \infty)$) and uniqueness of weak solutions on $\mathbb{R}$ can be proved.

## 9. LINK WITH OTHER KINETIC EMBEDDINGS

As mentioned in the introduction, there is a notion of kinetic spaces developed in [32] where embeddings similar to ours are proved without restriction on $\gamma$, while we are limited to $0 \le \gamma \le 2\beta$. It is worth explaining the links between the two works. For this, we need to introduce more spaces and make some extensions (mostly without proofs, which are adaptations of the previous arguments).

### 9.1. Homogeneous source spaces revisited.
We first extend the range of parameter $\gamma$ in our spaces $\dot{X}_\beta^\gamma$. We just mention at this stage that we did not do it in the article because it would not have helped to prove the existence of weak solutions in Theorem 1.5, where we only impose ellipticity in the $v$-variable.
In the proof of Lemma 2.10, we established the following isomorphism via the Fourier transform in $\mathbb{R}^{2d}$

$$ \dot{X}_{\beta}^{\gamma} \rightarrow L_{\varphi, \xi}^{2}(w^{2\gamma} d\varphi d\xi), $$

where $w(\varphi, \xi) = \sup(|\xi|, |\varphi|^{1/(1+2\beta)})$, provided $\gamma < d/2$. When $\gamma > 0$, the only reason for the restriction $\gamma < d/2$ is our definition of $\dot{X}_{\beta}^{\gamma}$ as the intersection of two spaces, separating the variables in some sense. We remark that a simple calculation shows for $\gamma \in \mathbb{R}$,

$$ w^{-\gamma} \in L_{\text{loc}}^{2}(\mathbb{R}^{2d}) \iff \gamma < (\beta + 1)d = \frac{(2\beta + 2)d}{2}. $$

Note that $(2\beta+2)d$ is the anisotropic homogeneous dimension for the scaling $(x, v) \mapsto (\delta^{2\beta+1}x, \delta v)$. This allows us to define $\dot{X}_{\beta}^{\gamma}$ for $\gamma < (\beta+1)d$ in the spirit of homogeneous Sobolev spaces by

$$ \dot{X}_{\beta}^{\gamma} = \{f \in S'(\mathbb{R}^{2d}) ; \exists g \in L^2(\mathbb{R}^{2d}), \hat{f} = w^{-\gamma} \hat{g}\} $$

with norm $\|f\|_{\dot{X}_{\beta}^{\gamma}} = \|g\|_{L_{\varphi, \xi}^{2}}$. In this range, it is a well-defined Hilbert space, contained in $S'(\mathbb{R}^{2d})$ with elements having Fourier transforms in $L_{\text{loc}}^{1}(\mathbb{R}^{2d})$.<fnref n="3" /> When $\gamma < d/2$, this space is equal, as a set and with equivalent norm, to the one defined in (2.2) and (2.3).

With this definition, our proofs on the Fourier transform side (where there is no restriction on exponents) show that the statements of Lemma 2.10, Proposition 2.16 hold for $\gamma < (\beta + 1)d$. Also the uniqueness as in Lemma 3.1 applies with $f \in L_t^2 \dot{X}_{\beta}^{\gamma}$ and $\gamma < (\beta + 1)d$, thanks to Remark 3.2.

**9.2. More homogeneous kinetic spaces.** We remark that even with this new definition of $\dot{X}_{\beta}^{\gamma}$, the upper bound on $\gamma$ in the definitions of our kinetic spaces $\dot{\mathcal{F}}_{\beta}^{\gamma}$, $\dot{\mathcal{G}}_{\beta}^{\gamma}$, $\dot{\mathcal{L}}_{\beta}^{\gamma}$ is $d/2$ because we impose them to be subspaces of $L_{t,x}^2 \dot{H}_{v}^{\gamma}$. Nevertheless, changing this condition to $f \in L_t^2 \dot{X}_{\beta}^{\gamma}$ allows to introduce when $\gamma < (\beta + 1)d$,

$$ (9.1) \qquad \dot{T}_{\beta}^{\gamma} = \{f \in \mathcal{D}'(\Omega) ; f \in L_t^2 \dot{X}_{\beta}^{\gamma} \ \& \ (\partial_t + v \cdot \nabla_x) f \in L_t^2 \dot{X}_{\beta}^{\gamma-2\beta}\} $$

with Hilbertian norm $\|f\|_{\dot{T}_{\beta}^{\gamma}}$ defined by

$$ \|f\|_{\dot{T}_{\beta}^{\gamma}}^{2} = \|f\|_{L_{t}^{2} \dot{X}_{\beta}^{\gamma}}^{2} + \|(\partial_{t} + v \cdot \nabla_{x})f\|_{L_{t}^{2} \dot{X}_{\beta}^{\gamma-2\beta}}^{2}, $$

and the larger space

$$ (9.2) \qquad \dot{U}_{\beta}^{\gamma} = \{f \in \mathcal{D}'(\Omega) ; f \in L_t^2 \dot{X}_{\beta}^{\gamma} \ \& \ (\partial_t + v \cdot \nabla_x) f \in L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{X}_{\beta}^{\gamma-\beta}\} $$

with norm $\|f\|_{\dot{U}_{\beta}^{\gamma}}$ defined by

$$ \|f\|_{\dot{U}_{\beta}^{\gamma}} = \|f\|_{L_t^2 \dot{X}_{\beta}^{\gamma}} + \|(\partial_t + v \cdot \nabla_x) f\|_{L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{X}_{\beta}^{\gamma-\beta}} $$

Note that the $x$-regularity is already included in the definition of these spaces, hence $\dot{T}_{\beta}^{\gamma} \subset \dot{\mathcal{G}}_{\beta}^{\gamma}$ and $\dot{U}_{\beta}^{\gamma} \subset \dot{\mathcal{L}}_{\beta}^{\gamma}$ when $0 \le \gamma < d/2$. When $\gamma \le 0$, the opposite inclusions hold.

There is an immediate observation that $(-\Delta_v)^\beta$ maps $L_t^2 \dot{X}_{\beta}^{\gamma}$ into $L_t^2 \dot{X}_{\beta}^{\gamma-2\beta}$ when $\gamma < (\beta + 1)d$. Indeed, writing $\hat{f} = w^{-\gamma} \hat{g}$ with $\hat{g} \in L_{t,\varphi,\xi}^2$, we have $|\xi|^{2\beta} \hat{f} =$

<footnote n="3">³One can give a definition a a space of tempered distribution also when $\gamma \ge (\beta + 1)d$; see [5].</footnote>
$w^{-\gamma+2\beta}(|\xi|^{2\beta}w^{-2\beta}\hat{g})$. As $|\xi|^{2\beta}w^{-2\beta}\hat{g}$ is in $L^2_{t,\varphi,\xi}$ and $\gamma - 2\beta < \gamma$, we have that $|\xi|^{2\beta}\hat{f}$ is a tempered distribution with inverse (partial) Fourier transform in $L^2_t \dot{X}^{\gamma-2\beta}_\beta$.

9.2.1. **Another homogeneous kinetic embedding.** In this scale, the kinetic embedding statement becomes the following one.

**Theorem 9.1 (Homogeneous kinetic embedding: extension).** Assume $\gamma < (\beta + 1)d$.

(i) $\dot{T}_\beta^\gamma \subset \dot{U}_\beta^\gamma \hookrightarrow C_0(\mathbb{R}_t; \dot{X}_\beta^{\gamma-\beta})$ with $\sup_{t \in \mathbb{R}} \|f(t)\|_{\dot{X}_\beta^{\gamma-\beta}} \lesssim_{d,\beta,\gamma} \|f\|_{\dot{U}_\beta^\gamma}$.

(ii) The subspaces in Theorem 2.2, (iii), are dense in all $\dot{T}_\beta^\gamma$ and $\dot{U}_\beta^\gamma$.

(iii) The family of spaces $(\dot{T}_\beta^\gamma)_\gamma$ has the complex interpolation property.

Again, this is only the definition of spaces that imposes the upper bound on $\gamma$. The inclusion (i) in this statement is weaker than (ii) in Theorem 2.2, but for a larger range.

We also have modifications of the statements of Theorem 1.16 and of Theorem 3.9 assuming $f \in \dot{U}_\beta^{\beta+\epsilon}$ and $\tilde{f} \in \dot{U}_\beta^{\beta-\epsilon}$, with only constraint $\beta \pm \epsilon < (\beta + 1)d$. In the latter statement, as there is no transfer of regularity, (i) and (ii) only contain the sup norm estimates. Next, (iii) has to be modified slightly with assumptions on the sums $S_1 + S_2, \tilde{S}_1 + \tilde{S}_2$ rather than individual assumptions. The dualities in the formula for the derivative also have to be adapted. This is left to the reader.

The isomorphism statement Lemma 3.5 is modified as follows: when $\gamma < (\beta + 1)d$, $K_\beta^\pm$ are isomorphisms from $L^2_t \dot{X}_\beta^{\gamma-2\beta}$ onto $\dot{T}_\beta^\gamma$ and from $L^2_t \dot{X}_\beta^{\gamma-2\beta} + L^1_t \dot{X}_\beta^{\gamma-\beta}$ onto $\dot{U}_\beta^\gamma$. This is what induces the interpolation property for the scale of spaces $\dot{T}_\beta^\gamma$.

The above discussion also extends to homogeneous spaces on $\Omega_+$.

**9.3. Inhomogeneous versions.** Consider the inhomogeneous versions $\mathcal{T}_\beta'(I)$, $\mathcal{U}_\beta'(I)$ on $\Omega, \Omega_+$ or on strips $(0, T) \times \mathbb{R}^d \times \mathbb{R}^d$ where $I = \mathbb{R}, (0, \infty)$ or $(0, T)$, respectively. With these spaces, there are no conditions on $\gamma \in \mathbb{R}$. The natural generalisations to Theorem 9.1 and absolute continuity hold. In particular, one has

$$ (9.3) \quad \mathcal{T}_{\beta}^{\gamma}(0, T) \hookrightarrow C([0, T]; X_{\beta}^{\gamma-\beta}) \text{ with } \sup_{t \in [0, T]} \|f(t)\|_{X_{\beta}^{\gamma-\beta}} \lesssim_{d, \beta, \gamma, T} \|f\|_{\mathcal{T}_{\beta}^{\gamma}(0, T)}. $$

If $T = \infty$, the homogeneous estimate is the one corresponding to (i) in Theorem 9.1 (with $\gamma < (\beta + 1)d$).

**Proposition 9.2.** If $0 \le \gamma \le 2\beta$, then $G_\beta^\gamma(0, T) = \mathcal{T}_\beta'(0, T)$. As a consequence, Theorem 6.1(i), and (9.3) are the same statements in this range of $\gamma$.

*Proof.* When $\gamma \ge 0$, the definition yields the inclusion $\mathcal{T}_\beta'(0, T) \subset G_\beta^\gamma(0, T)$. The inhomogeneous version of the transfer of regularity contained in Theorem 6.1(ii) shows that if $0 \le \gamma \le 2\beta$ then $G_\beta^\gamma(0, T) \subset \mathcal{T}_\beta'(0, T)$. $\square$

When $\gamma \notin [0, 2\beta]$, this equality of spaces does not seem true, hence the kinetic embedding for $G_\beta^\gamma(0, T)$ is not clear as well.
9.4. **Comparison with the embeddings in [32]**. We can now make a precise link between the estimate (9.3) and [32, Theorem 5.13]). More precisely, the spaces $T_β^γ(0, T)$ are introduced (with different notation and in the range $γ ≥ β$: a close examination shows this restriction was not necessary and their arguments work for all $γ$) as part of a more general framework (with temporal weights) for solving the kinetic Cauchy problem for the Kolmogorov equation with constant diffusion ($0 < β ≤ 1$) and (9.3) is a key ingredient in the argument. While the techniques used there relied on the Fourier transform as well, the method is different. It is proved by first establishing a weaker embedding in $C([0, T]; X)$ where $X$ is a Sobolev space with a large enough negative exponent. This argument is not possible when working with homogeneous spaces. Then, by an interpolation argument, the exact trace space was characterised as $X_β^{γ-β}$. Some estimates were mentioned to hold when $T = ∞$, but not the homogeneous embedding that would correspond to (9.3).

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UNIVERSITÉ PARIS-SACLAY, CNRS, LABORATOIRE DE MATHÉMATIQUES D'ORSAY, 91405 ORSAY, FRANCE, AND FRENCH-AUSTRALIAN MATHEMATICAL SCIENCES AND INTERACTIONS, AUSTRALIAN NATIONAL UNIVERSITY-CNRS, CANBERRA, ACT, 2601, AUSTRALIA

Email address: pascal.auscher@universite-paris-saclay.fr

DÉPARTEMENT DE MATHÉMATIQUES ET APPLICATIONS, ÉCOLE NORMALE SUPÉRIEURE, UNIVERSITÉ PSL, CNRS, 75005 PARIS, FRANCE

Email address: cyril.imbert@ens.psl.eu

INSTITUT FÜR ANALYSIS UND NUMERIK, WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER, ORLÉANS-RING 10, 48149 MÜNSTER, GERMANY.

Email address: lukas.niebel@uni-muenster.de