The following proposition quantifies the amount of decay needed on the auxiliary function $\psi$ in order to ensure that the $X_{\psi,A}^P$ quasinorm is equivalent to the $X_{\varphi,A}^P$ quasinorm whenever $\varphi$ has decay of arbitrarily high order at 0 and $\infty$.

**Proposition 4.4 (Independence on auxiliary function).** Let $\varphi \in \Psi_{\infty}^{\infty}$ and $\psi \in H^{\infty}$ be nondegenerate, let $\mathbf{p}$ be an exponent, and suppose that either

* $i(\mathbf{p}) \le 2$ and $\psi \in \Psi_{\theta(\mathbf{p})+}^{-(\theta(\mathbf{p})+n|\frac{1}{2}-j(\mathbf{p})|)+}$, or

* $i(\mathbf{p}) \ge 2$ and $\psi \in \Psi_{(\theta(\mathbf{p})+n|\frac{1}{2}-j(\mathbf{p})|)+}^{-\theta(\mathbf{p})+}$, or

Then $X_{\psi,A}^P = X_{\varphi,A}^P$ with equivalent quasinorms.

Observe that if $\psi \in \Psi_{\infty}^{\infty}$ then the requirements above are satisfied for all $\mathbf{p}$.

**PROOF.** First let $\nu \in \Psi_{\infty}^{\infty}$ be a Calderón sibling of $\psi$. Then for $f \in X_{\psi,A}^P$ we have

$$ \|f \mid X_{\varphi,A}^P\| = \|Q_{\varphi,A}f\|_{X^P} = \|Q_{\varphi,A}S_{\nu,A}Q_{\psi,A}f\|_{X^P} \lesssim \|Q_{\psi,A}f\|_{X^P} = \|f \mid X_{\psi,A}^P\|. $$

In the second equality we use the Calderón reproducing formula (3.26) along with the fact that $S_{\nu,A}Q_{\psi,A}f$ is defined for $f \in X_{\psi,A}^P$ (see Remark 4.2). The quasinorm estimate follows from Theorem 3.19, by the Standard Assumptions along with $\varphi, \nu \in \Psi_{\infty}^{\infty}$.

Now let $\nu \in \Psi_{\infty}^{\infty}$ be a Calderón sibling of $\varphi$ and repeat the previous argument with the roles of $\varphi$ and $\psi$ reversed, using the additional assumptions on $\psi$ to apply Theorem 3.19. This leads to the reverse estimate

$$ \|f \mid X_{\psi,A}^P\| \lesssim \|f \mid X_{\varphi,A}^P\|, $$

which completes the proof. $\square$

<figure><img src="image_13.png" /></figure>

**Definition 4.5.** For all exponents $\mathbf{p}$ we define

$$ X_A^P := X_{\psi,A}^P, $$

where any auxiliary function $\psi \in \Psi_{\infty}^{\infty}$ may be used to define the space and its corresponding quasinorm. We also define $\Psi(X_A^P)$ to be the set of all nondegenerate $\varphi \in H^{\infty}$ such that $X_{\varphi,A}^P = X_A^P$ with equivalent quasinorms. With this notation at hand, Proposition 4.4 tells us that

$$ \Psi_{\theta(\mathbf{p})+}^{(-\theta(\mathbf{p})+n|\frac{1}{2}-j(\mathbf{p})|)+} \cap H^{\infty} \subset \Psi(X_A^P) \quad (i(\mathbf{p}) \le 2), $$

$$ \Psi_{(\theta(\mathbf{p})+n|\frac{1}{2}-j(\mathbf{p})|)+}^{-\theta(\mathbf{p})+} \cap \Psi_{+}^{+} \subset \Psi(X_A^P) \quad (i(\mathbf{p}) > 2). $$

In Section 3.1 we considered the positive and negative spectral subspaces

$$ \overline{\mathcal{R}(A)}^{\pm} := \chi^{\pm}(A) \overline{\mathcal{R}(A)}. $$

These may be used to define corresponding positive and negative spectral subspaces of $X_A^P$.

**Definition 4.6.** Let $\mathbf{p}$ be an exponent. We define the positive and negative pre-Besov-Hardy-Sobolev spaces by

$$ X_A^{\mathbf{p},\pm} := X_A^{\mathbf{p}} \cap \overline{\mathcal{R}(A)}^{\pm}, $$

equipped with any of the equivalent $X_A^P$ quasinorms. Often we just refer to these as spectral subspaces. In Corollary 4.16 we characterise the spectral subspaces as images of the spectral projections $\chi^{\pm}(A)$.
The spaces $X_A^P$ may also be characterised in terms of the contraction maps $S_{\psi,A}$ for any $\psi \in \Psi_+^+$. Recall that $X^2 = T^2 = Z^2 = L^2(\mathbb{R}_+^{1+n})$.

**Proposition 4.7 (Characterisation by contraction maps).** Let $\mathbf{p}$ be an exponent and let $\psi \in \Psi_+^+$ be nondegenerate. Suppose furthermore that

$$ \psi \in \begin{cases} \Psi^{\theta(\mathbf{p})+}_{(-\theta(\mathbf{p})+n|\frac{1}{2}-j(\mathbf{p})|)+} & (i(\mathbf{p}) \le 2) \\ \Psi^{(\theta(\mathbf{p})+n|\frac{1}{2}-j(\mathbf{p})|)+}_{(-\theta(\mathbf{p}))+} & (i(\mathbf{p}) \ge 2) \end{cases} $$

(note that $\psi \in \Psi_\infty^\infty$ always satisfies this assumption). Then we have

$$ (4.1) \qquad X_A^P = S_{\psi,A}(X^2 \cap X^P), $$

and the mapping

$$ f \mapsto \inf\{\|F\|_{X^P} : F \in X^2 \cap X^P, S_{\psi,A}F = f\} $$

is an equivalent quasinorm on $X_A^P$.

**PROOF.** Fix a Calderón sibling $\varphi \in \Psi_\infty^\infty$ of $\psi$. First we show the equality (4.1). Suppose $f \in X_A^P$. Then $Q_{\varphi,A}f \in X^2 \cap X^P$, and by Theorem 3.26 we have $f = S_{\psi,A}(Q_{\varphi,A}f)$. Conversely, suppose that $f = S_{\psi,A}F$ for some $F \in X^2 \cap X^P$. Then $f \in \mathcal{R}(A)$, and $Q_{\varphi,A}f = Q_{\varphi,A}S_{\psi,A}F$. Theorem 3.19 then implies that $Q_{\varphi,A}f \in X^P$, which shows that $f \in X_A^P$. This proves (4.1).

Now for the quasinorm equivalence. Suppose $f \in X_A^P$. Then $f = S_{\psi,A}Q_{\varphi,A}f$ with $Q_{\varphi,A}f \in X^2 \cap X^P$, and so

$$ \inf\{\|F\|_{X^P} : F \in X^2 \cap X^P, S_{\psi,A}F = f\} \le \|Q_{\varphi,A}f\|_{X^P} \approx \|f\|_{X_A^P}. $$

Conversely, suppose $F \in X^2 \cap X^P$ and $S_{\psi,A}F = f$. Then

$$ \|F\|_{X^P} \gtrsim \|Q_{\varphi,A}S_{\psi,A}F\|_{X^P} = \|Q_{\varphi,A}f\|_{X^P} \approx \|f\|_{X_A^P} $$

using Theorem 3.19 again in the first line, and thus completing the proof. $\square$

<figure><img src="image_14.png" /></figure>

**Remark 4.8.** In Propositions 4.4 and 4.7 we quantify precisely how much decay (depending on $\mathbf{p}$) is sufficient for an auxiliary function $\varphi$ to characterise $X_A^P$ (either via $Q_{\varphi,A}$ or $S_{\psi,A}$). In both cases having $\varphi \in \Psi_\infty^\infty$ is sufficient, so in deducing abstract properties of the spaces $X_A^P$ we need only consider such functions. However, for applications to boundary value problems we must consider specific auxiliary functions without such decay, so this degree of precision is necessary.

**Corollary 4.9 (Density of intersections).** Let $\mathbf{p}$ and $\mathbf{q}$ be exponents, and suppose $X_1, X_2 \in \{\mathbb{H}, \mathbb{B}\}$. If $\mathbf{p}$ is finite then $(X_1)_A^P \cap (X_2)_A^q$ is dense in $(X_1)_A^P$. Otherwise, $(X_1)_A^P \cap (X_2)_A^q$ is weak-star dense in $(X_1)_A^P$.

**PROOF.** We suppose that $\mathbf{p}$ is finite; the same argument works for infinite $\mathbf{p}$, replacing limits with weak-star limits and norms with appropriate duality pairings.

Suppose $f \in (X_1)_A^P$, and fix $\psi \in \Psi_\infty^\infty$. By Proposition 4.7 we have $f = S_{\psi,A}F$ for some $F \in T^2 \cap (X_1)^P$, and by Proposition 2.36 we have $F = \lim_{k \to \infty} F_k$ (limit in $(X_1)^P$ for some sequence $(F_k)_{k \in \mathbb{N}}$ in $T^2 \cap (X_1)^P \cap (X_2)^q$. For all $k \in \mathbb{N}$ define

$$ f_k := S_{\psi,A}F_k \in (X_1)_A^P \cap (X_2)_A^q. $$

Then we have, again using Proposition 4.7,

$$ \lim_{k \to \infty} \|f - f_k\|_{(X_1)_A^P} \lesssim \lim_{k \to \infty} \|F - F_k\|_{(X_1)^P} = 0. $$

This proves the claimed density. $\square$

<figure><img src="image_15.png" /></figure>
The pre-Besov-Hardy-Sobolev spaces inherit a duality pairing from $\overline{\mathbb{R}}(A) \subset L^2(\mathbb{R}^n)$. However, we cannot identify $X_{A^*}^{P'}$ with the dual of $X_{A}^{P}$, because in general these spaces are incomplete, while the dual of a quasinormed space is always complete. We deal with completions in Section 4.3.

**Proposition 4.10 (Duality estimate).** Let $\mathbf{p}$ be an exponent. Then for all $f \in X_{A}^{P}$ and $g \in X_{A^*}^{P'}$ we have

$$ (4.2) \qquad |\langle f, g \rangle| \lesssim \|f\|_{X_A^P} \|g\|_{X_{A^*}^{P'}} $$

where $\langle \cdot, \cdot \rangle$ is the inner product on $L^2(\mathbb{R}^n)$.

**PROOF.** Let $\varphi, \psi \in \Psi_{\infty}^{\infty}$ be nondegenerate and suppose $\varepsilon > 0$. By Proposition 4.7 there exist $F \in X^2 \cap X^P$ and $G \in X^2 \cap X^{P'}$ such that $S_{\varphi,A}F = f$ and $S_{\psi,A^*}G = g$, with

$$ \|F\|_{X^P} \lesssim (1 + \varepsilon) \|f\|_{X_A^P} \quad \text{and} \quad \|G\|_{X^{P'}} \lesssim (1 + \varepsilon) \|g\|_{X_{A^*}^{P'}} $$

Since $S_{\varphi,A}^* = Q_{\tilde{\varphi},A^*}$, and using that the $L^2(\mathbb{R}_+^{1+n})$ inner product is a duality pairing for tent and Z-spaces, we thus have

$$ \begin{align} |\langle f, g \rangle| &= |\langle F, Q_{\tilde{\varphi},A^*} S_{\psi,A^*} G \rangle|_{L^2(\mathbb{R}_+^{1+n})} \nonumber \\ &\lesssim \|F\|_{X^P} \|Q_{\tilde{\varphi},A^*} S_{\psi,A^*} G\|_{X^{P'}} \nonumber \\ &\lesssim \|F\|_{X^P} \|G\|_{X^{P'}} \nonumber \\ &\lesssim (1 + \varepsilon)^2 \|f\|_{X_A^P} \|g\|_{X_{A^*}^{P'}} \tag{4.3} \end{align} $$

where (4.3) follows from Theorem 3.19. Since $\varepsilon > 0$ was arbitrary we obtain (4.2). $\square$

The tent space and Z-space embeddings of Chapter 2 immediately yield corresponding embeddings of pre-Besov-Hardy-Sobolev spaces.

**Proposition 4.11 (Mixed embeddings).** Let $\mathbf{p}$ and $\mathbf{q}$ be exponents with $\mathbf{p} \neq \mathbf{q}$ and $\mathbf{p} \mapsto \mathbf{q}$. Then we have the embedding

$$ (X_0)_A^P \hookrightarrow (X_1)_A^q, $$

where $X_0, X_1 \in \{\mathbb{H}, \mathbb{B}\}$, and the corresponding embedding holds for spectral subspaces.

**PROOF.** This follows directly from the definition of the spaces $X_A^P$ and from Theorem 2.34. Spectral subspace versions follow by intersecting with $\overline{\mathbb{R}}(A)^{\pm}$. $\square$

**Remark 4.12.** For $\mathbf{p} = (p, s)$ we sometimes write $X_A^P = X_{s,A}^p$, and for $\mathbf{p} = (\infty, s; \alpha)$ we may write $X_A^P = X_{s;\alpha,A}^\infty$. This notation is a bit heavy, so we avoid it whenever possible, except in the case of $X_{0,A}^2$, which we abbreviate as $X_A^2$ (as is standard).

**Remark 4.13.** There are other ways to define adapted Hardy-Sobolev (or more generally, Triebel-Lizorkin) and Besov spaces. See in particular the work of Kunstmann and Ullmann [62] and Ullmann's thesis [78], in which Triebel-Lizorkin spaces adapted to 'R_s-sectorial' operators are constructed via a vertical s-power function norm (a vertical square function when $s = 2$). We have not investigated possible links between our adapted function spaces and theirs.
## 4.2. Mapping properties of the holomorphic functional calculus

Just as we proved independence on auxiliary functions in the previous section, we can prove various mapping properties (including boundedness) of the holomorphic functional calculus between pre-Besov–Hardy–Sobolev spaces. The first result says heuristically that an operator of homogeneity $\delta$ decreases regularity by $\delta$.

**Theorem 4.14.** Let $\mathbf{p}$ be an exponent and $\delta \in \mathbb{R}$. Suppose $\eta \in \Psi_{-\delta}^{\delta}$. Then for all $f \in \mathcal{D}(\eta(A)) \cap \mathbb{X}_{A}^{\mathbf{p}}$,

$$ \|\eta(A)f\|_{\mathbb{X}_{A}^{\mathbf{p}+\delta}} \lesssim \|\eta\|_{\Psi_{-\delta}^{\delta}} \|f\|_{\mathbb{X}_{A}^{\mathbf{p}}}, $$

and likewise for spectral subspaces.

**PROOF.** Let $\varphi, \psi \in \Psi_{\infty}^{\infty}$ and let $\nu \in \Psi_{\infty}^{\infty}$ be a Calderón sibling of $\psi$. Then for all $f \in \mathcal{D}(\eta(A)) \cap \mathbb{X}_{A}^{\mathbf{p}}$ we have

$$ \|\eta(A)f\|_{\mathbb{X}_{A}^{\mathbf{p}+\delta}} \simeq \|\mathbb{Q}_{\varphi,A}\eta(A)\mathbb{S}_{\nu,A}\mathbb{Q}_{\psi,A}f\|_{\mathbb{X}^{\mathbf{p}+\delta}} \lesssim \|\eta\|_{\Psi_{-\delta}^{\delta}} \|\mathbb{Q}_{\psi}f\|_{\mathbb{X}^{\mathbf{p}}} \simeq \|\eta\|_{\Psi_{-\delta}^{\delta}} \|f\|_{\mathbb{X}_{A}^{\mathbf{p}}} $$

where the quasinorm estimate follows from Theorem 3.19. To incorporate spectral subspaces in this argument, write $f \in \mathcal{D}(\eta(A)) \cap \mathbb{X}_{A}^{\mathbf{p},\pm}$ as $f = \chi^{\pm}(A)f$ and

$$ \eta(A)f = \eta(A)\chi^{\pm}(A)f = \chi^{\pm}(A)\eta(A)f, $$

and note that this shows that $\eta(A)$ maps $\mathcal{D}(\eta(A)) \cap \overline{\mathbb{R}(A)}^{\pm}$ into $\overline{\mathbb{R}(A)}^{\pm}$. $\square$

Because the spaces $\mathbb{X}_{A}^{\mathbf{p}}$ may be incomplete, we cannot extend the operators $\eta(A)$ by boundedness without introducing completions. This is done in Section 4.3. Of course, when $\eta \in H^{\infty}$ we have $\mathcal{D}(\eta(A)) = \overline{\mathbb{R}(A)}$, and so we obtain bounded holomorphic functional calculus in the following sense.

**Corollary 4.15.** Let $\mathbf{p}$ be an exponent and $\eta \in H^{\infty}$. Then for all $f \in \mathbb{X}_{A}^{\mathbf{p}}$,

$$ \|\eta(A)f\|_{\mathbb{X}_{A}^{\mathbf{p}}} \lesssim \|\eta\|_{\infty} \|f\|_{\mathbb{X}_{A}^{\mathbf{p}}}, $$

and likewise for spectral subspaces.

This allows us to characterise the spectral subspaces $\mathbb{X}_{A}^{\mathbf{p},\pm}$ as images of spectral projections.

**Corollary 4.16.** Let $\mathbf{p}$ be an exponent. Then we have

$$ \mathbb{X}_{A}^{\mathbf{p},\pm} = \chi^{\pm}(A)\mathbb{X}_{A}^{\mathbf{p}}. $$

**PROOF.** If $f \in \mathbb{X}_{A}^{\mathbf{p},\pm}$ then by definition we have $f = \chi^{\pm}(A)f \in \chi^{\pm}(A)\mathbb{X}_{A}^{\mathbf{p}}$. Conversely, if $f \in \chi^{\pm}(A)\mathbb{X}_{A}^{\mathbf{p}}$, then $f$ is in $\overline{\mathbb{R}(A)}^{\pm}$, and by Corollary 4.15 we have $f \in \mathbb{X}_{A}^{\mathbf{p}}$. Therefore $f \in \overline{\mathbb{R}(A)}^{\pm} \cap \mathbb{X}_{A}^{\mathbf{p}} = \mathbb{X}_{A}^{\mathbf{p},\pm}$. $\square$

The power functions $A^{\lambda} := [z \mapsto z^{\lambda}](A)$ for $\lambda \in \mathbb{R} \setminus \{0\}$ (see Section 3.1) are generally unbounded on $\overline{\mathbb{R}(A)}$, but they do map between our adapted spaces with a shift in regularity (when we intersect with the domain). This is a direct consequence of Theorem 4.14 since $[z \mapsto z^{\lambda}] \in \Psi_{\lambda}^{-\lambda}$; the norm equivalence is obtained by applying Theorem 4.14 with both $\lambda$ and $-\lambda$.

**Corollary 4.17.** Let $\mathbf{p}$ be an exponent and $\lambda \in \mathbb{R} \setminus \{0\}$. Then for all $f \in \mathcal{D}(A^{\lambda}) \cap \mathbb{X}_{A}^{\mathbf{p}}$,

$$ \|A^{\lambda}f\|_{\mathbb{X}_{A}^{\mathbf{p}-\lambda}} \simeq \|f\|_{\mathbb{X}_{A}^{\mathbf{p}}}. $$
Since the operators $A^\lambda$ are all densely defined in $\overline{\mathbb{R}}(A)$, and since $A^{\lambda_0} A^{\lambda_1} = A^{\lambda_0 + \lambda_1}$ whenever this is meaningful, we have almost proven that $A^\lambda$ is an isomorphism from $X_A^P$ to $X_A^{P-\lambda}$. We must extend everything by boundedness to make this rigorous. As previously mentioned, this requires the introduction of completions.

### 4.3. Completions, interpolation, and inclusions

The spaces $X_A^P$ defined in the previous section are called pre-Besov-Hardy-Sobolev spaces because, with the exception of $X_A^2 = \overline{\mathbb{R}}(A)$, they need not be complete. One could try to solve this problem by taking arbitrary completions $(X_A^P)^c$ of $X_A^P$ and declaring these to be the Besov-Hardy-Sobolev spaces adapted to $A$. However, if we take this approach, then for different exponents $p_1$ and $p_2$, there may not exist a natural topological vector space in which the completions $(X_A^{p_1})^c$ both embed. This presents difficulties when discussing interpolants of these completions. The impact of this problem on abstract Hardy space theory seems to have first been discussed by the second author, McIntosh, and Morris [12]. We avoid this issue by introducing certain canonical completions within tent and $Z$-spaces (and hence within $L^0(\mathbb{R}_+^{1+n}))$. If another completion is possible—for example, within the space $Z'(\mathbb{R}^n)$ of distributions modulo polynomials, in which the classical smoothness spaces are embedded—then we are free to identify this with our canonical completion.

By a completion of a quasinormed space $Q$ we mean a continuous injective map $\iota: Q \to \tilde{Q}$, where $\tilde{Q}$ is a complete quasinormed space and $\iota(Q)$ is dense in $\tilde{Q}$. By a weak-star completion of $Q$, we mean $\iota: Q \to \tilde{Q}$ as above, where $\tilde{Q}$ is a dual space and where $\iota(Q)$ is weak-star dense in $\tilde{Q}$. We may refer to $\tilde{Q}$ as the completion, with the inclusion $\iota$ being implicit.

In this section, whenever $p$ is infinite, we interpret ‘completion’ to mean ‘weak-star completion’.

**Definition 4.18.** For an exponent $p$ and an auxiliary function $\psi \in \Psi(X_A^P)$, define the canonical completion

$$ \psi X_A^P := \overline{\mathbb{Q}_{\psi,A} X_A^P} \subset X^P $$

and likewise

$$ \psi X_A^{P, \pm} := \overline{\mathbb{Q}_{\psi,A} X_A^{P, \pm}} \subset \psi X_A^P $$

where the closures are taken in the $X^P$ quasinorm when $p$ is finite, and in the weak-star topology on $X^P$ when $p$ is infinite. We equip $\psi X_A^P$ and $\psi X_A^{P, \pm}$ with the $X^P$ quasinorm, so that $\psi X_A^P$ and $\psi X_A^{P, \pm}$ are quasi-Banach spaces.

**Remark 4.19.** Heuristically, what we are doing here is shifting viewpoint from functions to A-adapted $\psi$-expansions of functions, with each choice of $\psi$ giving a different image of the space $X_A^P$. This is like considering Fourier transforms of functions instead of functions themselves, except in this case we do not have a fixed ‘Fourier transform’; we only have a collection of ‘expanded Fourier multipliers’ (the operators $\mathbb{Q}_{\psi,A}$). This corresponds to the redundancy in the choice of auxiliary function used to define $X_A^P$. The advantage of this approach is that the space $\mathbb{Q}_{\psi,A} X_A^P$ of A-adapted $\psi$-expansions of $X_A^P$-functions has a natural completion, while (at least it seems that) the space $X_A^P$ does not.

**Proposition 4.20.** Fix $p$ and $\psi$ as in Definition 4.18. Then $\mathbb{Q}_{\psi,A}: X_A^P \to \psi X_A^P$ and $\mathbb{Q}_{\psi,A}: X_A^{P, \pm} \to \psi X_A^{P, \pm}$ are completions of $X_A^P$ and $X_A^{P, \pm}$.
**PROOF.** By construction the spaces $\psi X_{s,A}^p$ and $\psi X_{s,A}^{p, \pm}$ are complete and contain $\mathbb{Q}_{\psi,A} X_A^p$ and $\mathbb{Q}_{\psi,A} X_A^{p, \pm}$ respectively as dense subspaces (in the weak-star topology when $\mathbf{p}$ is infinite). The map $\mathbb{Q}_{\psi,A}: X_A^p \to \psi X_A^p$ is continuous since $\psi \in \Psi(X_A^p)$, and injective by Corollary 3.21. These properties automatically continue to hold for the restrictions of $\mathbb{Q}_{\psi,A}$ to the spectral subspaces $X_A^{p, \pm}$. $\square$

Completions are unique up to isomorphism, so any completion may be identified with any canonical completion. It is useful to make this identification precise.

**Proposition 4.21 (Identification of completions).** Fix $\mathbf{p}$ and $\psi$ as in Definition 4.18 and suppose that $\iota: X_A^p \to X$ is a completion of $X_A^p$. Then there exists a unique map $\mathbb{Q}_{\psi,A}^\iota: X \to X^p$ such that the triangle

<figure><img src="image_16.png" /></figure>

commutes, and this map is an isomorphism between $X$ and $\psi X_A^p$. Its inverse is the map $S_{\psi,A}^\iota: X^p \to X$ which is the unique continuous extension of the map $\iota \circ S_{\nu,A}^\iota: X^2 \cap X^p \to X$ (using the weak-star topology on $X^p$ when $\mathbf{p}$ is infinite) for any $\nu \in \Psi_\infty$ which is a Calderón sibling of $\psi$. The same results hold if we replace all spaces with corresponding spectral subspaces.

**PROOF.** Since $\mathbb{Q}_{\psi,A}: X_A^p \to \psi X_A^p$ is a completion of $X_A^p$, by the universal property of completions there exists a unique map $\tilde{\mathbb{Q}}_{\psi,A}^\iota: X \to \psi X_A^p$ such that the triangle

<figure><img src="image_17.png" /></figure>

commutes. Hence we have a commutative diagram

<figure><img src="image_18.png" /></figure>

Since

$$(\mathrm{id} \circ \tilde{\mathbb{Q}}_{\psi,A}^\iota) \circ \iota = \mathbb{Q}_{\psi,A} = \mathbb{Q}_{\psi,A}^\iota \circ \iota,$$

by uniqueness we must have $\mathbb{Q}_{\psi,A}^\iota = \mathrm{id} \circ \tilde{\mathbb{Q}}_{\psi,A}^\iota = \tilde{\mathbb{Q}}_{\psi,A}^\iota$. Therefore it suffices to show that $\tilde{\mathbb{Q}}_{\psi,A}^\iota$ satisfies the desired properties.
To show that $S_{\psi,A}^t \tilde{Q}_{\psi,A}^t = \text{id}_X$, observe that we have a commutative diagram

$$ (4.4) \qquad \begin{tikzcd}[column sep=2.8em, row sep=2.8em]
    \xymatrix@C=1.8em@R=1em{
        \mathbf{X} \ar@/^1.5em/[l]^{\tilde{\mathbf{Q}}_{\psi,A}^t} \arrow[r]_{\psi \mathbf{X}_A^P} &
        \mathbf{X} \ar@/^1.5em/[r]^{\mathbf{S}_{\psi,A}^t} \arrow[r]_{\psi(\mathbf{X}_A^P)} &
        \mathbf{X} \ar@/^1.5em/[l]^{\tilde{\mathbf{Q}}_{\psi,A}^t} \arrow[r]_{\psi \mathbf{X}_A^P} &
        \mathbf{X} \ar@/^1.5em/[r]^{\mathbf{S}_{\psi,A}^t} \arrow[r]_{\psi(\mathbf{X}_A^P)} &
        \mathbf{X} \ar@/^1.5em/[r]^{\mathbf{Q}_{\psi,A}^t} \arrow[r]_{\psi(\mathbf{X}_A^P)}
    }
    \qquad \text{id}_X \arrow@{}[c]{@{}c}{\psi(\mathbf{X}_A^P)} \arrow@{}[c]{@{}c}{\text{id}_X}
\end{tikzcd} $$

Thus we have

$$ S_{\psi,A}^{t} \tilde{Q}_{\psi,A}^{t} = \iota \circ \text{id} = \iota $$

and

$$ \tilde{Q}_{\psi,A}^t S_{\psi,A}^t|_{\psi(\mathbf{X}_A^P)} = \text{id}, $$

so by uniqueness of extensions we must have that $\tilde{Q}_{\psi,A}^t$ and $S_{\psi,A}^t$ are mutual inverses. The proof for spectral subspaces is identical. □

As a corollary of this argument we can show that $\psi \mathbf{X}_A^P$ is a retract of $X^P$. This is crucial in identifying interpolants.

**Corollary 4.22.** Fix $\mathbf{p}, \psi$, and $\nu$ as in Proposition 4.21, and let $\iota: \mathbf{X}_A^P \to \mathbf{X}$ be a completion of $\mathbf{X}_A^P$. Then the map $Q_{\psi,A}^t S_{\nu,A}^t: X^P \to \psi \mathbf{X}_A^P$ is the extension of the projection $Q_{\psi,A} S_{\nu,A}: X^2 \cap X^P \to Q_{\psi,A} \mathbf{X}_A^P$ in the appropriate topology (hence independent of $\iota$), and it is a projection onto $\psi \mathbf{X}_A^P$. The same statements hold for spectral subspaces.

We write $Q_{\psi,A} S_{\nu,A} := Q_{\psi,A}^t S_{\nu,A}^t$ to denote the extension considered above. Now that we have thought hard enough about completions, we can extend the duality and boundedness results of the previous sections.

**Proposition 4.23 (Duality).** Let $\mathbf{p}$ be a finite exponent, and let $\psi, \nu \in \Psi_{\infty}^{\infty}$ be Calder\'on siblings. Then the $L^2$ duality pairing identifies $\tilde{\nu} \mathbf{X}_{A^*}^{\mathbf{p}'}$ as the Banach space dual of $\psi \mathbf{X}_A^P$, and also identifies $\tilde{\nu} \mathbf{X}_{A^*}^{\mathbf{p}'} \pm \mathbf{X}_{A^*}^{\mathbf{p}}$ as the Banach space dual of $\psi \mathbf{X}_A^{\mathbf{p},\pm}$.

**PROOF.** If $f \in \psi \mathbf{X}_A^P$ and $g \in \tilde{\nu} \mathbf{X}_{A^*}^{\mathbf{p}'}$, then we immediately have

$$ |\langle f, g \rangle| \le \|f\|_{X^P} \|g\|_{X^{P'}} = \|f\|_{\psi \mathbf{X}_A^P} \|g\|_{\tilde{\nu} \mathbf{X}_{A^*}^{\mathbf{p}'}} $$

so every $g \in \tilde{\nu} \mathbf{X}_{A^*}^{\mathbf{p}'}$ induces a bounded linear functional on $\mathbf{X}_A^P$. Conversely, suppose $\varphi \in (\psi \mathbf{X}_A^P)'$. Then we can define a bounded linear functional $\Phi \in (X^P)'$ by

$$ \Phi(F) := \varphi(Q_{\psi,A} S_{\nu,A} F) $$

for all $F \in X^P$. By $X$-space duality, there exists a function $G_{\Phi} \in X^{P'}$ such that

$$ \langle F, G_{\Phi} \rangle = \Phi(F) $$

for all $F \in X^P$, satisfying

$$ \|G_{\Phi}\|_{X^{P'}} \approx \|\Phi\|_{(X^P)'} \lesssim \|\varphi\|_{(\psi \mathbf{X}_A^P)'} $$

Hence for all $f \in \psi \mathbf{X}_A^P$ we have

$$ \langle f, (Q_{\psi,A} S_{\nu,A})^* G_{\Phi} \rangle = \langle f, G_{\Phi} \rangle = \Phi(f) = \varphi(f) $$
since $f = \mathbf{Q}_{\psi,A}\mathbf{S}_{\nu,A}f$. Since $\mathbf{Q}_{\psi,A}\mathbf{S}_{\nu,A}$ is the continuous extension of $\mathbb{Q}_{\psi,A}\mathbb{S}_{\nu,A}$ from $X^2 \cap X^P$ to $X^P$, and since $(\mathbb{Q}_{\psi,A}\mathbb{S}_{\nu,A})^* = \mathbb{Q}_{\tilde{\nu},A^*}\mathbb{S}_{\tilde{\psi},A^*}$ on $X^2$, we find that $(\mathbf{Q}_{\psi,A}\mathbf{S}_{\nu,A})^* = \mathbb{Q}_{\tilde{\nu},A^*}\mathbb{S}_{\tilde{\psi},A^*}$. Therefore

$$ \varphi(f) = \langle f, G_{\varphi} \rangle $$

for all $f \in \psi\mathbf{X}_A^P$, where $G_{\varphi} = \mathbf{Q}_{\tilde{\nu},A^*}\mathbf{S}_{\tilde{\psi},A^*}G_{\Phi} \in \tilde{\nu}\mathbf{X}_{A^*}^{P'}$. Furthermore we have

$$ \|G_{\varphi}\|_{\tilde{\nu}\mathbf{X}_{A^*}^{P'}} = \|\mathbf{Q}_{\tilde{\nu},A^*}\mathbf{S}_{\tilde{\psi},A^*}G_{\Phi}\|_{X^{P'}} \lesssim \|\varphi\|_{(\psi\mathbf{X}_A^P)'} $$

As with every other result in this section, the same proof works for spectral subspaces. $\square$

**Proposition 4.24 (Boundedness of functional calculus).** Let $p$ be an exponent, $\delta \in \mathbb{R}$, and $\eta \in \Psi_{-\delta}^{\delta}$. Suppose $\iota_1: \mathbb{X}_A^P \to \mathbf{X}$ and $\iota_2: \mathbb{X}_A^{P+\delta} \to \mathbf{Y}$ are completions. Then $\eta(A)$ extends to a bounded operator $\widetilde{\eta}(A): \mathbf{X} \to \mathbf{Y}$, in the sense that the diagram

$$ \begin{tikzcd} \mathcal{D}(\eta(A)) \cap \mathbb{X}_A^P \arrow[r, "\eta(A)"'] & \mathbb{X}_A^{P+\delta} \arrow[d, "\iota_1"] \\ \mathbf{X} \arrow[u, "\iota_2"'] \arrow[r, "\widetilde{\eta}(A)"'] & \mathbf{Y} \end{tikzcd} $$

commutes, and that

$$ (4.5) \qquad \|\widetilde{\eta(A)}f\|_{\mathbf{Y}} \lesssim \|\eta\|_{\Psi_{-\delta}} \|f\|_{\mathbf{X}}. $$

for all $f \in \mathbf{X}$. Similar results hold for spectral subspaces.

**PROOF.** Since $\mathcal{D}(\eta(A))$ is dense in $\mathbb{X}_A^2 = \overline{\mathbb{R}(A)}$ and since $\mathbb{X}_A^2 \cap \mathbb{X}_A^P$ is dense in $\mathbb{X}_A^P$ (Corollary 4.9 for finite exponents, duality for infinite exponents using the weak-star topology), we have that $\mathcal{D}(\eta(A)) \cap \mathbb{X}_A^P$ is dense in $\mathbb{X}_A^P$. The result then follows from Theorem 4.14 and the universal property of completions. $\square$

**Remark 4.25.** Evidently ‘completed’ versions of Corollaries 4.15, 4.16, and 4.17 can be formulated.

**Remark 4.26.** In the situation of Proposition 4.24 we will write $\eta(A)$ to denote both the original operator $\mathcal{D}(\eta(A)) \cap \mathbb{X}_A^P \to \mathbb{X}_A^{P+\delta}$ and its extension to completions $\mathbf{X} \to \mathbf{Y}$. This will not cause any ambiguity, but one should be careful.

**Remark 4.27.** If $\mathbf{X}$ is a completion of $\mathbb{X}_A^P$, then the spectral projections $\chi^{\pm}(A)$ extend to projections on $\mathbf{X}$ by Proposition 4.24, so we get a spectral decomposition

$$ \mathbf{X} = \mathbf{X}^{+} \oplus \mathbf{X}^{-}. $$

It is immediate that $\mathbf{X}^+$ and $\mathbf{X}^-$ are completions of $\mathbb{X}_A^{P,+}$ and $\mathbb{X}_A^{P,-}$ respectively. That is, every completion of $\mathbb{X}_A^P$ induces natural completions of the spectral subspaces $\mathbb{X}_A^{P,\pm}$.

Finally we can present the interpolation theorem for canonical completions of pre-Besov-Hardy-Sobolev spaces. Having established so much abstract theory, this is now a simple consequence of the interpolation results for tent spaces and $Z$-spaces.
**Theorem 4.28 (Interpolation of completions).** Fix $0 < \eta < 1$ and $\psi \in \Psi_{\infty}^{\infty}$. Let $\mathbf{p}_0$ and $\mathbf{p}_1$ be exponents, and set $\mathbf{p}_{\eta} := [\mathbf{p}_0, \mathbf{p}_1]_{\eta}$.

(i) Suppose $j(\mathbf{p}_0), j(\mathbf{p}_1) \ge 0$, with equality for at most one exponent. Then we have the identification

$$[\psi \mathbf{H}_A^{\mathbf{p}_0}, \psi \mathbf{H}_A^{\mathbf{p}_1}]_{\eta} = \psi \mathbf{H}_A^{\mathbf{p}_{\eta}}.$$

(ii) Suppose at least one of $\mathbf{p}_0$ and $\mathbf{p}_1$ is finite. Then we have the identification

$$[\psi \mathbf{B}_A^{\mathbf{p}_0}, \psi \mathbf{B}_A^{\mathbf{p}_1}]_{\eta} = \psi \mathbf{B}_A^{\mathbf{p}_{\eta}}.$$

(iii) Suppose $\theta(\mathbf{p}_0) \neq \theta(\mathbf{p}_1)$. Then we have the identification

$$(\psi \mathbf{X}_A^{\mathbf{p}_0}, \psi \mathbf{X}_A^{\mathbf{p}_1})_{\eta, i(\mathbf{p}_\eta)} = \psi \mathbf{B}_A^{\mathbf{p}_{\eta}}.$$

All of these statements have analogues for spectral subspaces.

**PROOF.** Fix a Calderón sibling $\nu \in \Psi_{\infty}^{\infty}$ of $\psi$. By Corollary 4.22 the map $\mathbb{Q}_{\psi,A} \mathbb{S}_{\nu,A}$ extends to a map $\mathbf{Q}_{\psi,A} \mathbb{S}_{\nu,A}: X^{\mathbf{p}_0} + X^{\mathbf{p}_1} \to \psi \mathbf{X}_A^{\mathbf{p}_0} + \psi \mathbf{X}_A^{\mathbf{p}_1}$ which restricts to projections $X^{\mathbf{p}_0} \to \psi \mathbf{X}_A^{\mathbf{p}_0}$ and $X^{\mathbf{p}_1} \to \psi \mathbf{X}_A^{\mathbf{p}_1}$. Therefore by the retraction/coretraction interpolation theorem (see [75, §1.2.4]),<fnref n="1" /> for all interpolation functions $\mathcal{F}$ we have

$$\mathcal{F}(\mathbf{X}_A^{\mathbf{p}_0}, \mathbf{X}_A^{\mathbf{p}_1}) = \mathbf{Q}_{\psi,A} \mathbb{S}_{\nu,A} \mathcal{F}(X^{\mathbf{p}_0}, X^{\mathbf{p}_1}).$$

The results then follow from Corollary 4.22 and the interpolation theorems for tent- and Z-spaces (Theorems 2.12 and 2.30, and Propositions 2.31 and 2.32). □

**Remark 4.29.** The assumption $\psi \in \Psi_{\infty}^{\infty}$ is only for convenience. The argument still holds provided that $\psi \in \Psi(\mathbf{X}_A^{\mathbf{p}_0})$ for all the spaces $\mathbf{X}_A^{\mathbf{p}_0}$ involved. Furthermore, when interpolating between (e.g. positive) spectral subspaces, it suffices to have $\|f\|_{\mathbf{X}_A^{\mathbf{p}_0}} \simeq \|\mathbb{Q}_{\psi,A} f\|_{X^{\mathbf{p}}}$ for all $f \in \mathbf{X}_A^{\mathbf{p}_0}$. We will use this observation in Section 6.5.

We now present an application of this interpolation theorem. For applications to boundary value problems, it will be absolutely crucial to know when we have $\mathbf{X}_{DB}^{\mathbf{p}} = \mathbf{X}_D^{\mathbf{p}}$, where $D$ is the Dirac operator and $B$ is a multiplier as in Subsection 1.1.3. As a preliminary step, we consider inclusions of the form $\mathbf{X}_{A_0}^{\mathbf{p}} \hookrightarrow \mathbf{X}_{A_1}^{\mathbf{p}}$ for operators $A_0$ and $A_1$ satisfying the Standard Assumptions and with equal range closures (as is the case with $D$ and $DB$).

**Definition 4.30.** Suppose $A_0$ and $A_1$ are operators satisfying the Standard Assumptions, and additionally assume that $\overline{\mathcal{R}}(A_0) = \overline{\mathcal{R}}(A_1) =: \overline{\mathcal{R}}$. Define the inclusion region

$$i(\mathbf{X}, A_0, A_1) := \{ \mathbf{p} \in \mathbf{E}_{\text{fin}} : \|f\|_{\mathbf{X}_{A_1}^{\mathbf{p}}} \lesssim \|f\|_{\mathbf{X}_{A_0}^{\mathbf{p}}} \text{ for all } f \in \overline{\mathcal{R}} \}.$$

It is technically simpler for us to restrict to finite exponents in this definition, and this is good enough for our applications, but it is not essential.

**Lemma 4.31.** Suppose that $\mathbf{p}$ is a finite exponent, and let $A_0$ and $A_1$ be as in Definition 4.30. Then $\mathbf{p} \in i(\mathbf{X}, A_0, A_1)$ if and only if for all Calderón siblings $\varphi, \psi \in \Psi_{\infty}^{\infty}$ the composition

$$\mathbb{Q}_{\varphi, A_0} \mathbf{X}_{A_0}^{\mathbf{p}} \xrightarrow{S_{\psi, A_0}} \overline{\mathcal{R}} \xrightarrow{\mathcal{Q}_{\varphi, A_1}} L^2(\mathbb{R}_+ : L^2) \\ \text{is bounded from } \mathbb{Q}_{\varphi, A_0} \mathbf{X}_{A_0}^{\mathbf{p}} \text{ to } \mathbb{Q}_{\varphi, A_1} \mathbf{X}_{A_1}^{\mathbf{p}}.$$

<footnote n="1">¹This is only stated for Banach spaces in the given reference. The only property specific to Banach spaces which is needed is the validity of the closed graph theorem, which also holds for quasi-Banach spaces [54, §2].</footnote>
PROOF. For any pair of Calderón siblings $\varphi, \psi \in \Psi_{\infty}^{\infty}$, we have

$$ \|f\|_{X_{A_1}^{P}} \lesssim \|f\|_{X_{A_0}^{P}} \quad \text{for all } f \in \bar{\mathcal{R}} $$

if and only if

$$ (4.6) \qquad \|\mathbb{Q}_{\varphi, A_1} f\|_{X^P} \lesssim \|\mathbb{Q}_{\varphi, A_0} f\|_{X^P} \quad \text{for all } f \in \bar{\mathcal{R}} $$

if and only if

$$ \|\mathbb{Q}_{\varphi, A_1} \mathbb{S}_{\varphi, A_0} F\|_{X^P} \lesssim \|F\|_{X^P} \quad \text{for all } F \in \mathbb{Q}_{\varphi, A_0} X_{A_0}^{P}, $$

using that the right hand side of (4.6) is finite if and only if $f = \mathbb{S}_{\psi, A_0} F$ for some $F \in \mathbb{Q}_{\varphi, A_0} X_{A_0}^{P}$, and the Calderón reproducing formula (combining Proposition 4.7 and Theorem 3.20). $\square$

**Theorem 4.32 (Interpolation of inclusions of adapted spaces).** Let $A_0$ and $A_1$ be as in Definition 4.30. Let $\mathbf{p}_0$ and $\mathbf{p}_1$ be finite exponents, let $\eta \in (0, 1)$, and set $\mathbf{p}_{\eta} := [\mathbf{p}_0, \mathbf{p}_1]_{\eta}$.

(i) If $\mathbf{p}_0, \mathbf{p}_1 \in i(\mathbf{H}, A_0, A_1)$, then $\mathbf{p}_{\eta} \in i(\mathbf{H}, A_0, A_1)$.

(ii) If $\mathbf{p}_0, \mathbf{p}_1 \in i(\mathbf{B}, A_0, A_1)$, then $\mathbf{p}_{\eta} \in i(\mathbf{B}, A_0, A_1)$.

(iii) If $\theta(\mathbf{p}_0) \neq \theta(\mathbf{p}_1)$ and $\mathbf{p}_0, \mathbf{p}_1 \in i(\mathbf{X}, A_0, A_1)$, then $\mathbf{p}_{\eta} \in i(\mathbf{B}, A_0, A_1)$.

PROOF. Fix Calderón siblings $\varphi, \psi \in \Psi_{\infty}^{\infty}$. By Lemma 4.31, we have $\mathbf{p}_0, \mathbf{p}_1 \in I(\mathbf{X}; A_0, A_1)$ if and only if $\mathbb{Q}_{\varphi, A_1} \mathbb{S}_{\psi, A_0}$ is bounded from $\mathbb{Q}_{\varphi, A_0} X_{A_0}^{P}$ to $\mathbb{Q}_{\varphi, A_1} X_{A_1}^{P}$ for $i \in \{1, 2\}$, in which case $\mathbb{Q}_{\varphi, A_1} \mathbb{S}_{\psi, A_0}$ extends to a bounded operator $\varphi \mathbf{X}_{A_0}^{P} \to \varphi \mathbf{X}_{A_1}^{P}$ for $i \in \{1, 2\}$. The theorem then follows from the identification of interpolation spaces from Theorem 4.28 and a second application of Lemma 4.31. $\square$

## 4.4. The Cauchy operator on general adapted spaces

Recall the Cauchy extension operator, defined in (3.8) by

$$ C_A f(t) := e^{-tA} \chi^{\operatorname{sgn}(t)}(A) f = e^{-|t|^{[A]}} \chi^{\operatorname{sgn}(t)}(A) f = \operatorname{sgp}_{|t|}(A) \chi^{\operatorname{sgn}(t)}(A) f. $$

By Proposition 4.24, for any $\varphi \in H^{\infty}$, the operator $\varphi(A): X_A^P \to X_A^P$ extends to a bounded map on any completion $\mathbf{X}$ of $X_A^P$. Thus we can extend the Cauchy operator to a map $C_A: \mathbf{X} \to L^{\infty}(\mathbb{R} \setminus \{0\} : \mathbf{X})$.

**Proposition 4.33 (Properties of Cauchy extensions).** Let $\mathbf{p}$ be an exponent, and fix a completion $\mathbf{X}$ of $X_A^P$.<fnref n="2" /> Then for all $f \in \mathbf{X}$ the Cauchy extension $C_A f$ is in $C^{\infty}(\mathbb{R} \setminus \{0\} : \mathbf{X})$, and solves the Cauchy equation

$$ \partial_t C_A f + A C_A f = 0 $$

strongly in $C^{\infty}(\mathbb{R} \setminus \{0\} : \mathbf{X})$. Furthermore we have the limits

$$ (4.7) \qquad \lim_{t \to 0^{\pm}} C_A f(t) = \chi^{\pm}(A) f \quad \text{and} \quad \lim_{t \to \pm\infty} C_A f(t) = 0 $$

in $\mathbf{X}$.

<footnote n="2">²Recall that we mean a weak-star completion when $\mathbf{p}$ is infinite, and in this case we use the weak-star topology on $\mathbf{X}$.</footnote>
PROOF. First we prove the limit results, for which it suffices to consider $t > 0$ and the semigroup extension $e^{-|t|[A]}$ in place of $C_A$. Furthermore, it suffices to consider finite exponents $\mathbf{p}$, as otherwise we can deduce the limits (4.7) by testing against $X_{A_*}^{\mathbf{p}'}$. By density, it suffices to prove the limits

$$ \lim_{t \to 0} e^{-t[A]} f = f \quad \text{and} \quad \lim_{t \to \infty} e^{-t[A]} f = 0 $$

for $f \in X_{A_*}^{\mathbf{P}}$.

For $f \in \mathbb{H}_A^{\mathbf{P}}$, these follow from arguments almost identical to those of [19, Propositions 4.5 and 4.6], the only difference being the presence of the weight $\kappa^{-\theta(\mathbf{p})}$, which does not fundamentally change the argument. Now fix an exponent $\mathbf{q} \neq \mathbf{p}$ such that $\mathbf{q} \hookrightarrow \mathbf{p}$, so that $\mathbb{H}_A^{\mathbf{q}} \hookrightarrow \mathbb{B}_A^{\mathbf{P}}$ (Proposition 4.11). For $f \in \mathbb{H}_A^{\mathbf{q}}$ we then have

$$ \lim_{t \to 0} \|e^{-t[A]} f - f\|_{\mathbb{B}_A^{\mathbf{P}}} \lesssim \lim_{t \to 0} \|e^{-t[A]} f - f\|_{\mathbb{H}_A^{\mathbf{q}}} = 0 $$

and

$$ \lim_{t \to \infty} \|e^{-t[A]} f\|_{\mathbb{B}_A^{\mathbf{P}}} \lesssim \lim_{t \to \infty} \|e^{-t[A]} f\|_{\mathbb{H}_A^{\mathbf{q}}} = 0. $$

Since $\mathbb{H}_A^{\mathbf{q}}$ is dense in $\mathbb{B}_A^{\mathbf{P}}$ (Corollary 4.9), these limits hold for all $f \in \mathbb{B}_A^{\mathbf{P}}$.

Now we prove the smoothness result, and establish the Cauchy equation. Again it suffices to consider $t > 0$, as the result for $t < 0$ uses the same argument. First observe that the function $\Phi: \mathbb{R}_+ \to H^\infty$ defined by $\Phi(t) = [z \mapsto e^{-tz}\chi^+(z)]$ is smooth with Fréchet derivative $D_t\Phi: \mathbb{R} \to H^\infty$ given by $D_t\Phi(\tau) = [z \mapsto -\tau z e^{-tz}\chi^+(z)]$. Next, note that the map $\Omega_A: H^\infty \to \mathcal{L}(\mathbf{X})$ with $\Omega_A(f) = f(A)$ is linear and bounded in the strong topology (Proposition 4.24). By the chain rule, the composition of these maps is smooth, with Fréchet derivative

$$ D_t(\Omega_A \circ \Phi)(\tau) = \Omega_A \circ D_t \Phi(\tau) = -\tau A e^{-tA} \chi^+(A). $$

We can then write

$$ \partial_t C_A f(t) = D_t(\Omega_A \circ \Phi)(1)f = -A e^{-tA} \chi^+(A)f = -A C_A f(t), $$

which completes the proof. $\square$

<figure><img src="image_19.png" /></figure>

Now we must address whether $C_A|_{\mathbb{R}_+}$ maps $X_A^{\mathbf{P},+}$ into $X^{\mathbf{P}}$. This would imply that one could construct $C^\infty(\mathbb{R}_+ : X_A^{\mathbf{P}})$ solutions to Cauchy problems which are in $X^{\mathbf{P}}$ with initial data in $X_A^{\mathbf{P},+}$. It turns out that this is only reasonable when $\theta(\mathbf{p}) < 0$. For $i(\mathbf{p}) \le 2$ we already know everything we need to prove this, but for $i(\mathbf{p}) > 2$ we need more information (see Section 5.5). In fact, for $i(\mathbf{p}) \le 2$, we can work with the semigroup extension $\mathbb{Q}_{\text{sgp},A}$ instead of the Cauchy extension; that is, we need not incorporate spectral projections into the argument.

**Theorem 4.34 (Semigroup characterisation, $i(\mathbf{p}) \le 2$).** Let $\mathbf{p}$ be an exponent with $i(\mathbf{p}) \le 2$ and $\theta(\mathbf{p}) < 0$. Then for all $f \in \bar{\mathbb{R}}(A)$,

$$ \|f\|_{X_A^{\mathbf{P}}} \approx \|\mathbb{Q}_{\text{sgp},A} f\|_{X^{\mathbf{P}}}. $$

PROOF. By Proposition 4.4, we have

$$ \text{sgp} \in \Psi_0^\infty \subset \Psi_{\theta(\mathbf{p})+}^{(-\theta(\mathbf{p})+n|\frac{1}{2}-j(\mathbf{p})|)+} \cap H^\infty \subset \Psi(X_A^{\mathbf{P}}), $$

which yields the result. $\square$

<figure><img src="image_20.png" /></figure>
**Remark 4.35.** The estimate

$$ \|f\|_{X_A^p} \lesssim \|Q_{\text{sgp},A}f\|_{X^p} $$

holds for all $\mathbf{p}$, as can be shown by a Calderón reproducing argument as in the proof of Proposition 4.4. The reverse estimate need not hold in general.

**Corollary 4.36 (Cauchy characterisation of positive adapted spaces,** $i(\mathbf{p}) \le 2$). Let $\mathbf{p}$ be an exponent with $i(\mathbf{p}) \le 2$ and $\theta(\mathbf{p}) < 0$. Then for all $f \in \bar{\mathcal{R}}(A)^+$,

$$ \|f\|_{X_A^p} \approx \|C_A f\|_{X^p}. $$

**Remark 4.37.** We have only considered the positive spectral subspace here, but of course everything works just as well for the negative spectral subspace.

The following technical results will be needed in Section 6.3.

**Lemma 4.38.** For every $M > 0$, there exist functions $\varphi^+, \varphi^- \in H^\infty$ such that $(\varphi_s^\pm(A))_{s>0}$ satisfies off-diagonal estimates of order $M$, $\varphi_s^\pm(A) = e^{-s[A]}$ on the corresponding spectral subspace $X_A^{2,\pm}$, and $\lim_{s\to 0} \varphi_s^\pm(A) = I$ in the $L^2$-strong operator topology.

For a proof, see [16, Lemma 15.1], noting that $H_A^{2,\pm} = \mathbb{B}_A^{2,\pm}$. Although this result is stated for $A \in \{DB, BD\}$ there, the proof only uses the Standard Assumptions.

**Corollary 4.39.** Let $p \in (0, \infty]$. Suppose $f \in X_A^{2,+} \cap E^p$. Then $C_A f(t) \in E^p$ for each $t \in \mathbb{R}_+$, and if $p < \infty$ then $\lim_{t \to 0+} C_A f(t) = f$ in $E^p$. Similarly for $f \in X_A^{2,-} \cap E^p$ and $t \in \mathbb{R}_-$.

**Proof.** Choose functions $\varphi^+$ as in Lemma 4.38 such that $(\varphi_s^+(A))_{s>0}$ satisfies off-diagonal estimates of large order. By Proposition 3.9, the operators $\varphi_t^+(A)$ are bounded on $E^p$. Since $\varphi_t^+(A) = e^{-tA}$ on $X_A^{2,+}$, we have $C_A f(t) = \varphi_t^+(A) f \in E^p$ for all $t \in \mathbb{R}_+$. The limit statement follows from Lemma 4.38 (which gives strong convergence in $L^2$) and Proposition 3.10 (which improves this to $E^p$). The proof for the negative spectral subspace is identical. $\square$
<figure><img src="image_21.png" /></figure>
# Spaces Adapted to Perturbed Dirac Operators

In this chapter we consider the Dirac operator

$$D = \begin{bmatrix} 0 & \operatorname{div}_{\parallel} \\ -\nabla_{\parallel} & 0 \end{bmatrix}$$

and its perturbations DB and BD, where B is a bounded elliptic multiplier (see Subsection 1.1.3). These operators act on $C^{m(1+n)}$-valued functions, so from now on we take $N = m(1 + n)$ without further mention.

The operator D, initially defined distributionally, may be defined as an unbounded operator on $L^2 = L^2(\mathbb{R}^n : C^{m(1+n)})$ with maximal domain

$$\mathcal{D}(D) = \{f \in L^2 : Df \in L^2\},$$

and generally it is better to use this interpretation of D. The nullspace and the closure of the range of D may be simply characterised by means of the transversal/tangential notation: we have $f \in \mathcal{N}(D)$ if and only if $f_{\perp} = 0$ and $\operatorname{div}_{\parallel} f_{\parallel} = 0$, and $f \in \overline{\mathcal{R}(D)}$ if and only if $\operatorname{curl}_{\parallel} f_{\parallel} = 0$. Note that $\nabla_{\parallel} f_{\perp} = 0$ implies that $f_{\perp}$ is constant, which then forces $f_{\perp} = 0$ since $f \in L^2$.

The perturbed Dirac operators DB and BD satisfy the Standard Assumptions (Theorem 3.12), so pre-Besov-Hardy-Sobolev spaces $X_{DB}^p$ and $X_{BD}^p$ are defined. In this chapter we investigate these spaces, and in particular their connection with classical smoothness spaces.

Recall from Proposition 3.2 that every bisectorial operator A on $L^2$ induces a topological splitting

$$L^2 = \mathcal{N}(A) \oplus \overline{\mathcal{R}(A)},$$

and that $\mathbb{P}_A$ denotes the orthogonal projection of $L^2$ onto $\overline{\mathcal{R}(A)}$ along $\mathcal{N}(A)$. The operators $\mathbb{P}_D$, $\mathbb{P}_{DB}$, and $\mathbb{P}_{BD}$ will appear often in our analysis.

**Remark 5.1.** The operator D may instead be one of the first order differential operators considered by Hytönen, McIntosh, and Portal [49, 48, 50]; see [19, §2.3]. Our results apply to such generalisations. However, we shall stick with this choice of D, as this operator in particular is relevant to elliptic equations.

## 5.1. D-adapted spaces

The D-adapted pre-Besov-Hardy-Sobolev spaces $X_D^p$ play a central role in our theory. For all exponents p, they may be identified as projections of classical smoothness spaces (Theorem 5.3). Recall that we use the notation $X^p$ to denote classical smoothness spaces, as in Section 2.5. These spaces are all embedded in $Z' = Z'(\mathbb{R}^n)$, the space of tempered distributions modulo polynomials.<fnref n="1" />

<footnote n="1">¹We use the same notation to refer to $C^N$-valued tempered distributions modulo polynomials for different values of N. No ambiguity is caused by this abuse of notation.</footnote>
The projection $\mathbb{P}_D$ extends boundedly to $\mathbf{X}^p$ for all finite exponents $\mathbf{p}$ by virtue of being a Fourier multiplier within the scope of the Mikhlin multiplier theorem (see [76, Theorem 5.2.2] and [50, Proposition 4.4]). When $\mathbf{p}$ is infinite, $\mathbb{P}_D$ extends by duality to $\mathbf{X}^p$. Note also that we can write this projection as

$$ \mathbb{P}_D = -\Delta^{-1}D^2, $$

and that the transversal part of $\mathbb{P}_D$ is the identity operator.

**Definition 5.2.** For all exponents $\mathbf{p}$, define $\mathbf{X}_D^p := \mathbb{P}_D \mathbf{X}^p$. Note that $\mathbf{X}_D^p$ is a closed subspace of $\mathbf{X}^p$, hence also of $\mathcal{Z}'$.

**Theorem 5.3 (Identification of D-adapted spaces).** Let $\mathbf{p}$ be an exponent. Then $\mathbf{X}_D^p = \mathbb{P}_D(\mathbf{X}^p \cap L^2)$, with norm estimates

$$ \|\mathbb{P}_D f\|_{\mathbf{X}_D^p} \lesssim \|f\|_{\mathbf{X}^p} \quad (f \in \mathbf{X}^p \cap L^2), $$

$$ \|h\|_{\mathbf{X}^p} \simeq \|h\|_{\mathbf{X}_D^p} \quad (h \in \mathbf{X}_D^p). $$

In particular, $\mathbf{X}_D^p \hookrightarrow \mathbf{X}_D^p$ with dense image.

**PROOF.** Let $\psi \in \Psi_{\infty}^{\infty}(S_{\mu})$ be a nondegenerate even function, where $\mu \in (0, \pi/2)$. Then there exists a function $\Phi \in \Psi_{\infty}^{\infty}(S_{2\mu}^{+})$ such that $\psi(z) = \Phi(z^2)$ for all $z \in S_{\mu}$, where

$$ S_{2\mu}^{+} := \{z \in \mathbb{C} \setminus \{0\} : |\arg(z)| < 2\mu\} $$

is the sector of angle $2\mu$, and where $\Psi_{\infty}^{\infty}(S_{2\mu}^{+})$ is defined analogously to $\Psi_{\infty}^{\infty}(S_{\mu})$. We then have

$$ \Phi(-t^2\Delta)f = t^{-n}\varphi(t^{-1}\cdot) * f $$

for all $f \in L^2$, where $\varphi = \hat{\psi}$. Note that $\varphi \in \mathcal{Z}$, and that we may choose $\psi$ such that $\hat{\varphi}(\xi) > 0$ when $1/2 \le |\xi| \le 2$, so we may apply Theorem 2.57, which yields

$$ (5.1) \qquad \|f\|_{\mathbf{X}^p} \simeq \|t \mapsto \Phi(-t^2\Delta)f\|_{\mathbf{X}^p}. $$

Now we prove the claimed properties of $\mathbb{P}_D$. First let $f \in \mathbf{X}^p \cap L^2$. For all $t > 0$, note that

$$ \psi(tD)\mathbb{P}_D f = \Phi(t^2D^2)\mathbb{P}_D f = \Phi(-t^2\Delta)\mathbb{P}_D f. $$

Thus

$$ \|\mathbb{P}_D f\|_{\mathbf{X}_D^p} \simeq \|\mathbb{Q}_{\psi,D} \mathbb{P}_D f\|_{\mathbf{X}^p} = \|t \mapsto \Phi(-t^2\Delta) \mathbb{P}_D f\|_{\mathbf{X}^p} \simeq \|\mathbb{P}_D f\|_{\mathbf{X}^p} \lesssim \|f\|_{\mathbf{X}^p} $$

by (5.1) and boundedness of $\mathbb{P}_D$ on $\mathbf{X}^p$. On the other hand, suppose $h \in \mathbf{X}_D^p$. We will show that $h \in \mathbf{X}^p$; since $h = \mathbb{P}_D h \in L^2$, this will complete the proof. As before, for every $t > 0$ we have

$$ \Phi(-t^2\Delta)h = \Phi(-t^2\Delta)\mathbb{P}_D h = \Phi(t^2D^2)\mathbb{P}_D h = \psi(tD)h, $$

and hence

$$ \|h\|_{\mathbf{X}^p} \simeq \|t \mapsto \Phi(-t^2\Delta)h\|_{\mathbf{X}^p} = \|\mathbb{Q}_{\psi,D}h\|_{\mathbf{X}^p} \simeq \|h\|_{\mathbf{X}_D^p}. $$

Therefore $h \in \mathbf{X}^p$. $\square$

**Remark 5.4.** In [19] this result is proven for Hardy spaces $\mathbf{H}^{(p,0)}$ with $p > n/(n+1)$. This restriction arises because the argument there uses the atomic Hardy space $\mathbb{H}_{D,\text{ato},1}^p$. To get the full result by an atomic argument one should use atoms with a higher order of D-cancellation (as with classical Hardy spaces: for $p$ small, the atomic decomposition requires atoms of higher cancellation).
It follows from Theorem 5.3 that $\mathbf{X}_D^p$ is a completion of $\mathbf{X}_D^p$ for all exponents $p$. This is the most natural completion of $\mathbf{X}_D^p$ (we also have the 'canonical completions' $\varphi \mathbf{X}_D^p$ from Section 4.3, which are not so canonical in this case).

## 5.2. Similarity of functional calculus: going between DB and BD

The operators DB and BD are, unsurprisingly, related. However, again unsurprisingly, moving between the two operators introduces technicalities. The main fact to notice is that $\mathcal{R}(DB) = \mathcal{R}(D)$, and that the restrictions $DB|_{\overline{\mathcal{R}(DB)}}$ and $BD|_{\overline{\mathcal{R}(BD)}}$ are similar under conjugation by $B|_{\overline{\mathcal{R}(DB)}}$ [18, Proposition 2.1]. Consequently, whenever $f \in \mathcal{D}(D) \cap \overline{\mathcal{R}(BD)}$ and $\varphi \in H^\infty$ we have

$$D\varphi(DB)f = \varphi(DB)Df.$$

We refer to this principle as similarity of functional calculi. This will eventually imply that we have an identification like $DX_{BD}^{p+1} = \mathbf{X}_{DB}^p$ (this is just an informal statement; it is made precise in Corollary 5.9).

We need a local coercivity property of $B$, which is proven in [19, Lemma 5.14].

**Lemma 5.5.** For any $u \in L_{\text{loc}}^2$ with $Du \in L_{\text{loc}}^2$ and any ball $B(x, t) \in \mathbb{R}^n$,

$$\int_{B(x,t)} |Du|^2 \lesssim_{B,n,m} \int_{B(x,2t)} |BDu|^2 + t^{-2} \int_{B(x,2t)} |u|^2.$$

**Proposition 5.6 (Intertwining and regularity shift).** Let $\mathbf{p}$ be an exponent. Then $D: \mathbf{X}_{BD}^{p+1} \cap \mathcal{D}(D) \to \mathbf{X}_{BD}^p \cap \mathcal{R}(D)$ is bijective, and for all $f \in \mathbf{X}_{BD}^{p+1} \cap \mathcal{D}(D)$,

$$\|Df\|_{\mathbf{X}_{BD}^p} \approx \|f\|_{\mathbf{X}_{BD}^{p+1}}.$$

**PROOF.** We only consider $\mathbf{p}^p$ with $\mathbf{p} = (p, s)$ finite; all other cases are proven by the same argument. Let $\psi \in \Psi_\infty^\infty$ be nondegenerate and define $\tilde{\psi} \in \Psi_\infty^\infty$ by $\tilde{\psi}(z) = z\psi(z)$. Then $\tilde{\psi}(DB)$ maps $\overline{\mathcal{R}(DB)}$ into $\mathcal{D}((DB)^{-1})$. Since $f \in \mathcal{D}(D)$ we have $Df \in \mathcal{R}(D) = \mathcal{R}(DB)$. Using similarity of functional calculi, write

$$\begin{aligned}
\|Df\|_{\mathbb{H}_{BD}^{\mathbf{p}}} &\approx \|t \mapsto \psi(tDB)Df\|_{T_s^p} \\
&= \|t \mapsto D\psi(tBD)f\|_{T_s^p} \\
&= \|t \mapsto D(BD)^{-1}\tilde{\psi}(tBD)f\|_{T_{s+1}^p}.
\end{aligned}$$

For all $t > 0$ we have

$$(BD)^{-1}\tilde{\psi}(tBD)f \in L^2, (BD)^{-1}\tilde{\psi}(tBD)f = \psi(tDB)Df \in L^2,$$

so we can apply Lemma 5.5 for each $t > 0$ with $u = (BD)^{-1}\tilde{\psi}(tBD)f$ as follows: for all $x \in \mathbb{R}^n$,

$$\begin{aligned}
A(t \mapsto t^{-s}D(BD)^{-1}\tilde{\psi}(tBD)f)(x) \\
&= \left( \int_0^\infty t^{-2s} \int_{B(x,t)} |(D(BD)^{-1}\tilde{\psi}(tBD)f)(y)|^2 dy \frac{dt}{t^{n+1}} \right)^{1/2} \\
&\lesssim \left( \int_0^\infty t^{-2s} \left[ \int_{B(x,2t)} |(\tilde{\psi}(tBD)f)(y)|^2 dy + \int_{B(x,2t)} |(\psi(tBD)f)(y)|^2 dy \right] \frac{dt}{t^{n+1}} \right)^{1/2}
\end{aligned}$$
$$ \lesssim \mathcal{A}(\kappa^{-s} \mathcal{Q}_{\tilde{\psi},BD} f)(x) + \mathcal{A}(\kappa^{-s} \mathcal{Q}_{\psi,BD} f)(x). $$

Therefore

$$ \|Df\|_{\mathbb{H}_{BD}^{\mathbb{P}}} \lesssim \|\mathcal{Q}_{\tilde{\psi},BD}f\|_{T^{\mathbb{P}+1}} + \|\mathcal{Q}_{\psi,BD}f\|_{T^{\mathbb{P}+1}} \simeq \|f\|_{\mathbb{H}_{BD}^{\mathbb{P}+1}}. $$

To prove the reverse estimate, using that $f \in \mathcal{D}(D) = \mathcal{D}(BD)$ and $f \in \overline{\mathcal{R}(BD)}$, write

$$ \begin{aligned} \|f\|_{\mathbb{H}_{BD}^{\mathbb{P}+1}} &= \|(BD)^{-1}BDf\|_{\mathbb{H}_{BD}^{\mathbb{P}+1}} \\ &\lesssim \|BDf\|_{\mathbb{H}_{BD}^{\mathbb{P}}} \\ &\simeq \|t \mapsto \psi(tBD)BDf\|_{T^{\mathbb{P}}} \\ &= \|t \mapsto B\psi(tBD)Df\|_{T^{\mathbb{P}}} \\ &\lesssim \|\mathcal{Q}_{\psi,BD}Df\|_{T^{\mathbb{P}}} \\ &\simeq \|Df\|_{\mathbb{H}_{BD}^{\mathbb{P}}} \end{aligned} $$

using (4.17), boundedness of $B$, and similarity of functional calculi. We next prove surjectivity. Let $g \in \mathbb{H}_{BD}^{\mathbb{P}} \cap \mathcal{R}(D)$. Since $g \in \mathcal{R}(D)$ there exists $h \in \mathcal{D}(D)$ such that $g = Dh$. Set $f = \mathbb{P}_{BD}h$. By Lemma 5.22, we have $f \in \mathcal{D}(D) \cap \overline{\mathcal{R}(BD)}$ and $Df = D\mathbb{P}_{BD}h = Dh = g$. The calculation above shows that $f \in \mathbb{H}_{BD}^{\mathbb{P}+1}$. $\square$

**Remark 5.7.** The argument above proves more: it says that if $f \in \mathcal{D}(D) \cap \overline{\mathcal{R}(BD)}$ with $Df \in \mathbb{X}_{BD}^{\mathbb{P}}$, then $f \in \mathbb{X}_{BD}^{\mathbb{P}+1}$. This argument is in particular valid when $B = I$ (although it is simpler in this case).

Let us continue with a density lemma for adapted spaces.

**Lemma 5.8.** Let $\mathbf{p}$ be an exponent. Then $\mathbb{X}_{BD}^{\mathbb{P}} \cap \mathcal{D}(D)$ and $\mathbb{X}_{BD}^{\mathbb{P}} \cap \mathcal{R}(D)$ are dense in $\mathbb{X}_{BD}^{\mathbb{P}}$ and $\mathbb{X}_{BD}^{\mathbb{P}}$ respectively.<fnref n="2" />

**PROOF.** The result for $\mathbb{X}_{BD}^{\mathbb{P}} \cap \mathcal{D}(D)$ already appears in the proof of Proposition 4.24, since $\mathcal{D}(D) = \mathcal{D}(BD)$. Let $f \in \mathbb{X}_{BD}^{\mathbb{P}}$. By the argument in Proposition 4.33, we have that

$$ f_{\varepsilon} := e^{-\varepsilon[DB]}f - e^{-(1/\varepsilon)[DB]}f $$

belongs to $\mathbb{X}_{BD}^{\mathbb{P}}$ and converges to $f$ there as $\varepsilon \to 0$. Using that

$$ f_{\varepsilon} = - \int_{\varepsilon}^{1/\varepsilon} [DB]e^{-s[DB]}f ds $$

and using the spectral subspace decomposition, we see that $f_{\varepsilon}$ is in $\mathcal{R}(D)$. $\square$

The following is an immediate corollary of Proposition 5.6 and Lemma 5.8.

**Corollary 5.9.** Assume that $\mathbf{p}$ is an exponent. Let $\mathbf{X}$ and $\mathbf{Y}$ be completions of $\mathbb{X}_{BD}^{\mathbb{P}+1}$ and $\mathbb{X}_{BD}^{\mathbb{P}}$ respectively. Then $D$ extends to an isomorphism $\mathbf{X} \to \mathbf{Y}$ (which also holds for spectral subspaces), and

$$ (5.2) \qquad D\varphi(BD) = \varphi(DB)D $$

where we write $\varphi(BD)$ and $\varphi(DB)$ to denote the extended operators. In particular, for Cauchy extensions we have

$$ (5.3) \qquad DC_{BD} = C_{BD}D. $$

<footnote n="2">²As usual, we mean weak-star density for infinite exponents.</footnote>
### 5.3. Inclusions and identifications of DB- and D-adapted spaces

Now we discuss spaces adapted to DB, and the range of exponents for which they may be identified with spaces adapted to D. As shown in the introduction, the following ‘identification region’ plays a central role in the theorems of Section 6.3. The sets $i(\mathbf{X}, A_0, A_1)$ were defined in Definition 4.30.

**Definition 5.10.** We define the identification region for DB as

$$ (5.4) \quad I(\mathbf{X}, \mathcal{D}) := i(\mathbf{X}, \mathcal{D}, \mathcal{D}) \cap i(\mathbf{X}, \mathcal{D}, \mathcal{D}) \\ = \{ \mathbf{p} \in \mathbf{E}_{\text{fin}} : \|f\|_{\mathcal{X}_{\mathcal{D}}^{\mathbf{p}}} \approx \|f\|_{\mathcal{X}_{\mathcal{D}}^{\mathbf{p}}} \text{ for all } f \in \overline{\mathcal{R}(\mathcal{D})} = \overline{\mathcal{R}(\mathcal{D})} \}, $$

and for $s \in \mathbb{R}$,

$$ I_s(\mathbf{X}, \mathcal{D}) := \{i(\mathbf{p}) : \mathbf{p} \in I(\mathbf{X}, \mathcal{D}) \text{ with } \theta(\mathbf{p}) = s\} \subset (0, \infty). $$

Note that $I(\mathbf{X}, \mathcal{D})$ is defined to be a set of finite exponents. We could include infinite exponents in this definition, but it is technically more convenient to restrict ourselves to finite exponents. Whenever we deal with infinite exponents in applications, we always consider the space under consideration as a dual space, and the exponent corresponding to its predual will be in $I(\mathbf{X}, \mathcal{D}^*)$ (see for example Theorem 6.9).

We recall some key results of the second author and Stahlhut, which follow from [19, Proposition 4.17, Proposition 4.18, Theorem 5.1 and Remark 5.2]. An alternative proof of the second result has been given by Frey, McIntosh, and Portal [35, Corollary 3.2(3)].

**Theorem 5.11.**

(i) For $p \in (n/(n+1), 2]$ we have $(p, 0) \in i(\mathbf{H}, \mathcal{D}, \mathcal{D})$, and for $p \in [2, \infty)$ we have $(p, 0) \in i(\mathbf{H}, \mathcal{D}, \mathcal{D})$.

(ii) There exist $\varepsilon, \varepsilon' > 0$ (depending on B) such that
$$ (2n/(n+2) - \varepsilon, 2 + \varepsilon') \subset I_0(\mathbf{H}, \mathcal{D}). $$

(iii) If $n = 1$, then $I_0(\mathbf{H}, \mathcal{D}) = (1/2, \infty)$.

(iv) The set $I_0(\mathbf{H}, \mathcal{D})$ is an open interval.

**Remark 5.12.** The number $2 + \varepsilon'$ is the supremum of those $p$ for which DB has bounded $H^\infty$ functional calculus in $L^p$, and $\frac{2n}{n+2} - \varepsilon$ is at most the infimum of $\frac{2p}{p+n}$ over all $p$ such that DB has bounded $H^\infty$ functional calculus in $L^p$. The numbers $\varepsilon$ and $\varepsilon'$ depend only on $n$, $m$, and the ellipticity constants.

Following Theorem 5.11, we define the interval

$$ I_0^{\min}(\mathbf{H}, \mathcal{D}) := \begin{cases} \left(\frac{2n}{n+2} - \varepsilon, 2 + \varepsilon'\right) & (n \ge 2) \\ \left(\frac{1}{2}, \infty\right) & (n = 1), \end{cases} $$

where $\varepsilon$ and $\varepsilon'$ are the numbers appearing in Theorem 5.11, and the region of exponents

$$ \tilde{I}_0^{\min}(\mathbf{H}, \mathcal{D}) := \{ \mathbf{p} \in \mathbf{E} : \theta(\mathbf{p}) = 0, i(\mathbf{p}) \in I_0^{\min}(\mathbf{H}, \mathcal{D}) \}. $$

We now extend these results to allow for more general exponents of order $\theta(\mathbf{p}) \in [-1, 0]$, and also to incorporate Besov spaces. The $\mathcal{H}$-duality operation on exponents provides a link between the inclusion regions $i(\mathbf{X}, \mathcal{D}, \mathcal{D})$ and $i(\mathbf{X}, \mathcal{D}^*, \mathcal{D})$ (Proposition 5.13), hence also between the identification regions $I(\mathbf{X}, \mathcal{D})$ and $I(\mathbf{X}, \mathcal{D}^*)$ (Corollary 5.14).
**Proposition 5.13** (♥-duality of inclusion regions). If $\mathbf{p} \in i(\mathbf{X}, \mathcal{D}\mathcal{B}, \mathcal{D})$, then $\|f\|_{\mathbb{X}_{\mathcal{D}\mathcal{B}^*}^{\mathbf{P}}^{\heartsuit}} \lesssim \|f\|_{\mathbb{X}_{\mathcal{D}}^{\mathbf{P}}^{\heartsuit}}$ for all $f \in \overline{\mathcal{R}(\mathcal{D})}$. In particular, if $\mathbf{p}^{\heartsuit}$ is also finite, then $\mathbf{p}^{\heartsuit} \in i(\mathbf{X}, \mathcal{D}, \mathcal{D}\mathcal{B}^*)$. Likewise, if $\mathbf{p} \in i(\mathbf{X}, \mathcal{D}, \mathcal{D}\mathcal{B})$, then $\|f\|_{\mathbb{X}_{\mathcal{D}\mathcal{B}^*}^{\mathbf{P}}^{\heartsuit}} \gtrsim \|f\|_{\mathbb{X}_{\mathcal{D}}^{\mathbf{P}}^{\heartsuit}}$ for all $f \in \overline{\mathcal{R}(\mathcal{D})}$, and if $\mathbf{p}^{\heartsuit}$ is also finite then $\mathbf{p}^{\heartsuit} \in i(\mathbf{X}, \mathcal{D}\mathcal{B}^*, \mathcal{D})$.

**PROOF.** We only prove the first statement. The proof of the second statement is identical, with the $\lesssim$ replaced by a $\gtrsim$. Suppose $g \in \mathcal{D}(\mathcal{D}) \cap \mathbb{X}_{\mathcal{D}^*\mathcal{D}}^{\mathbf{P}'}$. Arguing by ‘♥-duality’,

$$
\begin{align}
\|Dg\|_{\mathbb{X}_{\mathcal{D}\mathcal{B}^*}^{\mathbf{P}^{\heartsuit}}} &\approx \sup_{h \in \mathcal{D}(\mathcal{D}) \cap \mathbb{X}_{\mathcal{D}}^{(\mathbf{P}^{\heartsuit})'}} |\langle Dg, h \rangle| \nonumber \\
&= \sup_{h \in \mathcal{D}(\mathcal{D}) \cap \mathbb{X}_{\mathcal{D}}^{\mathbf{P}^{+1}}} |\langle g, Dh \rangle| \tag{5.5} \\
&\approx \sup_{h \in \mathbb{X}_{\mathcal{D}}^{\mathbf{P}}} |\langle g, h \rangle| \nonumber \\
(5.6) \qquad &\lesssim \sup_{h \in \mathbb{X}_{\mathcal{D}}^{\mathbf{P}}} |\langle g, h \rangle| \nonumber \\
&\approx \|g\|_{\mathbb{X}_{\mathcal{D}}^{\mathbf{P}'}} \nonumber \\
(5.7) \qquad &\approx \|Dg\|_{\mathbb{X}_{\mathcal{D}}^{\mathbf{P}^{\heartsuit}}} \nonumber
\end{align}
$$

with all suprema taken over appropriately normalised elements. The equivalence (5.5) uses Proposition 5.6, and then (5.6) uses that $\mathbb{X}_{\mathcal{D}\mathcal{B}}^{\mathbf{P}} \subset \mathbb{X}_{\mathcal{D}}^{\mathbf{P}}$, i.e. that $\mathbf{p} \in i(\mathbf{X}, \mathcal{D}\mathcal{B}, \mathcal{D})$. The final equivalence (5.7) uses Corollary 4.17. Since $\mathcal{D}(\mathcal{D}) \cap \mathbb{X}_{\mathcal{D}^*\mathcal{D}}^{\mathbf{P}'}$ is dense in $\mathbb{X}_{\mathcal{D}\mathcal{B}^*}^{\mathbf{P}^{\heartsuit}}$ by Lemma 5.8 (weak-star dense when $\mathbf{p}'$ is infinite), we are done. $\square$

**Corollary 5.14.** If $\mathbf{p} \in I(\mathbf{X}, \mathcal{D}\mathcal{B})$, then $\|f\|_{\mathbb{X}_{\mathcal{D}\mathcal{B}^*}^{\mathbf{P}}^{\heartsuit}} \approx \|f\|_{\mathbb{X}_{\mathcal{D}}^{\mathbf{P}}^{\heartsuit}}$ for all $f \in \overline{\mathcal{R}(\mathcal{D})}$. In particular, if $\mathbf{p}^{\heartsuit}$ is also finite, then $\mathbf{p}^{\heartsuit} \in I(\mathbf{X}, \mathcal{D}\mathcal{B}^*)$.

The following results then follow from Theorem 5.11, Proposition 5.13, and Corollary 5.14.

**Corollary 5.15.**

(i) For $p \in (1, 2]$ we have $(p, -1) \in i(\mathbf{H}, \mathcal{D}\mathcal{B}, \mathcal{D})$, for $p \in [2, \infty)$ we have $(p, -1) \in i(\mathbf{H}, \mathcal{D}, \mathcal{D}\mathcal{B})$, and for $\alpha \in [0, -1)$ we have $(\infty, -1; \alpha) \in i(\mathbf{H}, \mathcal{D}, \mathcal{D}\mathcal{B})$.

(ii) There exist $\varepsilon, \varepsilon' > 0$ such that

$$
I_{-1}(\mathbf{H}, \mathcal{D}\mathcal{B}) \supset \begin{cases} (1, \infty) & (n=1) \\ (2-\varepsilon, \infty) & (n=2) \\ (2-\varepsilon, 2n/(n-2) + \varepsilon') & (n \ge 3). \end{cases}
$$

(iii) $I_{-1}(\mathbf{H}, \mathcal{D}\mathcal{B})$ is an open interval.

We can derive non-trivial regions of exponents within which we have embeddings $\mathbb{X}_{\mathcal{D}\mathcal{B}}^{\mathbf{P}} \hookrightarrow \mathbb{X}_{\mathcal{D}}^{\mathbf{P}}$ (or the reverse embedding), or identifications $\mathbb{X}_{\mathcal{D}\mathcal{B}}^{\mathbf{P}} = \mathbb{X}_{\mathcal{D}}^{\mathbf{P}}$, by combining Theorem 5.11 and Corollary 5.15 with Theorem 4.32.

**Definition 5.16.** Define the exponent region $I_{\max}$ by

$$
I_{\max} := \left\{ \mathbf{p} \in \mathbf{E} : \theta(\mathbf{p}) \in [-1, 0], j(\mathbf{p}) \in \left( \frac{-\theta(\mathbf{p})}{n}, \frac{n+1-\theta(\mathbf{p})}{n} \right) \right\}.
$$
<figure><img src="image_22.png" /><figcaption>FIGURE 2. The region $I_{\text{max}}$.</figcaption></figure>

The region $I_{\text{max}}$ is pictured in Figure 2. Note that $I_{\text{max}}$ is the convex hull of the union of lines

$$ \{ \mathbf{p} \in \mathbf{E} : \theta(\mathbf{p}) = 0, j(\mathbf{p}) \in \left(0, \frac{n+1}{n}\right) \} \cup \{ \mathbf{p} \in \mathbf{E} : \theta(\mathbf{p}) = -1, j(\mathbf{p}) \in \left(\frac{-1}{n}, 1\right) \} $$

in the $(j, \theta)$-plane. We also define a region $I_{\text{min}}$ pictorially in Figure 3. A formal definition can be made in terms of the numbers $\varepsilon, \varepsilon'$ from Theorem 5.11 and Corollary 5.15, but it is a bit awkward to do it this way. This diagram shows the case $n \ge 3$; the region $I_{-1}(\mathbf{H}, \mathbf{DB})$ is generally larger when $n \le 2$, as shown in Corollary 5.15.

**Theorem 5.17.**

(i) Suppose that $\mathbf{p} \in I_{\text{max}}$. If $i(\mathbf{p}) \le 2$, then $\mathbb{H}_{\mathbf{DB}}^{\mathbf{p}} \hookrightarrow \mathbb{H}_{\mathbf{D}}^{\mathbf{p}}$, and if $i(\mathbf{p}) \ge 2$, then $\mathbb{H}_{\mathbf{D}}^{\mathbf{p}} \hookrightarrow \mathbb{H}_{\mathbf{DB}}^{\mathbf{p}}$. Furthermore, if $\theta(\mathbf{p}) \in (-1, 0)$, then the same results hold with $\mathbf{H}$ replaced by $\mathbf{B}$.

(ii) The region $I_{\text{min}}$ is contained in $I(\mathbf{H}, \mathbf{DB})$, and its interior is contained in $I(\mathbf{B}, \mathbf{DB})$.

Heuristically, $I_{\text{max}}$ is the largest region of exponents $\mathbf{p}$ for which we expect to have $\mathbf{p} \in I(\mathbf{X}, \mathbf{DB})$ (it is certainly the largest region one can obtain by interpolation between the results of the second author and Stahlhut). Of course, we may have $\mathbf{p} \in I(\mathbf{X}, \mathbf{DB})$ for some $\mathbf{p} \notin I_{\text{max}}$, as in the case $B = I$. The second part of Theorem 5.17 tells us that there is a fixed open region, $I_{\text{min}}$, of exponents $\mathbf{p}$ for which $\mathbb{X}_{\mathbf{DB}}^{\mathbf{p}} = \mathbb{X}_{\mathbf{D}}^{\mathbf{p}}$ for all $B$. Generally this will be true for a larger range of exponents depending on $B$.

We can exploit the first part of Theorem 5.17 and the extrapolation theorem of Šneïberg [70] to prove an openness result. Šneïberg’s extrapolation theorem was
<figure><img src="image_23.png" /><figcaption>FIGURE 3. The region $I_{\min} \subset I(\mathbf{H}, \mathcal{D}\mathcal{B})$.</figcaption></figure>

extended to quasi-Banach spaces by Kalton and Mitrea [56, Theorem 2.7], and elaborated upon by Kalton, Mayboroda, and Mitrea [55, Theorem 8.1].

**Theorem 5.18.** The exponent regions $I(\mathbf{X}, \mathcal{D}\mathcal{B}) \cap \{\mathbf{p} \in I_{\max} : \theta(\mathbf{p}) \in (-1, 0)\}$ are open.

**PROOF.** We only prove the result for $I(\mathbf{H}, \mathcal{D}\mathcal{B})$, as that for $I(\mathbf{B}, \mathcal{D}\mathcal{B})$ follows by the same argument. Suppose $\mathbf{p} \in I(\mathbf{H}, \mathcal{D}\mathcal{B}) \cap \{\mathbf{p} \in I_{\max} : \theta(\mathbf{p}) \in (-1, 0)\}$. Since $I_{\min} \subset I(\mathbf{H}, \mathcal{D}\mathcal{B})$ is an open neighbourhood of $\{\mathbf{p} \in I_{\max} : i(\mathbf{p}) = 2\}$ (relatively to $I_{\max}$), we can assume $i(\mathbf{p}) \neq 2$. We will further assume $i(\mathbf{p}) < 2$, as the argument for $i(\mathbf{p}) > 2$ is the same.

There exists a ball $B_{\mathbf{p}}$ in the $(j, \theta)$-plane such that for all $\mathbf{q} \in B_{\mathbf{p}}$ we have $i(\mathbf{q}), i([\mathbf{p}, \mathbf{q}]_{-1}) < 2$, $\theta(\mathbf{p}) \in (-1, 0)$, and $\mathbf{q} \in I_{\max}$. For all such $\mathbf{q}$ write $\mathbf{q}_0 := \mathbf{q}$ and $\mathbf{q}_1 := [\mathbf{p}, \mathbf{q}]_{-1}$; we then have $\mathbf{p} = [\mathbf{q}_0, \mathbf{q}_1]_{1/2}$.

Fix $\mathbf{q} \in B_{\mathbf{p}}$. By part (i) of Theorem 5.17, we have $\mathbf{q}_0, \mathbf{q}_1 \in I(\mathbf{H}, \mathcal{D}\mathcal{B}, D)$. Let $\varphi, \psi \in \Psi_{\infty}^{\infty}$ be Calderón siblings. Then by Lemma 4.31 the composition $\mathbb{Q}_{\varphi, D}\mathbb{S}_{\psi, \mathcal{D}\mathcal{B}}$ is bounded from $\mathbb{Q}_{\varphi, D}\mathbb{H}_{\mathcal{D}\mathcal{B}}^{\mathbf{q}_i}$ to $\mathbb{Q}_{\varphi, D}\mathbb{H}_{\mathcal{D}}^{\mathbf{q}_i}$ for $i \in \{0, 1\}$, and thus its extension, which we call $\mathbb{Q}_{\varphi, D}\mathbb{S}_{\psi, \mathcal{D}\mathcal{B}}$, is bounded between the completions $\varphi\mathbb{H}_{\mathcal{D}\mathcal{B}}^{\mathbf{q}_i} \to \varphi\mathbb{H}_{\mathcal{D}}^{\mathbf{q}_i}$. Furthermore, the assumption on $\mathbf{p}$ implies that $\mathbb{Q}_{\varphi, D}\mathbb{S}_{\psi, \mathcal{D}\mathcal{B}}$ is an isomorphism from $\varphi\mathbb{H}_{\mathcal{D}\mathcal{B}}^{\mathbf{p}}$ to $\varphi\mathbb{H}_{\mathcal{D}}^{\mathbf{p}}$. For $\nu \in (-1/2, 1/2)$ define $\mathbf{p}_{\nu} := [\mathbf{q}_0, \mathbf{q}_1]_{1/2+\nu}$. Since

$$ \varphi \mathbf{H}_{A}^{\mathbf{p}_{\nu}} = [\varphi \mathbf{H}_{A}^{\mathbf{q}_{0}}, \varphi \mathbf{H}_{A}^{\mathbf{q}_{1}}]_{1/2+\nu} $$

(Theorem 4.28), Šneïberg's extrapolation theorem implies that there exists $\varepsilon > 0$ independent of $\mathbf{q} \in B_{\mathbf{p}}$ (see Remark 5.19 below) such that for all $\nu \in (-\varepsilon, \varepsilon)$,

$$ \mathbb{Q}_{\varphi, D}\mathbb{S}_{\psi, \mathcal{D}\mathcal{B}} : \varphi \mathbb{H}_{\mathcal{D}\mathcal{B}}^{\mathbf{p}_{\nu}} \to \varphi \mathbb{H}_{\mathcal{D}}^{\mathbf{p}_{\nu}} $$

is an isomorphism. Since for each $\nu \in (-\varepsilon, \varepsilon)$ the map $\mathbb{Q}_{\varphi, D}\mathbb{S}_{\psi, \mathcal{D}\mathcal{B}}$ extends $\mathbb{Q}_{\varphi, D}\mathbb{S}_{\psi, \mathcal{D}\mathcal{B}}$ and since by the Calderón reproducing formula (Theorem 3.20) its
inverse (if it exists) extends $Q_{\varphi, DB}S_{\psi, D}$, this implies that $\mathbf{p}_{\nu} \in I(\mathbf{H}, DB)$, and thus there is a neighbourhood $O_{\mathbf{p}}$ of $\mathbf{p}$ such that

$$O_{\mathbf{p}} \subset B_{\mathbf{p}} \subset I(\mathbf{H}, DB) \cap \{\mathbf{p} \in I_{\max} : \theta(\mathbf{p}) \in (-1, 0)\}.$$

This completes the proof. □

**Remark 5.19.** The fact that $\varepsilon$ is independent of $\mathbf{q}$ in the proof above is not obvious. Inspection of the Kalton–Mitrae proof of Šneïberg’s theorem [56, Theorem 2.7] shows that $\varepsilon$ depends only on the norms of $Q_{\varphi, D}S_{\psi, DB}: \varphi\mathbf{H}_{DB}^{\mathbf{q}_i} \to \varphi\mathbf{H}_{D}^{\mathbf{q}_i}$ and the constants in the norm equivalences $\varphi\mathbf{H}_{A}^{[\mathbf{q}_0, \mathbf{q}_1]_{\theta}} = [\varphi\mathbf{H}_{A}^{\mathbf{q}_0}, \varphi\mathbf{H}_{A}^{\mathbf{q}_1}]_{\theta}$. Having fixed $\varphi$ and $\psi$, these constants are bounded independently of $\mathbf{q}$, provided that we restrict $\mathbf{q}$ to a small ball $B_{\mathbf{p}}$, as we have done.

**Remark 5.20.** If $\mathbf{p} \in I(\mathbf{X}, DB)$, then we identify $\mathbf{X}_{D}^{\mathbf{p}} = \mathbb{P}_{D}\mathbf{X}^{\mathbf{p}}$ (see Definition 5.2) as a completion of $\mathbf{X}_{DB}^{\mathbf{p}}$ via the extension of the identity map $\mathbf{X}_{DB}^{\mathbf{p}} \to \mathbf{X}_{D}^{\mathbf{p}}$. If $\mathbf{p}$ is infinite and $\mathbf{p}^{\heartsuit} \in I(\mathbf{X}, DB^{*})$, then by Proposition 5.13 we may identify $\mathbf{X}_{D}^{\mathbf{p}}$ as a weak-star completion of $\mathbf{X}_{DB}^{\mathbf{p}}$. We write $\mathbf{X}_{DB}^{\mathbf{p}}$ for these completions.

Note that we do not have equality of $\mathbf{X}_{DB}^{\mathbf{p}+}$ and $\mathbf{X}_{D}^{\mathbf{p}+}$. The first of these spaces is defined via the spectral projection $\chi^{+}(DB)$, while the second is defined via $\chi^{+}(D)$. However, we do of course have $\mathbf{X}_{DB}^{\mathbf{p}+} \subset \mathbf{X}_{D}^{\mathbf{p}}$. This will be important in applications to boundary value problems.

**Remark 5.21.** For a coefficient matrix $A$ as in the introduction, if $B = \hat{A}$, then $\hat{A}^{*} = \tilde{B} := NB^{*}N$, where

$$N := \begin{bmatrix} I & 0 \\ 0 & -I \end{bmatrix}.$$

Since $DN = -ND$ and $N$ acts on $\overline{\mathcal{R}(D)}$, the operators $DB^{*}$ and $-D\tilde{B}$ are similar on $\overline{\mathcal{R}(D)} = \overline{\mathcal{R}(DB^{*})} = \overline{\mathcal{R}(D\tilde{B})}$. Thus all functional calculus properties of $DB^{*}$ can be transferred to $D\tilde{B}$, and vice versa. This gives natural isomorphisms between $\mathbf{X}_{DB^{*}}^{\mathbf{p}}$ and $\mathbf{X}_{D\tilde{B}}^{\mathbf{p}}$, and in particular we have $I(\mathbf{X}, DB^{*}) = I(\mathbf{X}, D\tilde{B})$. However, note that this isomorphism interchanges spectral subspaces: $\mathbf{X}_{DB^{*}}^{\mathbf{p}, \pm} = \mathbf{X}_{D\tilde{B}}^{\mathbf{p}, \mp}$. For further details see [19, §12.2].

## 5.4. Projections and BD-adapted spaces

In the previous section we discussed exponents $\mathbf{p}$ for which $\mathbf{X}_{DB}^{\mathbf{p}} = \mathbf{X}_{D}^{\mathbf{p}}$. For such exponents, we show that the projections $\mathbb{P}_{D}$ and $\mathbb{P}_{BD}$ can be used to move between $\mathbf{X}_{D}^{\mathbf{p}+1}$ and the BD-adapted space $\mathbf{X}_{BD}^{\mathbf{p}+1}$. Note that we must now use the exponent $\mathbf{p} + 1$. We begin with a basic lemma.

### Lemma 5.22.

(i) For all $h \in L^2$, we have $D\mathbb{P}_{BD}h = Dh$ in the distributional sense. Consequently the projection $\mathbb{P}_{BD}$ preserves $\mathcal{D}(D)$, and $\|D\mathbb{P}_{BD}h\|_2 = \|Dh\|_2$ (note that this holds in particular for $B = I$).

(ii) The restrictions $\mathbb{P}_{BD}: \overline{\mathcal{R}(D)} \to \overline{\mathcal{R}(BD)}$ and $\mathbb{P}_{D}: \overline{\mathcal{R}(BD)} \to \overline{\mathcal{R}(D)}$ are bounded isomorphisms and mutually inverse.

(iii) The restrictions

$$\mathbb{P}_{BD}: \mathcal{D}(D) \cap \overline{\mathcal{R}(D)} \to \mathcal{D}(D) \cap \overline{\mathcal{R}(BD)}$$

$$\mathbb{P}_{D}: \mathcal{D}(D) \cap \overline{\mathcal{R}(BD)} \to \mathcal{D}(D) \cap \overline{\mathcal{R}(D)}$$
are mutually inverse.

**PROOF.** To see the first statement, use that $\mathbb{P}_{BD}$ is a projection along $\mathcal{N}(BD) = \mathcal{N}(D)$ to write $D(h - \mathbb{P}_{BD}h) = 0$. As $Dh$ and $D\mathbb{P}_{BD}h$ can be computed distribu-
tionally, the equality follows. The second statement is classical, and holds because
the projections $\mathbb{P}_{BD}$ and $\mathbb{P}_D$ have same nullspace. The third follows by combining
the first and second.
□

**Proposition 5.23.** Assume that **p** is an exponent for which $X_{DB}^{\mathbf{P}} = X_{D}^{\mathbf{P}}$. Then we have commutative diagrams of bounded bijective maps

<figure><img src="image_24.png" /></figure>

In particular, the projections are mutually inverse.

**PROOF.** Commutativity of the diagrams comes from Lemma 5.22. All we need to show is boundedness of both projections. First consider $f \in X_D^{\mathbf{P}+1} \cap \mathcal{D}(D)$. Then $\mathbb{P}_{BD}f \in \bar{\mathcal{R}}(BD)$ and $D\mathbb{P}_{BD}f = Df$, hence

$$ \|D\mathbb{P}_{BD}f\|_{X_{DB}^{\mathbf{P}}} = \|Df\|_{X_{DB}^{\mathbf{P}}} \simeq \|Df\|_{X_D^{\mathbf{P}}} \simeq \|f\|_{X_D^{\mathbf{P}+1}} $$

where we used the assumption in the first equivalence and then Proposition 5.6 for $D$ in the second. This shows that $\mathbb{P}_{BD}f \in X_{BD}^{\mathbf{P}+1}$ with the desired estimate. Now consider $f \in X_{BD}^{\mathbf{P}+1} \cap \mathcal{D}(D)$. Then we have $D\mathbb{P}_{D}f = Df$, so

$$ \|D\mathbb{P}_{D}f\|_{X_{D}^{\mathbf{P}}} = \|Df\|_{X_{D}^{\mathbf{P}}} \simeq \|Df\|_{X_{DB}^{\mathbf{P}}} \simeq \|f\|_{X_{BD}^{\mathbf{P}+1}} $$

by the same argument as before.

<figure><img src="image_25.png" /></figure>

The following corollaries are then immediate.

**Corollary 5.24.** Assume that **p** is an exponent for which $X_{DB}^{\mathbf{P}} = X_{D}^{\mathbf{P}}$. Then for $\varphi \in H^{\infty}$ we have

$$ (5.8) \qquad D\varphi(BD)\mathbb{P}_{BD} = \varphi(DB)D $$

where $\varphi(BD)$ and $\varphi(DB)$ are extensions to (any) completions. In particular, for Cauchy extensions,

$$ (5.9) \qquad DC_{BD}\mathbb{P}_{BD} = C_{DB}D. $$

**Corollary 5.25.** Let **p** be an exponent for which $X_{DB}^{\mathbf{P}} = X_{D}^{\mathbf{P}}$. Then for any completion **Y** of $X_{BD}^{\mathbf{P}+1}$ we have $\mathbb{P}_{D}\mathbf{Y} = X_{D}^{\mathbf{P}+1}$ and $\mathbb{P}_{BD}X_{D}^{\mathbf{P}+1} = \mathbf{Y}$. In particular, taking the transversal component, we have

$$ (5.10) \qquad (\mathbb{P}_D \mathbf{Y})_{\perp} = (X_D^{\mathbf{P}+1})_{\perp} = X_{\perp}^{\mathbf{P}+1}. $$

**PROOF.** The first equality in (5.10) follows from the equalities in the statement. The second is because the projection $\mathbb{P}_D$ is the identity on the $\perp$ component. □
## 5.5. The Cauchy operator on DB-adapted spaces

This section is devoted to the proof of the following theorem.

**Theorem 5.26 (Cauchy characterisation of adapted Besov-Hardy— Sobolev spaces, $i(\mathbf{p}) > 2$).** Let $\mathbf{p}$ be such that $i(\mathbf{p}) > 2$, $\theta(\mathbf{p}) \in (-1, 0)$ and $\mathbf{p}^{\diamond} \in I(\mathbf{X}, \mathcal{D}\mathcal{B}^*)$. Then for all $f \in \overline{\mathcal{R}(\mathcal{D}\mathcal{B})}$,

$$ \|C_{DB}^{+} f\|_{X^{\mathbf{p}}} \lesssim \|f\|_{X_{D}^{\mathbf{p}}}. $$

**Remark 5.27.** The condition $\mathbf{p}^{\diamond} \in I(\mathbf{X}, \mathcal{D}\mathcal{B}^*)$ is equivalent to $\mathbf{p} \in I(\mathbf{X}, \mathcal{D}\mathcal{B})$ when $\mathbf{p}$ is finite (Proposition 5.13). Note that the norm on the right hand side is that of $X_{D}^{\mathbf{p}}$, which is a completion of $X_{\mathcal{D}\mathcal{B}}^{\mathbf{p}}$ from the hypothesis and Corollary 5.14 (changing $B$ to $B^*$ and $\mathbf{p}$ to $\mathbf{p}^{\diamond}$).

**Remark 5.28.** The reverse estimate

$$ \|f\|_{X_{A}^{\mathbf{p}}} \lesssim \|C_{A}^{+} f\|_{X^{\mathbf{p}}}. $$

holds for all operators $A$ satisfying the Standard Assumptions, all $\mathbf{p}$, and all $f \in X_{A}^{\mathbf{p},+}$ (see Remark 4.35). However, we do not know whether Theorem 5.26 holds with $\mathcal{D}\mathcal{B}$ replaced by $A$ and without the assumption on $\mathbf{p}^{\diamond}$.

In contrast with Theorem 4.34, the proof of this theorem is quite long, and requires concrete arguments. First we establish a technical lemma. Recall that

$$ A(x, r_0, r_1) := B(x, r_1) \setminus B(x, r_0) $$

was defined in Section 1.3.

**Lemma 5.29.** Suppose $\theta \in (-1, 0)$, $g \in \mathcal{D}(D)$, and $f = \mathrm{Dg}$. Then for all $\xi \in \mathbb{R}^n$, $\tau > 0$, and $M \in \mathbb{N}$, we have

$$ \iint_{T(B(\xi, \tau))} |t^{-\theta}(I + it\mathcal{D}\mathcal{B})^{-2}f(x)|^2 \frac{dx dt}{t} \\ \lesssim_M \int_{B(\xi, 4\tau)} \left( \int_{B(\xi, 4\tau)} + \sum_{j=2}^{\infty} 2^{-2j(M-\frac{n}{2}-(1+\theta))} \int_{A(\xi, 2^{j-1}\tau, 2^{j+2}\tau)} \right) G(x, y) \, dx \, dy, $$

where

$$ G(x, y) := \frac{|g(x) - g(y)|^2}{|x - y|^{n+2(1+\theta)}}. $$

**PROOF.** Fix $\chi_1, \chi \in C_c^{\infty}(\mathbb{R}^n)$ such that

$$ \supp \chi_1 \subset B(\xi, 4\tau), \qquad \chi_1|_{B(\xi, 2\tau)} \equiv \text{const}, $$

$$ \supp \chi \subset A(\xi, \tau/2, 4\tau) \qquad \chi|_{A(\xi, \tau, 2\tau)} \equiv \text{const}. $$

For all $j \ge 2$ define $\chi_j(x) := \chi(2^{-j}x)$, so that $\supp \chi_j \subset A(\xi, 2^{j-1}\tau, 2^{j+2}\tau)$ and $\chi_j|_{A(\xi, 2^j\tau, 2^{j+1}\tau)} \equiv 1$. We can choose the functions $\chi_1$ and $\chi$ such that $\sum_{j=1}^{\infty} \chi_j \equiv 1$. Let

$$ c := \int_{B(\xi, \tau/2)} g. $$

Then we have

$$ f = D(g-c) = \sum_{j=1}^{\infty} D(g-c)\chi_j. $$

Here, and throughout the proof, we write $D(g-c)\chi_j$ to mean $D[(g-c)\chi_j]$.
First we prove that

$$ (5.11) \qquad \iint_{T(B(\xi, \tau))} |t^{-\theta}(I + itDB)^{-2}D(g-c)\chi_1(x)|^2 \frac{dx dt}{t} \\ \lesssim \int_{B(\xi, 4\tau)} \int_{B(\xi, 4\tau)} G(x, y) \, dx dy. $$

Since $\Psi_0^2 \in \Psi_{\theta+}^{-\theta} \cap H^\infty \subset \Psi(X_{DB}^{(2,\theta)})$ and since $(2,\theta) \in I(H, DB)$, we have

$$ \begin{aligned} \iint_{T(B(\xi, \tau))} & |t^{-\theta}(I + itDB)^{-2}D(g-c)\chi_1(x)|^2 \frac{dx dt}{t} \\ & \le \iint_{\mathbb{R}_+^{1+n}} |t^{-\theta}(I + itDB)^{-2}D(g-c)\chi_1(x)|^2 \frac{dx dt}{t} \\ & \approx \|D(g-c)\chi_1\|_{\mathbb{H}_{DB}^{(2,\theta)}}^2 \\ & \approx \|(g-c)\chi_1\|_{\dot{H}_{\theta+1}^2}^2 \\ & \approx \int_{\mathbb{R}^n} |\mathcal{D}_{\theta+1}^2(g-c)\chi_1(x)|^2 dx \end{aligned} $$

using Theorem 2.52 (which is valid since $2 > 2n/(n+1+\theta)$) in the last line.

We claim that

$$ (5.12) \qquad \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{|(g-c)\chi_1(z) - (g-c)\chi_1(y)|^2}{|z-y|^{n+2(\theta+1)}} dz dy \\ \lesssim \iint_{B(\xi, 4\tau)^2} \frac{|g(z) - g(y)|^2}{|z-y|^{n+2(\theta+1)}} dz dy, $$

from which estimate (5.11) will follow. First observe that if $y \in B(\xi, \tau)$ and $z \in B(\xi, \tau/2)$, then $\chi_1(z) = \chi_1(y) = 1$ and the estimate (5.12) (restricted to such $y$ and $z$) follows immediately. Next, we can estimate

$$ \begin{aligned} & \int_{B(\xi, \tau)^c} \int_{B(\xi, \tau/2)} \frac{|(g-c)\chi_1(z) - (g-c)\chi_1(x)|^2}{|z-x|^{n+2(\theta+1)}} dz dx \\ & \lesssim \int_{A(\xi, \tau, 4\tau)} \int_{B(\xi, \tau/2)} \frac{|(g-c)(z)(1-\chi_1(x))|^2}{|z-x|^{n+2(\theta+1)}} dz dx + A \\ & \lesssim \chi \int_{A(\xi, \tau, 4\tau)} \int_{B(\xi, \tau/2)} |z-x|^{-n-2(\theta+1)} \left( \int_{B(\xi, \tau/2)} |g(z)-g(y)| dy \right)^2 dz dx + A \\ & \lesssim r^n \int_{B(\xi, \tau/2)} |z-y|^{-n-2(\theta+1)} \left( \int_{B(\xi, \tau/2)} |g(z)-g(y)| dy \right)^2 dz + A \\ & \lesssim r^n \int_{B(\xi, \tau/2)} \int_{B(\xi, \tau/2)} \frac{|g(z)-g(y)|^2}{|z-y|^{n+2(\theta+1)}} dz dy + A \\ & \lesssim \iint_{B(\xi, 4\tau)^2} \frac{|g(z)-g(y)|^2}{|z-y|^{n+2(\theta+1)}} dz dy, \end{aligned} $$

where

$$ A \lesssim \iint_{B(\xi, 4\tau)^2} \frac{|g(z) - g(y)|^2}{|z - y|^{n+2(\theta+1)}} dz dy, $$

and where we used $|z-x| \gtrsim |z-y|$ on the region of integration.
Finally, we estimate

$$
\begin{align*}
& \int_{\mathbb{R}^n} \int_{B(\xi, \tau/2)^c} \frac{|(g-c)\chi_1(z) - (g-c)\chi_1(x)|^2}{|z-x|^{n+2(\theta+1)}} dz \, dx \\
& \lesssim \int_{B(\xi, 4\tau)} \int_{A(\xi, \tau/2, 4\tau)} \frac{|(g-c)(x)(\chi_1-1)(z) - (g-c)(x)(\chi_1-1)(x)|^2}{|z-x|^{n+2(\theta+1)}} dz \, dx + A \\
& \lesssim \chi \int_{B(\xi, 4\tau)} \int_{A(\xi, \tau/2, 4\tau)} |z-x|^{-n-2\theta} \left( \int_{B(\xi, \tau/2)} |g(x)-g(y)| \, dy \right)^2 dz \, dx + A \\
& \lesssim \int_{B(\xi, 4\tau)} r^{-2\theta} \left( \int_{B(\xi, \tau/2)} |g(x)-g(y)| \, dy \right)^2 dz \, dx + A \\
& \lesssim \iint_{B(\xi, 4\tau)^2} \frac{|g(x)-g(y)|^2}{|x-y|^{n+2(\theta+1)}} dx \, dy
\end{align*}
$$

with $A$ as before, using $r \gtrsim |x-y|$ and $|x-y|^{n+2\theta} \gtrsim |x-y|^{n+2(\theta+1)}$ on the region of integration.

Now we handle the remaining $\chi_j$ terms. For $j \ge 2$, by local coercivity (Lemma 5.5), the equality $BD(I + itBD)^{-1} = (I - (I + itBD)^{-1})/it$, and off-diagonal estimates of the families $(I + itBD)^{-1}$ and $(I + itBD)^{-2}$ of order $M$, we can estimate

$$
\begin{align*}
\iint_{T(B(\xi, \tau))} |t^{-\theta}(I + itDB)^{-2}D(g-c)\chi_j(x)|^2 \frac{dx \, dt}{t} & \\
& \lesssim \int_0^\tau t^{-2\theta} \int_{B(\xi, \tau)} |D(I + itBD)^{-2}(g-c)\chi_j(x)|^2 dx \frac{dt}{t} \\
& \lesssim \int_0^\tau t^{-2\theta} \left[ \int_{B(\xi, 2\tau)} |(BD(I + itBD)^{-2}(g-c)\chi_j(x)|^2 dx \right. \\
& \qquad \left. + \tau^{-2} \int_{B(\xi, 2\tau)} |(I + itBD)^{-2}(g-c)\chi_j(x)|^2 dx \right] \frac{dt}{t} \\
& \lesssim \int_0^\tau t^{-2\theta} \left( \frac{2^j \tau}{t} \right)^{-2M} (t^{-2} + \tau^{-2}) \|(g-c)\chi_j\|_2^2 \frac{dt}{t} \\
& \lesssim \frac{2^{-2jM}}{\tau^{2(1+\theta)}} \|(g-c)\chi_j\|_2^2.
\end{align*}
$$

Furthermore, for each $j \ge 2$ we have

$$
\begin{align*}
\|(g-c)\chi_j\|_2^2 &\le \int_{A(\xi, 2^{j-1}\tau, 2^{j+2}\tau)} \left( \int_{B(\xi, \tau/2)} |g(x) - g(y)| \, dy \right)^2 \, dx \\
&\le \tau^{-n} \int_{B(\xi, \tau/2)} \int_{A(\xi, 2^{j-1}\tau, 2^{j+2}\tau)} |g(x) - g(y)|^2 \, dx \, dy \\
&\lesssim \tau^{2(1-\theta)} 2^{j(n+2(1+\theta))} \int_{B(\xi, 4\tau)} \int_{A(\xi, 2^{j-1}\tau, 2^{j+2}\tau)} \frac{|g(x) - g(y)|^2}{|x-y|^{n+2(1+\theta)}} \, dx \, dy.
\end{align*}
$$

Putting these estimates together completes the proof of the lemma. $\square$

<figure><img src="image_26.png" /></figure>

PROOF OF THEOREM 5.26.

**Step 1: Reduction to a resolvent estimate.**
As stated in [16, Proof of Lemma 15.1], there exists $\rho \in H^\infty$ of the form

$$ \rho(z) = \sum_{m=1}^{N} c_m (1 + imz)^{-2} $$

for some scalars $c_1, \dots, c_N \in \mathbb{C}$, and $\psi \in \Psi_N^2$ nondegenerate, such that

$$ e^{-z} = \rho(z) + \psi(z) \quad \text{for all } z \in S_{\mu}^{+}. $$

Therefore we have

$$ (5.13) \quad \|C_{DB}^{+} f\|_{X^{\mathbf{P}}} \lesssim_N \|t \mapsto (I + itDB)^{-2} \chi^{+}(DB)f\|_{X^{\mathbf{P}}} + \|\mathbb{Q}_{\psi, DB} \chi^{+}(DB)f\|_{X^{\mathbf{P}}}. $$

For $N$ sufficiently large we have

$$ \psi \in \Psi_N^2 \subset \Psi_{(\theta(\mathbf{p})+n|\frac{1}{2}-j(\mathbf{p})|)+}^{-\theta(\mathbf{p})+} \cap \Psi_{+}^{+} \subset \Psi(\mathbb{X}_{DB}^{\mathbf{P}}), $$

and so

$$ \|\mathbb{Q}_{\psi, DB} \chi^{+}(DB)f\|_{X^{\mathbf{P}}} \lesssim \|f\|_{X_{DB}^{\mathbf{P}}} \approx \|f\|_{X_{D}^{\mathbf{P}}}. $$

by Proposition 5.13 and using that $\mathbf{p}^{\diamond} \in I(\mathbf{X}, DB^*)$. Therefore it suffices to prove the estimate

$$ (5.14) \quad \|t \mapsto (I + itDB)^{-2} f\|_{X^{\mathbf{P}}} \lesssim \|f\|_{X_{D}^{\mathbf{P}}}. $$

for all $f \in \overline{\mathbb{R}(DB)}$. Applying this inequality to $\chi^+(DB)f$ and invoking the boundedness of $\chi^+(DB)$ on $X_{DB}^P = X_D^P$ will yield

$$ \|C_{DB}^{+} f\|_{X^{\mathbf{P}}} \lesssim \|\chi^{+}(DB)\|_{X_{D}^{\mathbf{P}}} \approx \|f\|_{X_{D}^{\mathbf{P}}}. $$

To prove (5.14), by density (Corollary 4.9 and density of $\mathcal{R}(D)$ in $X_D^2$),<fnref n="3" /> it suffices to consider $f = Dg$ for $g \in \mathcal{D}(D) \cap \overline{\mathbb{R}(D)}$ such that

$$ \|f\|_{X_D^{\mathbf{P}}} \approx \|f\|_{X^{\mathbf{P}}} \approx \|g\|_{X^{\mathbf{P}+1}}. $$

We will now prove (5.14) by splitting cases depending on the function space under consideration.

**Step 2a: Completing the proof for Hardy–Sobolev spaces.**

Suppose $i(\mathbf{p}) < \infty$ and $(X, \mathbb{X}) = (T, \mathbb{H})$. Lemma 5.29 and a crude estimate give

$$ \iint_{T(B(\xi, \tau))} |t^{-\theta(\mathbf{p})}(I + itDB)^{-2} f(x)|^2 \frac{dx dt}{t} \\ \lesssim_M \left( 1 + \sum_{j=2}^{\infty} 2^{-2j(M-\frac{n}{2}-(1+\theta(\mathbf{p})))} \right) \int_{B(\xi, 4\tau)} |\mathcal{D}_{1+\theta(\mathbf{p})}^2 g(x)|^2 dx, $$

so by taking $M > \frac{n}{2} - (1 + \theta(\mathbf{p}))$ we get

$$ \iint_{T(B(\xi, \tau))} |t^{-\theta(\mathbf{p})}(I + itDB)^{-2} f(x)|^2 \frac{dx dt}{t} \lesssim \int_{B(\xi, 4\tau)} |\mathcal{D}_{1+\theta(\mathbf{p})}^2 g(x)|^2 dx. $$

Hence for all $\xi \in \mathbb{R}^n$ we have

$$ C_0(t \mapsto t^{-\theta(\mathbf{p})}(I + itDB)^{-2} f)(\xi)^2 \lesssim \sup_{\tau>0} \int_{B(\xi, 4\tau)} |\mathcal{D}_{1+\theta(\mathbf{p})}^2 g|^2 = \mathcal{M}_2(\mathcal{D}_{1+\theta(\mathbf{p})}^2 g)(\xi)^2, $$

<footnote n="3">³When $\mathbf{p}$ is infinite, we use weak-star density.</footnote>
where $\mathcal{M}_2(f) := \mathcal{M}(|f|^2)^{1/2}$, with $\mathcal{M}$ the Hardy-Littlewood maximal operator. By Theorem 2.9, boundedness of $\mathcal{M}_2$ on $L^i(\mathbf{p})$ (since $i(\mathbf{p}) > 2$), Theorem 2.52 (using $1 + \theta(\mathbf{p}) \in (0, 1)$), $Dg = f$, and $\mathbf{p} \in I(\mathbf{H}, \mathcal{D})$, we get

$$ \|t \mapsto (I + itDB)^{-2} f\|_{T\mathbf{P}} \lesssim \|\mathcal{D}_{1+\theta(\mathbf{p})}^2 g\|_{L^i(\mathbf{p})} \approx \|g\|_{\dot{H}^{p+1}} \approx \|f\|_{\mathbb{H}_D^p}, $$

which completes the proof in the Hardy-Sobolev case.

**Step 2b: Completing the proof for BMO-Sobolev spaces.** Suppose $\mathbf{p} = (\infty, \theta; 0)$ and $(X, \mathcal{X}) = (T, \mathbb{H})$. For all $(t, x) \in \mathbb{R}_+^{1+n}$, Lemma 5.29 and Strichartz's characterisation of $B\dot{M}O_{1+\theta}$ (Theorem 2.55) yield

$$ \begin{aligned} & \left(t^{-n} \iint_{T(B(x,t))} |\tau^{-\theta}(I + i\tau DB)^{-2} f(\xi)|^2 d\xi \frac{d\tau}{\tau}\right)^{1/2} \\ & \lesssim_M t^{-n/2} \left( t^n \|g\|_{B\dot{M}O_{1+\theta}}^2 + \sum_{j=2}^{\infty} 2^{-2j(M-\frac{n}{2}-(1+\theta))} (2^{j+2}t)^n \|g\|_{B\dot{M}O_{1+\theta}}^2 \right)^{1/2} \\ & \approx \|g\|_{B\dot{M}O_{1+\theta}} \left( 1 + \sum_{j=2}^{\infty} 2^{-2j(M-n-(1+\theta))} \right)^{1/2} \\ & \approx \|g\|_{B\dot{M}O_{1+\theta}} \end{aligned} $$

provided that $M$ is sufficiently large. Therefore as in the previous step we have

$$ \|\tau \mapsto (I + i\tau DB)^{-2} f\|_{T\mathbf{P}} \lesssim \|g\|_{\mathbb{H}^{p+1}} \approx \|f\|_{\mathbb{H}_D^p}, $$

which completes the proof in the BMO-Sobolev case.

**Step 2c: Completing the proof for Hölder spaces.** Let $\mathbf{p} = (\infty, \theta; \alpha)$. First we prove the result for $X = T$. By the definition of the Hölder norm we have

$$ G(x, y) \le \|g\|_{\dot{H}_{1+\theta+\alpha}}^2 |x-y|^{2\alpha-n}, $$

so by Lemma 5.29,

$$ \begin{aligned} & t^{-\alpha} \left( t^{-n} \iint_{T(B(x,t))} |\tau^{-\theta}(I + i\tau DB)^{-2} f(\xi)|^2 d\xi \frac{d\tau}{\tau} \right)^{1/2} \\ & \lesssim_M t^{-\alpha - \frac{n}{2}} \|g\|_{\dot{H}_{1+\theta+\alpha}} \left( \iint_{B(x,4t)^2} \frac{d\xi d\eta}{|\xi - \eta|^{n-2\alpha}} \right. \\ & \qquad \left. + \sum_{j=2}^{\infty} 2^{-2j(M-\frac{n}{2}-(1+\theta))} \int_{B(x,4t)} \int_{A(x,2^{j-1}t,2^{j+2}t)} \frac{d\xi d\eta}{|\xi - \eta|^{n-2\alpha}} \right)^{1/2} \\ & \lesssim t^{-\alpha - \frac{n}{2}} \|g\|_{\dot{H}_{1+\theta+\alpha}} \left( t^{n+2\alpha} + \sum_{j=2}^{\infty} 2^{-2j(M-\frac{n}{2}-(1+\theta))} 2^{-j(n-2\alpha)} t^{n+2\alpha} \right)^{1/2} \\ & = \|g\|_{\dot{H}_{1+\theta+\alpha}} \end{aligned} $$

for $M$ sufficiently large. Therefore, by the same concluding argument as in the previous steps,

$$ \|\tau \mapsto (I + itDB)^{-2} f\|_{T\mathbf{P}} \lesssim \|f\|_{\dot{H}^{\theta+\alpha}}. $$

In the case that $X = Z$, since $\mathbf{p}$ is infinite, Lemma 2.35 yields $T\mathbf{P} \hookrightarrow Z\mathbf{P}$, and so by previous estimate we have

$$ \|\tau \mapsto (I + itDB)^{-2} f\|_{Z\mathbf{P}} \lesssim \|\tau \mapsto (I + itDB)^{-2} f\|_{T\mathbf{P}} \lesssim \|f\|_{\dot{H}^{\theta+\alpha}}. $$

This completes the proof in the Hölder space case.
Step 2d: Completing the proof for Besov spaces.

Let $\mathbf{p} = (p, \theta)$. We use a slightly different argument here. Fix cutoff functions $\chi_1, \chi \in C_c^\infty(\mathbb{R}^n)$ with

$$ \text{supp } \chi_1 \subset B(0, 4), \qquad \chi_1|_{B(0,2)} \equiv \text{const}, $$

$$ \text{supp } \chi \subset A(0, 1/2, 4) \qquad \chi|_{A(0,1,2)} \equiv \text{const}, $$

for all integers $j \ge 2$ define $\chi_j(x) := \chi(2^{-j}x)$, and for all $j \ge 1$ define

$$ \eta_j(t, x, \xi) := \chi_j \left( \frac{x - \xi}{t} \right) \quad ((t, x) \in \mathbb{R}_+^{1+n}, \xi \in \mathbb{R}^n). $$

As before, these functions can be chosen such that $\sum_{j=1}^\infty \eta_j = 1$. Also define

$$ \tilde{g}(t, x, \xi) := g(\xi) - \int_{B(x,t)} g(\zeta) \, d\zeta \quad ((t, x) \in \mathbb{R}_+^{1+n}, \xi \in \mathbb{R}^n)). $$

By the triangle inequality we have

$$ \|t \mapsto (I + itDB)^{-2} f\|_{Z_{\theta}^{p}} $$

$$ (5.15) \quad \lesssim \sum_{j=1}^{\infty} \iint_{\mathbb{R}_{+}^{1+n}} \left( \iint_{\Omega(t,x)} |\tau^{-\theta}(I + i\tau DB)^{-2} D(\tilde{g}\eta_j)(t, x, \xi)|^2 d\xi \, d\tau \right)^{p/2} dx \frac{dt}{t}, $$

where the operators involving $D$ and $B$ act in the $\xi$ variable. By using local coercivity (Lemma 5.5) as in the proof of Lemma 5.29, the $j$-th term in (5.15) can be estimated by

$$ \begin{aligned} & \iint_{\mathbb{R}_+^{1+n}} \left( \iint_{\Omega(t,x)} |\tau^{-\theta}(I + i\tau DB)^{-2} D(\tilde{g}\eta_j)(t, x, \xi)|^2 d\xi \, d\tau \right)^{p/2} dx \frac{dt}{t} \\ & \lesssim \iint_{\mathbb{R}_+^{1+n}} \left( \iint_{\Omega_c(t,x)} |\tau^{-\theta-1}(I + i\tau BD)^{-1}(\tilde{g}\eta_j)(t, x, \xi)|^2 d\xi \, d\tau \right)^{p/2} dx \frac{dt}{t} \\ & + \iint_{\mathbb{R}_+^{1+n}} \left( \iint_{\Omega_c(t,x)} |\tau^{-\theta-1}(I + i\tau BD)^{-2}(\tilde{g}\eta_j)(t, x, \xi)|^2 d\xi \, d\tau \right)^{p/2} dx \frac{dt}{t} \end{aligned} $$

with Whitney parameter $c = (2, 2)$. The two terms in this sum differ only in the power of the resolvent. The resolvent families $(I+i\tau DB)^{-1}$ and $(I+i\tau DB)^{-2}$ both satisfy off-diagonal estimates of arbitrarily large order $M$ (as off-diagonal estimates may be composed); we will use this to estimate the terms above, making reference only to $(I+i\tau DB)^{-1}$.

For $j = 1$ we estimate

$$ \begin{aligned} & \iint_{\mathbb{R}_+^{1+n}} \left( \iint_{\Omega_c(t,x)} |\tau^{-\theta-1}(I + i\tau BD)^{-1}(\tilde{g}\eta_1)(t, x, \xi)|^2 d\xi \, d\tau \right)^{p/2} dx \frac{dt}{t} \\ & \lesssim \iint_{\mathbb{R}_+^{1+n}} \left( t^{-2\theta-2-n} \int_{B(x,4t)} |\tilde{g}(t, x, \xi)|^2 d\xi \right)^{p/2} dx \frac{dt}{t} \\ & \lesssim \iint_{\mathbb{R}_+^{1+n}} \left( \int_{B(x,4t)} \int_{B(x,t)} \left| \frac{g(\xi) - g(\zeta)}{t^{\theta+1}} \right|^2 d\zeta \, d\xi \right)^{p/2} dx \frac{dt}{t} \\ & \le \iint_{\mathbb{R}_+^{1+n}} \int_{B(x,4t)} \int_{B(x,t)} \left| \frac{g(\xi) - g(\zeta)}{t^{\theta+1}} \right|^p d\zeta \, d\xi \, dx \frac{dt}{t} \end{aligned} $$
$$
\begin{align*}
&= \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \int_0^\infty \int_{B(\xi, 4t) \cap B(\zeta, t)} dx \frac{1}{t^{2n+p(\theta+1)}} \frac{dt}{t} |g(\xi) - g(\zeta)|^p d\zeta d\xi \\
&\le \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \int_{|\zeta-\xi|/5}^\infty \frac{1}{t^{n+p(\theta+1)}} \frac{dt}{t} |g(\xi) - g(\zeta)|^p d\zeta d\xi \\
&\approx \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{|g(\xi) - g(\zeta)|^p}{|\zeta - \xi|^{n+p(\theta+1)}} d\zeta d\xi \\
&\approx \|g\|_{\dot{B}_{\theta+1}^{p,p}} \\
&\approx \|f\|_{\dot{B}_{\theta}^{p,p}},
\end{align*}
$$

using that $\eta_1(t, x, \cdot)$ is supported in $B(x, 4t)$, that $p/2 > 1$, that $B(\xi, 4t) \cap B(\zeta, t)$ is nonempty only if $t > |\zeta - \xi|/5$, and the Besov norm characterisation from Theorem 2.53.

For $j \ge 2$ we have, using off-diagonal estimates,

$$
\begin{align*}
& \iint_{\mathbb{R}_+^{1+n}} \left( \iint_{\Omega_c(t,x)} |\tau^{-\theta-1}(I + i\tau BD)^{-1}(\tilde{g}\eta_j)(t, x, \xi)|^2 d\xi d\tau \right)^{p/2} dx \frac{dt}{t} \\
&\lesssim 2^{-j(Mp-(np)/2)} \\
& \qquad \cdot \iint_{\mathbb{R}_+^{1+n}} \left( t^{-2\theta-2} \int_{A(x,2^{j-1}t, 2^{j+2}t)} \int_{B(x,t)} |g(\xi) - g(\zeta)|^2 d\zeta d\xi \right)^{p/2} dx \frac{dt}{t} \\
&\le 2^{-j(Mp-(np)/2)} \iint_{\mathbb{R}_+^{1+n}} \int_{A(x,2^{j-1}t, 2^{j+2}t)} \int_{B(x,t)} \left| \frac{g(\xi) - g(\zeta)}{t^{\theta+1}} \right|^p d\zeta d\xi dx \frac{dt}{t} \\
&= 2^{-j(Mp-(np)/2+n)} \\
& \qquad \cdot \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \int_0^\infty \frac{1}{t^{2n+p(\theta+1)}} \int_{B(\zeta,t) \cap A(\xi, 2^{j-1}t, 2^{j+1}t)} dx \frac{dt}{t} |g(\xi) - g(\zeta)|^p d\xi d\zeta \\
&\lesssim 2^{-j(Mp-(np)/2+2n)} \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \int_{2^{-j}|\zeta-\xi|}^\infty \frac{1}{t^{n+p(\theta+1)}} \frac{dt}{t} |g(\xi) - g(\zeta)|^p d\xi d\zeta \\
&\approx 2^{-j(pM-(np)/2+n-p(\theta+1))} \|f\|_{\dot{B}_\theta^{p,p}}
\end{align*}
$$

arguing similarly to before. For $M$ sufficiently large, we can thus estimate (5.15)
by summing a geometric series, yielding

$$
\|t \mapsto (I + itDB)^{-2} f\|_{Z_{\theta}^{p}} \lesssim \|f\|_{\dot{B}_{\theta}^{p,p}}
$$

as required. This completes the proof.

<figure><img src="image_27.png" /></figure>

**5.6. Appendix: Technicalities involving extensions and projections**

In previous sections we were careful to keep track of which completions (of
spaces and operators) we used. However, in previous work—in particular in [19]—
such care was not taken, and so there are some ambiguities which we must address.
Here we tie up these loose ends. This appendix may be skipped on first and subse-
quent readings.

Here, for various holomorphic functions $\varphi$, we will define operators $\tilde{\varphi}(BD)$, $\varphi(BD)$, and $\varphi(\mathbb{P}_D BD)$, which are (related to) extensions of $\varphi(BD)$ to various completions. In this section we take an arbitrary completion $\dot{D}(D)$ of $D(D)$ with respect to the seminorm $\|Dh\|_2 \simeq \|BDh\|_2$. Note that $\mathbb{H}_{BD}^{(2,1)} = D(D) \cap \overline{\mathbb{R}(BD)}$,
and thus we can pick a completion $\mathbf{H}_{BD}^{(2,1)}$ of $\mathbb{H}_{BD}^{(2,1)}$ within $\dot{D}(D)$. This applies in particular when $B = I$. By Lemma 5.22, $\mathbb{P}_D$ extends to an isomorphism from $\mathbf{H}_{BD}^{(2,1)}$ into $\mathbf{H}_D^{(2,1)}$ whose inverse is the extension of $\mathbb{P}_{BD}f$.

**Proposition 5.30.** Let $\varphi \in H^\infty$. Then there exists a bounded operator

$$ \widetilde{\varphi(BD)}: \dot{D}(D) \to \mathbf{H}_{BD}^{(2,1)} $$

such that $D\widetilde{\varphi(BD)} = \varphi(BD)D$. Moreover, for $h \in \mathcal{D}(D) \cap \overline{\mathbb{R}(BD)}$, we have $\widetilde{\varphi(BD)}h = \varphi(BD)h$ and for $h \in \mathcal{D}(D) \cap \overline{\mathbb{R}(D)}$, we have $\widetilde{\varphi(BD)}h = \varphi(BD)\mathbb{P}_{BD}h$.

**PROOF.** We argue by density. Consider $h \in \mathcal{D}(D)$. As $\varphi(DB)Dh \in \overline{\mathbb{R}(D)}$, there exists $g \in \mathbf{H}_D^{(2,1)}$ such that $Dg = \varphi(DB)Dh$. Set $\widetilde{\varphi(BD)}h = \mathbb{P}_{BD}g \in \mathbf{H}_{BD}^{(2,1)}$. We then have $D\widetilde{\varphi(BD)}h = D\mathbb{P}_{BD}g = Dg = \varphi(DB)Dh$. Taking extensions shows the first claim.

Next, let $h \in \mathcal{D}(D) \cap \overline{\mathbb{R}(BD)}$. Then

$$ D\widetilde{\varphi(BD)}h = \varphi(DB)Dh = D\varphi(BD)h, $$

where the second equality follows from Corollary 5.9. As both $\widetilde{\varphi(BD)}h$ and $\varphi(DB)h$ belong to $\mathbf{H}_{BD}^{(2,1)}$, we deduce their equality.

Finally, let $h \in \mathcal{D}(D) \cap \overline{\mathbb{R}(D)}$. Then

$$ D\widetilde{\varphi(BD)}h = \varphi(DB)Dh = \varphi(DB)D\mathbb{P}_{BD}h = D\varphi(BD)\mathbb{P}_{BD}h, $$

where the last equality is from the one above applied to $\mathbb{P}_{BD}h$. Again we deduce $\widetilde{\varphi(BD)}h = \varphi(BD)\mathbb{P}_{BD}h$. $\square$

In the next proposition we consider an extension of $\varphi(BD)$ from $\overline{\mathbb{R}(BD)}$ to all of $L^2$ given by imposing $\varphi(BD)|_{\mathcal{N}(BD)} = 0$, which is reasonable for certain $\varphi$.

**Proposition 5.31.** Let $\varphi \in H^\infty$ be such that the limit $\lim_{z \to 0, z \in S_\theta} \varphi(z)$ either vanishes or does not exist. In either case, set $\varphi(0) = 0$, and define $\overline{\varphi(BD)}: L^2 \to \overline{\mathbb{R}(BD)}$ by

$$ \overline{\varphi(BD)}h = \varphi(BD)\mathbb{P}_{BD}h. $$

Then $\overline{\varphi(BD)}$ is bounded and its restriction to $\mathcal{D}(D)$ agrees with $\widetilde{\varphi(BD)}$, and thus extends boundedly from $\mathbf{H}_D^{(2,1)}$ into $\mathbf{H}_{BD}^{(2,1)}$.

**PROOF.** Well-definedness and boundedness of $\overline{\varphi(BD)}$ on $L^2$ follow from the splitting $L^2 = \overline{\mathbb{R}(BD)} \oplus \mathcal{N}(BD)$. The construction in the previous result shows that $\overline{\varphi(BD)}$ agrees with $\varphi(BD)$ on $\mathcal{D}(D)$. $\square$

**Remark 5.32.** The assumption on $\varphi$ is necessary to avoid functions such as the resolvent $(1+iz)^{-1}$, which naturally extends to 1 by continuity at 0. In that case we would not have agreement with $\widetilde{\varphi(BD)}$ on the domain of $D$. This proposition is adapted to characteristic functions $\varphi = \chi^\pm$ and products involving characteristic functions.

One can consider the operator $\mathbb{P}_D BD$ on $\overline{\mathbb{R}(D)}$ and then establish an adapted $L^2$-Sobolev space theory. This was done in the work of the second author with McIntosh and Mourgoglou [13] and also pursued in [19, Section 11].
**Proposition 5.33.** Consider the operator $\mathbb{P}_{D}BD$ defined on $\overline{\mathcal{R}(D)}$ with do-main $\mathcal{D}(D) \cap \overline{\mathcal{R}(D)}$. Then for all $\varphi \in H^{\infty}$ we have the similarity of operator extensions

$$ \varphi(\mathbb{P}_{D}BD) = \mathbb{P}_{D}\varphi(BD)\mathbb{P}_{BD} $$

on $\mathbf{H}_{D}^{(2,1)}$.

**PROOF.** First note that $\mathbb{P}_{D}BD$ coincides with $\mathbb{P}_{D}BD\mathbb{P}_{BD}$. Thus $\mathbb{P}_{D}BD$ is similar to $BD$, restricted to $\mathcal{D}(D) \cap \overline{\mathcal{R}(BD)}$. Lemma 5.22 yields the equality $\mathbb{P}_{D}BD = \mathbb{P}_{D}BD\mathbb{P}_{BD}$ on $\mathcal{D}(D) \cap \overline{\mathcal{R}(D)}$, as $\mathbb{P}_{BD}$ restricted to $\mathcal{D}(D) \cap \overline{\mathcal{R}(D)}$ is the inverse of $\mathbb{P}_{D}$ restricted to $\mathcal{D}(D) \cap \overline{\mathcal{R}(BD)}$. The extension in the statement follows. $\square$

We remark that as $\mathbb{P}_{D}BD$ does not satisfy the Standard Assumptions (because of the presence of $\mathbb{P}_{D}$) there is no available Besov-Hardy-Sobolev theory and only the adapted Sobolev spaces $\mathcal{H}_{\mathbb{P}_{D}BD}^{s}$ of [19, §11] can be developed. These are similar to the spaces $\mathbf{H}_{BD}^{(2,s)}$ when $0 \le s \le 1$. Thus the operator $\mathbb{P}_{D}BD$ is of limited interest in our context.

The conclusion of this appendix is that the operators $\varphi(BD)$ of [19, §12.3] should be replaced by $\varphi(BD)\mathbb{P}_{BD}$ or even by $\mathbb{P}_{D}\varphi(BD)\mathbb{P}_{BD}$ to be fully defined on various extensions. But as long as they are applied to $L^2$ functions, this ambiguity is resolved by using $\overline{\varphi(BD)}$ instead. All results in [19, §12.3] are nevertheless correct.
<figure><img src="image_28.png" /></figure>
Classification of Solutions to Cauchy–Riemann
Systems and Elliptic Equations

In this chapter we implicitly work with a fixed $m \in \mathbb{N}$, meaning that we consider the equation $L_A u = 0$ where $u$ is a $\mathbb{C}^m$-valued function. All of our arguments are independent of $m$. Most of the function spaces in this chapter are spaces of $\mathbb{C}^{m(1+n)}$-valued functions, but we do not reference this in the notation (as the notation is complicated enough already). Therefore we write $L^2(\mathbb{R}^n)$ in place of $L^2(\mathbb{R}^n : \mathbb{C}^{m(1+n)})$, and so on. As in the previous chapter, we fix the Dirac operator $D$ and multipliers $B$ from Subsection 1.1.3.

**6.1. Basic properties of solutions**

We use the following properties of solutions to $L_A u = 0$.

**Proposition 6.1.** Suppose that $u$ solves $L_A u = 0$. Then the following are true.

(1) The transversal derivative $\partial_t u$ solves $L_A(\partial_t u) = 0$.

(2) The function $t \mapsto \nabla_A u(t, \cdot)$ is in $C^\infty(\mathbb{R}_+ : L_{\text{loc}}^2(\mathbb{R}^n))$, and for all Whitney parameters $c = (c_0, c_1)$ and $t \in \mathbb{R}_+$ we have

$$
\int_{B(x,c_0t)} |\nabla_A u(t,x)|^2 dx \lesssim \iint_{\Omega_c(t,x)} |\nabla_A u(s,y)|^2 ds dy.
$$

(3) For all exponents $\mathbf{p}$, all $k \in \mathbb{N}$, and all $C \ge 1$ we have

$$
\begin{align*}
\sup_{\substack{t,t' \in \mathbb{R}_+ \\ C^{-1} \le t/t' \le C}} \|\partial_t^k \nabla_A u(t, \cdot)\|_{E^{\mathbf{p}-k}(t')} &\lesssim_C \|\partial_t^k \nabla_A u\|_{X^{\mathbf{p}-k}} \\
&= \|\nabla_A \partial_t^k u\|_{X^{\mathbf{p}-k}} \\
&\lesssim \|\nabla_A u\|_{X^{\mathbf{p}}}.
\end{align*}
$$

In particular, if $\nabla_A u$ is in $X^{\mathbf{P}}$, then the function $t \mapsto \nabla_A u(t, \cdot)$ is in $C^\infty(\mathbb{R}_+ : E^{\mathbf{P}})$

PROOF. Property (1) follows from $t$-independence of the coefficients. The remaining statements are consequences of the Caccioppoli inequality, and are proven in [16, §5] for tent spaces. The corresponding $Z$-space statements are proven in the same way. $\square$

**Remark 6.2.** By Theorem 1.5, if $F$ is a solution to a Cauchy–Riemann system $(\mathrm{CR})_{\mathrm{DB}}$, then parts (2) and (3) of Proposition 6.1 hold with $\nabla_A u$ replaced by $F$. Furthermore, suppose that $G$ solves the anti-Cauchy–Riemann system

$$
(a \mathrm{CR})_{\mathrm{DB}} : \begin{cases} \partial_t G - \mathrm{DBG} = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \mathrm{curl}_{\|} G_{\|} = 0 & \text{in } \mathbb{R}_+^{1+n} \end{cases}
$$
defined analogously to $(CR)_{DB}$ but with a sign change. Then the reflection $F(t) := G(-t)$ solves $(CR)_{DB}$ on the lower half-space $\mathbb{R}_-^{1+n}$. By using $X$-spaces associated with the lower half-space rather than the upper half-space, parts (2) and (3) of Proposition 6.1 hold with $\nabla_A u$ replaced by $F$ and with $\mathbb{R}_+$ replaced by $\mathbb{R}_-$. A simple reflection argument then shows that parts (2) and (3) of Proposition 6.1 hold for $G$.

As a consequence we obtain the following technical lemma, which is analogous to [16, Lemma 10.2].

**Lemma 6.3.** Fix $\mathbf{p}$ with $i(\mathbf{p}) < 2$ and $\theta(\mathbf{p}) < 0$, suppose $M \in \mathbb{N}$, and let $f \in \mathbb{X}_{DB}^{\mathbf{p}}$. Then for all $t > 0$ we have that $(tDB)^M e^{-t[DB]} \chi^{\pm}(DB)f \in E^{\mathbf{p}}$, with

$$ \sup_{t>0} \|(tDB)^M e^{-t[DB]} \chi^{\pm}(DB)f\|_{E^{\mathbf{p}}(t)} \lesssim \|f\|_{\mathbb{X}_{DB}^{\mathbf{p}}}. $$

**PROOF.** We estimate

$$ \sup_{t>0} \|(tDB)^M e^{-t[DB]} \chi^{\pm}(DB)f\|_{E^{\mathbf{p}}(t)} \lesssim \|t \mapsto (tDB)^M e^{-t[DB]} \chi^{\pm}(DB)f\|_{X^{\mathbf{p}}} \\ \simeq \|f\|_{\mathbb{X}_{DB}^{\mathbf{p}}}. $$

The first line comes from Proposition 6.1, using that $(DB)^M e^{-t[DB]} \chi^{\pm}(DB)f$ solves either $(CR)_{DB}$ or $(aCR)_{DB}$. The second line is due to the fact that $[z \mapsto z^M e^{-[z]}] \in \Psi(\mathbb{X}_{DB}^{\mathbf{p}})$ when $i(\mathbf{p}) < 2$ and $\theta(\mathbf{p}) < 0$. $\square$

## 6.2. Decay of solutions at infinity

In the boundary value problems introduced in Subsection 1.1.2, we impose the decay condition

$$ (6.1) \qquad \lim_{t \to \infty} \nabla_{\|} u(t, \cdot) = 0 \quad \text{in } \mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{mn}) $$

for a solution $u$ to $L_A u$ with $\nabla u$ in $X^{\mathbf{p}}$. This is equivalent to

$$ (6.2) \qquad \lim_{t \to \infty} \langle u(t, \cdot), \varphi \rangle = 0 \quad \text{for all } \varphi \in S(\mathbb{R}^n : \mathbb{C}^m) \text{ with } \int x^\alpha \varphi(x) dx = 0 \text{ when } \alpha \neq 0 $$

because the divergence operator is surjective from the set of test functions in (6.2) onto $\mathcal{Z}(\mathbb{R}^n : \mathbb{C}^{mn})$. This is a convenient generalisation of the usual pointwise vanishing limit at $\infty$, since solutions to the elliptic equations that we consider may not have pointwise values. In this section we show that this decay condition is redundant for certain exponents $\mathbf{p}$ (quantified in terms of $A$). In fact, our results give decay not only in $\mathcal{Z}'$, but also in the slice space $E^\infty$ (Lemma 6.4) or in $L^2$ (in Lemma 6.7). Demanding decay in $\mathcal{Z}'$ is really just an artefact of having identified the classical smoothness spaces $X^{\mathbf{p}}$ as subspaces of $\mathcal{Z}'$.

By the 'hole filling technique' attributed to Widman (see [36, page 23]) there exists a number $\lambda(A) \in (0, n+1)$ such that for all $\lambda \in [0, \lambda(A))$, for all $(t_0, x_0) \in \mathbb{R}_+^{n+1}$ and $0 < r < R < \infty$, and for all weak solutions $u$ to $L_A u = 0$, we have

$$ (6.3) \qquad \iint_{B((t_0, x_0), r)} |\nabla u(t, x)|^2 dx dt \lesssim_{\lambda} \left(\frac{r}{R}\right)^{\lambda} \iint_{B((t_0, x_0), R)} |\nabla u(t, x)|^2 dx dt, $$

where $B((t_0, x_0), r)$ and $B((t_0, x_0), R)$ denote open balls in $\mathbb{R}^{1+n}$, and where $R$ is such that $B((t_0, x_0), R) \subset \mathbb{R}_+^{1+n}$. These balls can be taken with respect to any norm on $\mathbb{R}^{1+n}$, keeping in mind that the implicit constant in (6.3) will depend on
<figure><img src="image_29.png" /><figcaption>FIGURE 4. The exponent region in Lemma 6.4.</figcaption></figure>

the chosen norm. By ellipticity we may replace the gradient $\nabla$ with the conormal gradient $\nabla_A$ in (6.3).

**Lemma 6.4.** Suppose that the exponent **p** lies in the shaded region pictured in Figure 4, which depends on $\lambda(A)$. Let $u$ be a solution to $L_A u = 0$ on $\mathbb{R}_+^{1+n}$ such that $\nabla_A u \in X^P$. Then $\lim_{t\to\infty} \nabla_A u(t, \cdot) = 0$ in $E^\infty$ (and therefore also in $Z'$).

**Remark 6.5.** The shaded region in Figure 4 is the open half-plane determined by the equation $j(\mathbf{p}) > \frac{\theta(\mathbf{p})}{n} - \frac{n+1-\lambda(A)}{2n}$. Note that $\frac{n+1-\lambda(A)}{2n} \ge \frac{1}{2}$ if and only if $\lambda(A) \le 1$; we have pictured the case $\lambda(A) > 1$. In Lemma 6.7 we handle exponents **p** with $i(\mathbf{p}) \le 2$ and $\theta(\mathbf{p}) < 0$ independently of $\lambda(A)$, so this distinction is not important.

PROOF OF LEMMA 6.4. The region pictured in Figure 4 is precisely the set of exponents **p** such that there exists an infinite exponent **q** with $\mathbf{p} \mapsto \mathbf{q}$ and $r(\mathbf{q}) < \frac{\lambda(A)-(n+1)}{2}$. Fix such a **q**. For all $\lambda < \lambda(A)$ we can estimate

$$ \|\nabla_A u(t, \cdot)\|_{E_0^\infty(1)} \approx \sup_{x \in \mathbb{R}^n} \left( \int_{B(x,1)} |\nabla_A u(t, y)|^2 dy \right)^{1/2} $$

$$ (6.4) \qquad \lesssim \sup_{x \in \mathbb{R}^n} \left( \int_{t-\frac{1}{2}}^{t+\frac{1}{2}} \int_{B(x,1)} |\nabla_A u(s, y)|^2 dy ds \right)^{1/2} $$

$$ \begin{align} (6.5) \qquad & \lesssim \sup_{x \in \mathbb{R}^n} \left( t^{-\lambda} \int_{t-\frac{1}{2}}^{t+\frac{1}{2}} \int_{B(x,t)} |\nabla_A u(s, y)|^2 dy ds \right)^{1/2} \nonumber \\ & \lesssim t^{\frac{(n+1)-\lambda}{2}} + r(\mathbf{q}) \left( \int_{t-\frac{1}{2}}^{t+\frac{1}{2}} \|\nabla_A u(s, \cdot)\|_{E^{\mathbf{q}}(s)}^2 ds \right)^{1/2} \nonumber \\ & \lesssim t^{\frac{(n+1)-\lambda}{2}} + r(\mathbf{q}) \|\nabla_A u\|_{X^P} \end{align} $$

where (6.4) follows from Proposition 6.1, (6.5) follows from (6.3),<fnref n="1" /> and the last line follows from the embedding $E^P(s) \hookrightarrow E^{\mathbf{q}}(s)$ and another application of Proposition 6.1. For $\lambda$ sufficiently close to $\lambda(A)$ we have $(n+1-\lambda)/2 + r(\mathbf{q}) < 0$, so we find that $\lim_{t\to\infty} \nabla_A u(t, \cdot) = 0$ in $E^\infty$. □

**Remark 6.6.** It is known that $\lambda(A) > n-1$ if and only if $A$ satisfies the De Giorgi–Nash–Moser condition (7.2) of all exponents less than $\alpha = (\lambda(A)-(n-1))/2$.

<footnote n="1">¹Here we use the balls $B((t, x), r) := (t - r/2, t + r/2) \times B(x, r)$.</footnote>
<figure><img src="image_30.png" /><figcaption>FIGURE 5. The region in Lemma 6.4 in the case that A satisfies the De Giorgi–Nash–Moser condition with exponent α.</figcaption></figure>

In this case we have $\frac{\lambda(A)-(n+1)}{2} = \alpha - 1$ and $\frac{n+1-\lambda(A)}{2n} = \frac{1-\alpha}{n}$, and Lemma 6.4 then holds for the shaded region pictured in Figure 5. Evidently this region increases as the De Giorgi–Nash–Moser exponent $\alpha$ increases.

As a concrete example, consider A satisfying the De Giorgi–Nash–Moser condition of some exponent $\alpha$ close to 1, and let $p \in (1, \infty)$ and $\theta \in (-1, 0)$ with $\theta$ close to 0. Then whenever $u$ solves the elliptic equation $L_A u = 0$ on $\mathbb{R}^n$ and $\nabla_A u \in T^{(p,\theta)}$, we have that $\lim_{t \to \infty} \nabla_A u(t, \cdot) = 0$ in $E^\infty$ provided that $p$ is not too large. However, for $p$ sufficiently large (depending on $\theta$ and $\alpha$), this decay is not guaranteed.

An easier argument can be used to deduce decay in $L^2$ for exponents $p$ with $i(\mathbf{p}) \le 2$ and $\theta(\mathbf{p}) < 0$.

**Lemma 6.7.** Let $\mathbf{p} = (p, s)$ with $i(\mathbf{p}) \le (0, 2]$ and $\theta(\mathbf{p}) < 0$, and suppose $F \in X^\mathbf{P}$ solves $(CR)_{DB}$ or $(aCR)_{DB}$. Then $\lim_{t \to 0} F(t) = 0$ in $L^2$.

*PROOF.* By Proposition 6.1, for all $t \in \mathbb{R}_+$ we have $\|F(t)\|_{EP(t)} \lesssim \|F\|_{XP}$, and so

$$ \|F(t)\|_2 \lesssim \|F(t)\|_{E_0^{i(\mathbf{p})}(t)} = t^{\theta(\mathbf{p})} \|F(t)\|_{EP(t)} \lesssim t^{\theta(\mathbf{p})} \|F\|_{XP} $$

using the embedding $E_0^{i(\mathbf{p})}(t) \hookrightarrow E_0^2(t) = L^2$ (the equality $E_0^2(t) = L^2$ is a consequence of Fubini's theorem). Since $\theta(\mathbf{p}) < 0$, we have

$$ \lim_{t \to \infty} F(t) = 0 $$

in $L^2$.

□

### 6.3. Classification and representation theorems

The following classification theorems (Theorems 6.8 and 6.9) for solutions to $(CR)_{DB}$ (as formulated in Subsection 1.1.3) will be proven in the next section. For the convenience of the reader we state them both here. We remind the reader that more ‘concrete’ versions of these theorems are stated in Subsection 1.1.5 of the introduction, and that a table detailing the possible interpretations of $(X^\mathbf{P}, X^\mathbf{P})$ is in Section 2.7.

**Theorem 6.8 (Classification of solutions to $(CR)_{DB}$, $i(\mathbf{p}) \le 2$).** Let $\mathbf{p}$ be an exponent with $i(\mathbf{p}) \le 2$ and $\theta(\mathbf{p}) < 0$, and fix a completion $\mathbf{X}$ of $X_{DB}^\mathbf{P}$.
(i) For all $F_0 \in \mathbf{X}^+$, $\mathbf{C}_{DB}^+ F_0$ solves (CR)$_{DB}$, and $\|\mathbf{C}_{DB}^+ F_0\|_{\mathbf{X}^+} \lesssim \|F_0\|_{\mathbf{X}}$.

(ii) Conversely, if $F \in \mathbf{X}^+$ solves (CR)$_{DB}$, then there exists a unique $F_0 \in \mathbf{X}^+$ such that $F = \mathbf{C}_{DB}^+ F_0$. Furthermore, $\|F_0\|_{\mathbf{X}} \lesssim \|F\|_{\mathbf{X}^+}$.

The boundary limit $F_0$ is necessarily in a completion of $\mathbf{X}_{DB}^{\mathbf{P},+}$. When we identify such a completion with a classical function space (as for example when $\mathbf{p} \in I(\mathbf{X}, \mathbf{DB}))$, the boundary limit exists in a classical sense.

When $i(\mathbf{p}) > 2$ the argument is much more complicated. In this case we must restrict attention to exponents $\mathbf{p}$ with $\theta(\mathbf{p}) \in (-1, 0)$, and such that the adapted space $\mathbf{X}_{DB}^{\mathbf{P},*}$ may be identified with the classical space $\mathbf{X}_{D}^{\mathbf{P},*}$, plus we need an additional decay condition on $F$. Recall that for such $\mathbf{p}$ we have identified $\mathbf{X}_{DB}^{\mathbf{P},+}$ as a subspace of $\mathbf{X}_{D}^{\mathbf{P},+}$, and that if $\mathbf{p}$ is finite, then $\mathbf{p}^{\vee} \in I(\mathbf{X}, \mathbf{DB}^*)$ if and only if $\mathbf{p} \in I(\mathbf{X}, \mathbf{DB})$ (Proposition 5.13).

**Theorem 6.9 (Classification of solutions to (CR)$_{DB}$, $i(\mathbf{p}) > 2$).** Let $\mathbf{p}$ be an exponent with $i(\mathbf{p}) > 2$, $\theta(\mathbf{p}) \in (-1, 0)$, and $\mathbf{p}^{\vee} \in I(\mathbf{X}, \mathbf{DB}^*)$. Furthermore, if $\mathbf{p}$ is infinite, suppose that $r(\mathbf{p}) < 0$.

(i) If $F_0 \in \mathbf{X}_{DB}^{\mathbf{P},+}$, then $\mathbf{C}_{DB}^+ F_0$ solves (CR)$_{DB}$, $\lim_{t \to \infty} \mathbf{C}_{DB}^+ F_0(t)_{\|} = 0$ in $\mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{nm})$, and $\|\mathbf{C}_{DB}^+ F_0\|_{\mathbf{X}^+} \lesssim \|F_0\|_{\mathbf{X}_{DB}^{\mathbf{P},+}} \simeq \|F_0\|_{\mathbf{X}_{D}^{\mathbf{P},+}}$.

(ii) Conversely, if $F \in \mathbf{X}^+$ solves (CR)$_{DB}$ and $\lim_{t \to \infty} F(t)_{\|} = 0$ in $\mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{nm})$, then there exists a unique $F_0 \in \mathbf{X}_{DB}^{\mathbf{P},+}$ such that $F = \mathbf{C}_{DB}^+ F_0$. Furthermore, $\|F_0\|_{\mathbf{X}_{D}^{\mathbf{P},+}} \simeq \|F_0\|_{\mathbf{X}_{DB}^{\mathbf{P},+}} \lesssim \|F\|_{\mathbf{X}^+}$.

Note that the conditions on $\mathbf{p}$ in Theorem 6.9 imply that $\mathbf{p} \in I_{\max}$. The conditions in Theorems 6.8 and 6.9, along with those in [16, Theorems 1.1 and 1.3], suggest the definition of the following classification region, which we will use in stating our results in the following sections.

**Definition 6.10.** We define the classification region for DB as

$$ J(\mathbf{X}, \mathbf{DB}) := \{ \mathbf{p} \in I_{\max} : [i(\mathbf{p}) \le 2 \text{ and } \mathbf{p} \in I(\mathbf{X}, \mathbf{DB})] \\ \text{ or } [i(\mathbf{p}) > 2 \text{ and } \mathbf{p}^{\vee} \in I(\mathbf{X}, \mathbf{DB}^*)] \}. $$

It is not strictly necessary to impose $\mathbf{p} \in I_{\max}$ in this definition, but it is technically convenient and in our results we lose nothing from doing so. Unlike the identification region $I(\mathbf{X}, \mathbf{DB})$, we do not restrict to finite exponents in this definition. Note however that

$$ \{\mathbf{p} \in J(\mathbf{X}, \mathbf{DB}) \cap \mathbf{E}_{\text{fin}} : \theta(\mathbf{p}) \in [-1, 0]\} = I(\mathbf{X}, \mathbf{DB}) $$

follows from Proposition 5.13.

For the moment let us assume Theorems 6.8 and 6.9. Although we have proven more in the case $i(\mathbf{p}) \le 2$, the following result is a good summary of what we can deduce for solutions of the elliptic equation.

**Theorem 6.11 (First representation theorem).** Let $\mathbf{p}$ be an exponent in the classification region $J(\mathbf{X}, \mathbf{DB})$ with $B = \hat{A}$. Let $u$ be a solution to $L_A u = 0$ in $\mathbb{R}_{+}^{1+n}$ with $\nabla_A u \in \mathbf{X}^+$ and $\lim_{t \to \infty} \nabla_{\|} u(t) = 0$ in $\mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{mn})$. Then there exists a unique $F_0 \in \mathbf{X}_{DB}^{\mathbf{P},+}$ such that $\nabla_A u = \mathbf{C}_{DB}^+(F_0)$ and $F_0 = \lim_{t \to 0} \nabla_A u(t, \cdot)$ in $\mathbf{X}^+$.

We often refer to the element $F_0$ obtained in this theorem as $\nabla_A u|_{t=0}$.
PROOF. The results for $\theta(\mathbf{p}) = 0$ and $\theta(\mathbf{p}) = -1$ correspond to [16, Theorems 1.1 and 1.3], so we need only consider $\theta(\mathbf{p}) \in (-1, 0)$. By Theorem 1.5, solutions $u$ to $L_A u = 0$ are in bijective correspondence (up to an additive constant) to solutions $F$ to $(CR)_{DB}$, with $F = \nabla_A u$. By Theorems 6.8 and 6.9 and by our assumptions, $F = \mathbf{C}_{DB}^+ F_0$ for a unique $F_0 \in \mathbf{X}_{DB}^{\mathbf{P},+}$, and by Proposition 4.33,

$$F_0 = \lim_{t \to 0} \nabla_A u(t, \cdot)$$

with limit in $\mathbf{X}^{\mathbf{P}}$.

□

**Remark 6.12.** We stress again that that since the exponent $\mathbf{p}$ is in the classification region, the space of elements $F_0$ obtained in the previous theorem is a subspace of $\mathbf{X}_D^{\mathbf{P}}$, which is a space of distributions modulo polynomials. We obtain a posteriori that $\lim_{t \to \infty} \nabla_A u(t) = 0$ in $\mathbf{X}^{\mathbf{P}}$, which encodes behaviour at infinity of the conormal derivative. Note also that if $\mathbf{p}$ is in the region given by Lemma 6.4 then the decay condition on $\nabla_{\|} u$ is redundant. In particular the condition may be eliminated from the requirements when $i(\mathbf{p}) \le 2$, which is in agreement with the statement of Theorem 6.8.

Through Theorem 6.11 we may obtain a representation for solutions themselves, rather than for their conormal gradients.

**Theorem 6.13 (Second representation theorem).** Let $\mathbf{p}$ be an exponent in $J(\mathbf{X}, DB)$ with $B = \hat{A}$. Let $u$ be a solution to $L_A u = 0$ in $\mathbb{R}_+^{1+n}$ with $\nabla_A u \in \mathbf{X}^{\mathbf{P}}$ and $\lim_{t \to \infty} \nabla_{\|} u(t) = 0$ in $Z'(\mathbb{R}^n : \mathbb{C}^{mn})$. Then there exists a unique $\tilde{F}_0 \in \mathbb{P}_D \mathbf{X}_{BD}^{\mathbf{P},+1,+}$ such that

$$(6.6) \qquad u = -(\mathbb{P}_D C_{BD}^{\mathbf{P}} \mathbb{P}_{BD} \tilde{F}_0)_{\perp}$$

up to an additive constant. Moreover, $\tilde{F}_0 = \nabla_A u|_{t=0}$ and $(\tilde{F}_0)_{\perp} = -u|_{t=0}$ up to an additive constant in $(\mathbf{X}_D^{\mathbf{P},+1})_{\perp}$, where $\nabla_A u|_{t=0}$ is the trace obtained in the first representation theorem. All limits are taken in the weak-star topology when $\mathbf{p}$ is infinite.

PROOF. By Corollary 5.24, there exist unique $\tilde{F}_0 \in \mathbf{X}_D^{\mathbf{P},+1}$ and $F_0^{\|} \in \mathbf{X}_{BD}^{\mathbf{P},+1}$ such that $\tilde{F}_0 = DF_0^{\|} = \nabla_A u|_{t=0}$. Moreover, $\tilde{F}_0 = \mathbb{P}_D F_0^{\|}$ and $F_0^{\|} = \mathbb{P}_{BD} \tilde{F}_0 \in \mathbf{X}_{BD}^{\mathbf{P},+1,+}$ by Corollary 5.9. We will show (6.6) for this choice of $\tilde{F}_0$.

For simplicity assume $\nabla_A u|_{t=0} \in \mathbf{X}_{BD}^{\mathbf{P},+} \cap \mathcal{R}(D)$ (a dense class) so that $\tilde{F}_0$ and $F_0^{\|}$ belong to $\mathcal{D}(D)$, and likewise $(C_{BD}^{\mathbf{P}} \mathbb{P}_{BD} \tilde{F}_0)(t) \in \mathcal{D}(D)$ for all $t > 0$ (see Proposition 5.23). Set $v = -(\mathbb{P}_D C_{BD}^{\mathbf{P}} \mathbb{P}_{BD} \tilde{F}_0)_{\perp}$. By (5.9) we have

$$\partial_t v = (BD \mathbb{P}_D C_{BD}^{\mathbf{P}} \mathbb{P}_{BD} \tilde{F}_0)_{\perp} = (BDC_{BD}^{\mathbf{P}} \mathbb{P}_{BD} \tilde{F}_0)_{\perp} = (BC_{DB}^{\mathbf{P}} (\nabla_A u|_{t=0}))_{\perp}$$

and

$$\nabla_x v = (D \mathbb{P}_D C_{BD}^{\mathbf{P}} \mathbb{P}_{BD} \tilde{F}_0)_{\|} = (DC_{BD}^{\mathbf{P}} \mathbb{P}_{BD} \tilde{F}_0)_{\|} = (C_{DB}^{\mathbf{P}} (\nabla_A u|_{t=0}))_{\|}.$$

By the relation $B = \hat{A}$, these two equalities are equivalent to $\nabla_A v = C_{DB}^{\mathbf{P}} (\nabla_A u|_{t=0})$, hence $\nabla_A v = \nabla_A u$ on $\mathbb{R}_+^{1+n}$ and $v = u$ up to an additive constant. The general case follows by a limiting argument. □

**Remark 6.14.** We obtain from the proof that $\lim_{t \to \infty} u(t) = 0$ in $(\mathbf{X}_D^{\mathbf{P},+1})_{\perp}$. As $(\mathbf{X}_D^{\mathbf{P},+1})_{\perp}$ can be embedded in the space of distributions modulo constants, this is stronger than decay in $Z'$.
We can consider $-C_{BD}^{+} \mathbb{P}_{BD} \tilde{F}_0$ as a conjugate system to $u$, because it solves the equation $\partial_t F + BDF = 0$. Its transversal component, morally speaking, should be $u$ (up to a constant); this point of view was taken by the second author and Stahlhut [19]. However this is a purely abstract object (unless $\tilde{F}_0 \in L^2$), and as such does not have a well-defined transversal component. Nevertheless things can be made concrete: $-\mathbb{P}_D C_{BD}^{+} \mathbb{P}_{BD} \tilde{F}_0(t)$ is an element of $\mathbf{X}_D^{\mathbf{P}+1}$ (hence a tempered distribution modulo polynomials) for all $t > 0$, its transversal component is exactly $u(t)$ (up to an additive constant), and its tangential component can be used as a conjugate vector. This was done by Hofmann, Kenig, Mayboroda, and Pipher [41] in a different manner and in a different formulation.

## 6.4. Proofs of classification theorems

Now we prove Theorems 6.8 and 6.9. We deal with both theorems simultaneously.

**6.4.1. Construction of solutions via Cauchy extension.** We begin with the proof of part (i) of both theorems.

Let $F_0 \in \mathbf{X}_{DB}^{\mathbf{P},+}$. Then the estimate $\|C_{DB}^{+} F_0\|_{X^{\mathbf{P}}} \lesssim \|F_0\|_{\mathbf{X}_{D}^{\mathbf{P}}}^*$ follows from either Theorem 4.34 or Theorem 5.26.

In Proposition 4.33 we showed that $C_{DB}^{+} F_0$ solves $(CR)_{DB}$ strongly in $\mathbf{X}_{DB}^{\mathbf{P},+}$. Generally $\mathbf{X}_{DB}^{\mathbf{P},+}$ need not be contained in $L_{\text{loc}}^2(\mathbb{R}^n)$, and so these two solution concepts need not coincide. We must argue differently here. If $F_0 \in \overline{\mathbb{R}(DB)}$, then Proposition 3.6 implies that $C_{DB}^{+} F_0$ solves $(CR)_{DB}$ strongly in $C^{\infty}(\mathbb{R}_+ : L^2)$, and this implies that $C_{DB}^{+} F_0$ solves $(CR)_{DB}$. It remains to deal with $F_0 \in \mathbf{X}_{DB}^{\mathbf{P}} \setminus \mathbf{X}_{DB}^{\mathbf{P},+}$. For such an $F_0$, let $(F_0^k)_{k \in \mathbb{N}}$ be a sequence in $\mathbf{X}_{DB}^{\mathbf{P}}$ which converges to $F_0$ as $k \to \infty$ (in the weak-star topology when $\mathbf{p}$ is infinite). Then, again using either Theorem 4.34 or Theorem 5.26, we have

$$ \lim_{k \to \infty} C_{DB}^{+} F_0^k = C_{DB}^{+} F_0 \quad \text{in } \mathbf{X}^{\mathbf{P}}, $$

and hence also in $L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$. It follows that $C_{DB}^{+} F_0$ solves $(CR)_{DB}$.

It remains to show that $\lim_{t \to \infty} C_{DB}^{+} F_0(t) \| = 0$ in $Z'(\mathbb{R}^n : \mathbb{C}^{nm})$ when $i(\mathbf{p}) > 2$. This follows from Proposition 4.33, since we have $\lim_{t \to \infty} C_{DB}^{+} F_0(t) = 0$ in $\mathbf{X}_{DB}^{\mathbf{P}} \hookrightarrow Z'$.

**6.4.2. Initial limiting arguments.** We now begin preparation for the proof of part (ii) of Theorems 6.8 and 6.9. This section is a rephrasing of the start of [16, §8]. There are no fundamentally new ideas, but the notation and the flow of ideas are simplified.

For $t_0 \in \mathbb{R}_+$ we write $\mathbb{R}_{t_0} := \mathbb{R} \setminus \{t_0\}$ and $\mathbb{R}_{+,t_0} := \mathbb{R}_+ \setminus \{t_0\}$.

**Definition 6.15.** For $t_0 \in \mathbb{R}_+$ and $\varphi \in L^2(\mathbb{R}^n)$, we define the test function $G_{t_0, \varphi} \in C^\infty(\mathbb{R}_{+, t_0} : D(B^*D))$ by

$$ G_{t_0, \varphi}(t) := \operatorname{sgn}(t_0 - t) e^{-[(t_0 - t)B^*D]} \chi^{\operatorname{sgn}(t_0 - t)}(B^*D) \mathbb{P}_{B^*D} \varphi \\ = \begin{cases} e^{-(t_0 - t)B^*D} \chi^+(B^*D) \mathbb{P}_{B^*D} \varphi & \text{if } t < t_0 \\ -e^{-(t - t_0)B^*D} \chi^-(B^*D) \mathbb{P}_{B^*D} \varphi & \text{if } t > t_0 \end{cases} $$

for all $t \in \mathbb{R}_{+, t_0}$.
Note that $\partial_t \mathbf{G}_{t_0, \varphi} = B^* D \mathbf{G}_{t_0, \varphi}$. Also observe that since $D$ annihilates the nullspace $\mathcal{N}(B^*D)$ and since $L^2(\mathbb{R}^n) = \mathcal{N}(B^*D) \oplus \overline{\mathbb{R}(B^*D)}$, whenever $\varphi \in \mathcal{D}(D)$,

$$ (6.7) \qquad D\mathbf{G}_{t_0, \varphi}(t) = \operatorname{sgn}(t_0 - t)e^{-[(t_0 - t)DB^*]} \chi^{\operatorname{sgn}(t_0 - t)}(DB^*)D\varphi. $$

The following lemma is a rewording of [16, Lemma 7.4].

**Lemma 6.16.** Let $F$ solve $(\mathrm{CR})_{DB}$. Fix $\varphi \in L^2(\mathbb{R}^n)$, $t_0 \in \mathbb{R}_+$, and let $\eta \in \mathrm{Lip}(\mathbb{R}_+ : \mathbb{R})$ and $\chi \in \mathrm{Lip}(\mathbb{R}^n : \mathbb{R})$ be compactly supported in $\mathbb{R}_{+,t_0}$ and $\mathbb{R}^n$ respectively. Then we have, with absolutely convergent integrals,

$$ (6.8) \qquad \iint_{\mathbb{R}_+^{1+n}} \langle \eta'(t)\chi(x)B^*D\mathbf{G}_{t_0,\varphi}(t,x), F(t,x) \rangle \,dx\,dt \\ = \iint_{\mathbb{R}_+^{1+n}} \langle \eta(t)B^*[D, m_\chi]\partial_t\mathbf{G}_{t_0,\varphi}(t,x), F(t,x) \rangle \,dx\,dt $$

where $m_\chi$ denotes the multiplication operator on $L^2(\mathbb{R}^n)$ with symbol $\chi$.

As a corollary, under an integrability condition involving $F$ and $\varphi$, we can obtain the following.

**Corollary 6.17.** Let $F$, $\varphi$, and $t_0$ be as in the statement of Lemma 6.16. Suppose also that for all compact $K \subset \mathbb{R}_{+,t_0}$ we have

$$ (6.9) \qquad \mathbf{1}_K(t)|B^*D\mathbf{G}_{t_0,\varphi}(t,x)||F(t,x)| \in L^1(\mathbb{R}_+^{1+n}). $$

Then for all $\eta \in \mathrm{Lip}(\mathbb{R}_+ : \mathbb{R})$ compactly supported in $\mathbb{R}_{+,t_0}$, we have the absolutely convergent integral

$$ (6.10) \qquad \iint_{\mathbb{R}_+^{1+n}} \langle \eta'(t)B^*D\mathbf{G}_{t_0,\varphi}(t,x), F(t,x) \rangle \,dt\,dx = 0. $$

**PROOF.** Fix $\chi \in \mathrm{Lip}(\mathbb{R}^n : \mathbb{R})$ with $\chi(x) = 1$ for all $x \in B(0, 1)$, and for $R > 0$ define $\chi_R(x) := \chi(x/R)$. Then $\chi_R \to 1$ and $[D, m_{\chi_R}] \to 0$ pointwise as $R \to \infty$,$^2$ since $\|[D, m_{\chi_R}]\|_{\infty} \lesssim R^{-1}\|\nabla\chi\|_{\infty}$. Condition (6.9) applied with $K = \mathrm{supp}\,\eta$, the fact that $\partial_t \mathbf{G}_{t_0, \varphi} = B^* D \mathbf{G}_{t_0, \varphi}$, and boundedness of $\eta$ and $\eta'$ imply

$$ \begin{aligned} & |\eta'(t)B^*D\mathbf{G}_{t_0,\varphi}(t,x)||F(t,x)| \in L^1(\mathbb{R}_+^{1+n}) \quad \text{and} \\ & |\eta(t)\partial_t\mathbf{G}_{t_0,\varphi}(t,x)||F(t,x)| \in L^1(\mathbb{R}_+^{1+n}). \end{aligned} $$

This allows us to deduce (6.10) from the equality of Lebesgue integrals (6.8) and dominated convergence. $\square$

Now, assuming that (6.9) holds, we can conclude the following.

**Corollary 6.18.** Let $F$, $\varphi$, and $t_0$ be as in the statement of Lemma 6.16. Assume also that condition (6.9) is satisfied. Then for sufficiently small $\varepsilon > 0$ we have

$$ (6.11) \qquad \int_{t_0+\varepsilon}^{t_0+2\varepsilon} \int_{\mathbb{R}^n} \langle B^* D \mathbf{G}_{t_0, \varphi}(t,x), F(t,x) \rangle \,dx\,dt \\ = \int_{t_0+(2\varepsilon)-1}^{t_0+\varepsilon-1} \int_{\mathbb{R}^n} \langle B^* D \mathbf{G}_{t_0, \varphi}(t,x), F(t,x) \rangle \,dx\,dt $$

<footnote>$^2$More precisely, $[D, m_{\chi_R}]$ is given by multiplication with a function that tends to 0 pointwise.</footnote>
<figure><img src="image_31.png" /><figcaption>FIGURE 6. The functions $\eta_1$ and $\eta_2$.</figcaption></figure>

and

$$ (6.12) \qquad -\int_{\varepsilon}^{2\varepsilon} \int_{\mathbb{R}^n} \langle B^* D \mathbf{G}_{t_0, \varphi}(t_0 - t, x), F(t_0 - t, x) \rangle \, dx \, dt \\ = \int_{\varepsilon}^{2\varepsilon} \int_{\mathbb{R}^n} \langle B^* D \mathbf{G}_{t_0, \varphi}(t, x), F(t, x) \rangle \, dx \, dt. $$

These are all absolutely convergent integrals.

**PROOF.** As in [16, §8, Step 1b] this follows from applying Corollary 6.17 with the piecewise linear functions $\eta_1, \eta_2 \in \text{Lip}(\mathbb{R}_+ : \mathbb{R})$ drawn in Figure 6, where we impose $\varepsilon < \min(t_0/4, 1/4, 1/t_0)$ (we have carried out a change of variables in the left hand side of (6.12)). $\square$

**6.4.3. Proof of Theorem 6.8.** Recall that part (i) has already been proven in Subsection 6.4.1; here we prove part (ii).

All of the results in this section are valid for $\mathbf{p} = (p, s)$ such that $p \le 2$ and $s < 0$. We do not 'fix' such a $\mathbf{p}$, however, because in the final step we will invoke prior results with a different choice of $\mathbf{p}$.

**Step 1: Verification and application of initial limiting arguments.**

**Lemma 6.19.** Let $\varphi \in L^2(\mathbb{R}^n)$. Then $\mathbf{1}_{K \times \mathbb{R}^n} B^* D \mathbf{G}_{t_0, \varphi} \in X^{\mathbf{P}'}$ for all compact $K \subset \mathbb{R}_+, t_0$, with

$$ (6.13) \qquad \| \mathbf{1}_{K \times \mathbb{R}^n} B^* D \mathbf{G}_{t_0, \varphi} \|_{X^{\mathbf{P}'}} \lesssim \| \varphi \|_2 \operatorname{dist}(K, t_0)^{-1} K_-^{s+n\delta_{p,2}} \\ \text{where } K_- = \inf(K). $$

**PROOF.** First note that the estimate

$$ \| \mathbf{1}_{K \times \mathbb{R}^n} B^* D \mathbf{G}_{t_0, \varphi} \|_{X_{-s-n\delta_{p,2}}^2} \lesssim \| \varphi \|_2 \operatorname{dist}(K, t_0)^{-1} K_-^{s+n\delta_{p,2}} $$

can be shown by writing

$$ \| \mathbf{1}_{K \times \mathbb{R}^n} B^* D \mathbf{G}_{t_0, \varphi} \|_{X_{-s-n\delta_{p,2}}^2} = \left( \int_{K_-}^{K_+} \| t^{s+n\delta_{p,2}} B^* D \mathbf{G}_{t_0, \varphi} \|_2^2 \frac{dt}{t} \right)^{1/2} $$
$$
(6.14) \qquad \lesssim \|\varphi\|_2 \left( \int_{K_-}^{K_+} t^{2(s+n\delta_{p,2})} \operatorname{dist}(K, t_0)^{-2} \frac{dt}{t} \right)^{1/2}
$$

$$
(6.15) \qquad \lesssim \|\varphi\|_2 \operatorname{dist}(K, t_0)^{-1} K_-^{s+n\delta_{p,2}}.
$$

The estimate (6.14) follows by writing

$$
B^* D \mathbf{G}_{t_0, \varphi} = \frac{\operatorname{sgn}(t_0 - t)}{t_0 - t} (t_0 - t) B^* D e^{-(t_0 - t)B^*D} \chi^{\operatorname{sgn}(t_0 - t)}(B^*D) \mathbb{P}_{B^*D} \varphi
$$

and noting that the operator

$$
(t_0 - t) B^* D e^{-(t_0 - t) B^* D} \chi^{\operatorname{sgn}(t_0 - t)} (B^* D) \mathbb{P}_{B^* D}
$$

is bounded on $L^2(\mathbb{R}^n)$ uniformly in $t \in \mathbb{R}_{+}, t_0$, and that $|(t_0 - t)^{-1}| \lesssim \operatorname{dist}(K, t_0)^{-1}$ for $t \in K$. Then (6.15) follows because $s + n\delta_{p,2}$ is negative whenever $s < 0$ and $p < 2$. Now use the $X$-space embeddings to write

$$
X_{-s-n\delta_{p,2}}^2 \hookrightarrow X^{\mathbf{P}'},
$$

from which follows (6.13).

<figure><img src="image_32.png" /></figure>

**Corollary 6.20.** Let $\varphi \in L^2(\mathbb{R}^n)$ and suppose that $F \in X^P$ solves (CR)$_{DB}$.
Then

$$
(6.16) \quad \lim_{\varepsilon \to 0} \int_{t_0 + \varepsilon}^{t_0 + 2\varepsilon} \int_{\mathbb{R}^n} \langle B^* D \mathbf{G}_{t_0, \varphi}(t, x), F(t, x) \rangle \, dx \, dt = 0.
$$

PROOF. For $\varepsilon > 0$ small the previous lemma yields

$$
\| \mathbf{1}_{[t_0 + (2\epsilon)^{-1}, t_0 + \epsilon^{-1}]} \times \mathbb{R}^n B^* D \mathbf{G}_{t_0, \varphi} \|_{X^{\mathbf{P}'}} \lesssim \| \varphi \|_2 (2\epsilon)(t_0 + (2\epsilon)^{-1})^{s+n\delta_{p,2}},
$$

which decays as $\varepsilon \to 0$ since $s + n\delta_{p,2}$ is negative when $s < 0$ and $p \le 2$. Therefore in particular, by $X$-space duality, condition (6.9) is satisfied, and by boundedness of the above quasinorms as $\varepsilon \to 0$ we can take the $\varepsilon \to 0$ limit in (6.11) to obtain (6.16). $\square$

<figure><img src="image_33.png" /></figure>

**Step 2: Semigroup property of F.**

**Lemma 6.21.** Suppose $F \in X^P$ solves (CR)$_{DB}$. Then $F \in C^\infty(\mathbb{R}_+ : \mathbb{H}_{DB}^2)$,
$F(t) \in \mathcal{D}(DB)$ for all $t > 0$, and $\partial_t F + DBF = 0$ holds strongly in $C^\infty(\mathbb{R}_+ : \mathbb{H}_{DB}^2)$.

PROOF. We already have that $F \in C^\infty(\mathbb{R}_+ : L^2_{\text{loc}}(\mathbb{R}^n))$ from Proposition 6.1, and furthermore that $\partial_t F \in X^{\mathbf{P}}^{-1}$. Hence we have $F(t_0)$, $(\partial_t F)(t_0) \in E^{\mathbf{P}}$ for all $t_0 \in \mathbb{R}_+$, and therefore by the slice space containments of Proposition 2.43 we obtain $F(t_0)$, $(\partial_t F)(t_0) \in L^2$ for all $t_0 \in \mathbb{R}_+$. Therefore $F(t_0) \in \mathcal{D}(DB)$ for all $t_0 \in \mathbb{R}_+$, and $\partial_t F + DBF = 0$ holds in $L^2$. We can iterate this argument by reapplying $\partial_t$, as this preserves the property of solving (CR)$_{DB}$ as well as the previously stated $L^2$ containments, so we obtain $F \in C^\infty(\mathbb{R}_+ : L^2)$.

Now since $\lim_{t_0 \to \infty} F(t_0) = 0$ in $L^2$ (Lemma 6.7), we can write

$$
F(t_0) = - \int_{t_0}^{\infty} (\partial_t F)(\tau) \, d\tau = - \int_{t_0}^{\infty} DB(F(\tau)) \, d\tau \in \overline{\mathcal{R}(DB)}
$$

by the fundamental theorem of calculus. Therefore $F(t_0) \in \mathbb{H}_{DB}^2$ for all $t_0$, and since the $\mathbb{H}_{DB}^2$-norm is equivalent to the $L^2$-norm when restricted to $\overline{\mathcal{R}(DB)}$, this completes the proof. $\square$

<figure><img src="image_34.png" /></figure>
**Lemma 6.22.** Suppose that $F \in X^P$ solves $(CR)_{DB}$. Then for all $t_0 > 0$ and $\tau \ge 0$ we have $F(t_0) \in \mathbb{H}_{DB}^{2,+} = \overline{R(DB)}^+$ and

$$ (6.17) \qquad F(t_0 + \tau) = e^{-\tau DB}(F(t_0)) $$

(recall that $e^{-\tau DB}$ is defined on the positive spectral subspace).

**PROOF.** For all $\varphi \in L^2(\mathbb{R}^n)$, the function $t \mapsto B^*DG_{t_0,\varphi}(t)$ is smooth in $t \in \mathbb{R}_{+,t_0}$ with values in $\mathbb{H}_{B^*D}^2$ and with

$$ \lim_{t \downarrow t_0} B^* D G_{t_0, \varphi}(t) = -B^* D \chi^{-}(B^* D) \mathbb{P}_{B^*D} \varphi $$

in $\mathbb{H}_{B^*D}^2$. Furthermore, by Lemma 6.21, $t \mapsto F(t)$ is smooth in $t \in \mathbb{R}_+$ with values in $\mathbb{H}_{DB}^2$. Therefore we may write for all $\varphi \in L^2(\mathbb{R}^n)$, using (6.16) from Corollary 6.20,

$$ \begin{align} 0 &= \lim_{\epsilon \to 0} \int_{t_0+\epsilon}^{t_0+2\epsilon} \int_{\mathbb{R}^n} \langle B^* D G_{t_0, \varphi}(t, x), F(t, x) \rangle dx dt \nonumber \\ &= \lim_{\epsilon \to 0} \int_{t_0+\epsilon}^{t_0+2\epsilon} \langle B^* D G_{t_0, \varphi}(t), F(t) \rangle_{\mathbb{H}_{B^*D}^2} dt \nonumber \\ &= -\langle B^* D \chi^{-}(B^* D) \mathbb{P}_{\overline{R(B^*D)}} \varphi, F(t_0) \rangle_{\mathbb{H}_{B^*D}^2}. \tag{6.18} \end{align} $$

Hence for all $\phi \in \overline{R(B^*D)}$ and all $\delta > 0$, since $e^{-\delta[B^*D]}$ maps $\mathbb{H}_{B^*D}^{2,-}$ into itself, applying (6.18) to $\varphi = e^{-\delta[B^*D]}\phi$ yields

$$ (6.19) \qquad \langle B^* D e^{\delta B^* D} \chi^{-}(B^* D) \phi, F(t_0) \rangle_{\mathbb{H}_{B^*D}^2} = 0. $$

The subspace

$$ \{B^* D e^{\delta B^* D} \chi^{-}(B^* D) \phi : \phi \in \overline{R(B^*D)}\} \subset L^2(\mathbb{R}^n) $$

is dense in $\mathbb{H}_{B^*D}^{2,-}$ (see [16, p. 28]), so (6.19) and the decomposition $\mathbb{H}_{DB}^2 = \mathbb{H}_{DB}^{2,+} \oplus \mathbb{H}_{DB}^{2,-}$ imply that $F(t_0) \in \mathbb{H}_{DB}^{2,+}$.

Now we derive the semigroup equation (6.17). For all $\delta \ge 0$ and $\varphi \in \mathbb{H}_{B^*D}^2$, define

$$ \varphi_{\delta} := e^{-\delta[B^*D]}\varphi $$

and

$$ I_{t_0, \varphi}^{\epsilon, \delta} := \int_{\epsilon}^{2\epsilon} \langle B^* D G_{t_0, \varphi_{\delta}}(t), F(t) \rangle_{\mathbb{H}_{B^*D}^2} dt. $$

By (6.12), using the same argument as before to write everything in terms of $\mathbb{H}_{B^*D}^2$-duality, we have

$$ \begin{align*} \lim_{\epsilon \to 0} I_{t_0, \varphi}^{\epsilon, \delta} &= - \lim_{\epsilon \to 0} \int_{\epsilon}^{2\epsilon} \langle B^* D e^{-tB^*D} e^{-\delta B^*D} \chi^{+}(B^*D) \varphi, F(t_0 - t) \rangle_{\mathbb{H}_{B^*D}^2} dt \\ &= - \langle B^* D e^{-\delta B^*D} \chi^{+}(B^*D) \varphi, F(t_0) \rangle_{\mathbb{H}_{B^*D}^2}. \end{align*} $$

Therefore for all $\tau \ge 0$, $\delta \ge 0$, and $\varphi \in \mathbb{H}_{B^*D}^2$, using $I_{t_0, \varphi}^{\epsilon, \delta+\tau} = I_{t_0+\delta, \varphi}^{\epsilon, \tau}$, we have

$$ \begin{align*} \langle B^* D e^{-\delta B^* D} \chi^{+}(B^* D) \varphi, e^{-\tau DB}(F(t_0)) \rangle_{\mathbb{H}_{B^*D}^2} &= \langle B^* D e^{-(\delta+\tau)B^*D} \chi^{+}(B^* D) \varphi, F(t_0) \rangle_{\mathbb{H}_{B^*D}^2} \\ &= - \lim_{\epsilon \to 0} I_{t_0, \varphi}^{\epsilon, \delta+\tau} \\ &= - \lim_{\epsilon \to 0} I_{t_0+\tau, \varphi}^{\epsilon, \delta} \end{align*} $$
$$= \langle B^* D e^{-\delta B^* D} \chi^+(B^* D) \varphi, F(t_0 + \tau) \rangle_{\mathbb{H}_{B^* D}^2}.$$

As before, the subspace {$B^*D e^{-\delta B^* D} \chi^+(B^*D)\varphi : \varphi \in \mathbb{H}_{B^*D}^2$} is dense in $\mathbb{H}_{B^*D}^{2,+}$, so by duality we have $F(t_0+\tau) = e^{-\tau DB}F(t_0)$ in $\mathbb{H}_{DB}^{2,+}$ for all $t_0 > 0$ and all $\tau \ge 0$. $\square$

### Step 3: Completing the proof.

**Proposition 6.23 (Existence of boundary trace).** Suppose that $F \in X^p$ solves (CR)$_{DB}$, and let $X$ be a completion of $X_{DB}^p$. Then there exists a unique $F_0 \in X^+$ such that $F = C_{DB}^+ F_0$. Furthermore, $\|F_0\|_{X} \lesssim \|F\|_{X^p}$.

**PROOF.** Fix an exponent $\tilde{p}$ with $i(\tilde{p}) \in (1,2]$ and $\theta(\tilde{p}) < 0$ such that $p \mapsto \tilde{p}$ (when $p > 1$ we may take $\tilde{p} = p$). By Lemma 6.22 we have $F(t_0) \in \mathbb{H}_{DB}^{2,+} \cap \mathcal{D}(DB)$ for all $t_0 > 0$. We can then estimate

$$(6.20) \qquad \|F(t_0)\|_{X_{DB}^{\tilde{p}}} \simeq \|DBF(t_0)\|_{X_{DB}^{\tilde{p}-1}}$$

$$(6.21) \qquad = \|\tau \mapsto e^{-\tau DB}(DBF)(t_0)\|_{X^{\tilde{p}-1}}$$

$$(6.22) \qquad = \|\tau \mapsto DBF(t_0 + \tau)\|_{X^{\tilde{p}-1}} \\
= \|S_{t_0} DBF\|_{X^{\tilde{p}-1}}$$

$$(6.23) \qquad \lesssim \|DBF\|_{X^{\tilde{p}-1}} \\
= \|\partial_t F\|_{X^{\tilde{p}-1}}$$

$$(6.24) \qquad \lesssim \|F\|_{X^{\tilde{p}}}$$

$$(6.25) \qquad \lesssim \|F\|_{X^p}.$$

The first line (6.20) is from Corollary 4.17. Line (6.21) comes from Theorem 4.34. Line (6.22) comes from Lemma 6.22, (6.23) comes from Proposition 2.37 because $i(\tilde{p} - 1) \le 2$ and $\theta(\tilde{p} - 1) \le -1/2$, (6.24) comes from Proposition 6.1, and finally (6.25) follows from $X$-space embeddings by $p \mapsto \tilde{p}$. Therefore $F(t_0) \in X_{DB}^{\tilde{p},+}$ uniformly in $t_0 > 0$.

Let $\tilde{X}$ be a completion of $X_{DB}^{\tilde{p}}$, so that $X \hookrightarrow \tilde{X}$. Since $\tilde{X}^+$ can be identified with the dual of $\tilde{Y}^+$ for any completion $\tilde{Y}$ of $X_{B^*D}^{\tilde{p}'}$, there exists a sequence $t_k \downarrow 0$ and an $F_0 \in \tilde{X}^+$ such that $F(t_k)$ converges weakly to $F_0$ in $\tilde{X}^+$ as $k \to \infty$. We thus have for all $\varphi \in X_{B^*D}^{\tilde{p}',+}$ and for all $\tau > 0$,

$$\begin{align*}
\langle \varphi, e^{-\tau DB} F_0 \rangle_{\tilde{Y}} &= \langle e^{-\tau B^* D} \varphi, F_0 \rangle_{\tilde{Y}} \\
&= \lim_{k \to \infty} \langle e^{-\tau B^* D} \varphi, F(t_k) \rangle_{\tilde{Y}} \\
&= \lim_{k \to \infty} \langle \varphi, e^{-\tau DB} F(t_k) \rangle_{\tilde{Y}} \\
(6.26) \qquad &= \lim_{k \to \infty} \langle \varphi, F(t_k + \tau) \rangle_{\tilde{Y}} \\
&= \langle \varphi, F(\tau) \rangle_{\tilde{Y}},
\end{align*}$$

using Lemma 6.22 in (6.26). Therefore by density we have $C_{DB}^+ F_0 = F$.

It only remains to show that $F_0$ is in $X^+$, with the correct quasinorm estimate, and uniquely determined. Recall that $C_{DB}^+$ is equal to the restriction of $Q_{sgp,DB}$ to the positive spectral subspace ($Q_{sgp,DB}$ is the extension of $Q_{sgp,DB}$ to the completion $X$). Let $\varphi \in \Psi_{\infty}^{\infty}$ be a Calderón sibling of sgp. Then $F_0 = S_{\varphi,DB}Q_{sgp,DB}F_0 = S_{\varphi,DB}F$, so by Proposition 4.21 we have $F_0 \in X$ with $\|F_0\|_{X} \lesssim \|F\|_{X^p}$. In fact,
since $F(t_0) \in \mathbb{H}_{DB}^{2,+}$ for all $t_0 > 0$, we find that $F$ is in the positive spectral subspace $\mathbf{X}^+$. Uniqueness follows by injectivity of $\mathbf{Q}_{sgp, DB}$ (Proposition 4.21). $\square$

This completes the proof of Theorem 6.8.

**6.4.4. Proof of Theorem 6.9.** Recall that part (i) has already been proven in Subsection 6.4.1; here we prove part (ii). Our proof roughly follows that of [16, Theorem 1.3], arguing via a series of rather technical lemmas. In this section we continually assume that $\mathbf{p}$ satisfies the assumptions of Theorem 6.9. Most of the lemmas work without assuming $\mathbf{p}^{\heartsuit} \in I(\mathbf{X}, DB^*)$, but we gain nothing from dropping this assumption.

**Step 1: Establishing a good class of test functions.**

We define the following class of test functions for $\mathbb{X}_{DB}^{\mathbf{P}}$:

$$ \mathbb{D}^{\mathbf{P}}(X) := \{\varphi \in \mathcal{D}(D) : D\varphi \in \mathbb{X}_{DB^*}^{\mathbf{P}^{\heartsuit}}, \chi^{\pm}(DB^*)D\varphi \in E^{\mathbf{P}^{\heartsuit}}\}. $$

This is large enough to contain the Schwartz functions and to be stable under the action of various operators, yet it is restrictive enough to let us exploit slice space containments.

**Lemma 6.24.** The Schwartz class $\mathcal{S}(\mathbb{R}^n : \mathbb{C}^{m(1+n)})$ is contained in $\mathbb{D}^{\mathbf{P}}(X)$.

This is a modification of the argument of [16, Lemma 8.10].

**PROOF.** Suppose $\varphi \in \mathcal{S}$. Then $\varphi \in \mathcal{D}(D)$ and $D\varphi \in \mathbb{X}_{D}^{\mathbf{P}^{\heartsuit}} = \mathbb{X}_{DB^*}^{\mathbf{P}^{\heartsuit}}$ by the assumption on $\mathbf{p}$. It remains to show that $\chi^{\pm}(DB^*)D\varphi \in E^{\mathbf{P}^{\heartsuit}}$, and this takes some work. Since $D\varphi \in \mathcal{S} \subset E^{\mathbf{P}^{\heartsuit}}$ (Proposition 2.44) and since

$$ D\varphi = \chi^{+}(DB^{*})D\varphi + \chi^{-}(DB^{*})D\varphi $$

it suffices to show that $\chi^{+}(DB^{*})D\varphi$ is in $E^{\mathbf{P}^{\heartsuit}}$.

Define $\psi \in \Psi_N^{\infty}$, with $N$ large to be chosen later, by

$$ \psi(z) := \frac{[z]^N e^{-[z]}}{N!}. $$

Then for all $t \in \mathbb{R} \setminus \{0\}$ we have

$$ \int_0^\infty \psi(st) \frac{ds}{s} = \frac{1}{N!} \int_0^\infty s^N e^{-s} \frac{ds}{s} = 1, $$

so by holomorphy we have

$$ \int_0^\infty \psi(sz) \frac{ds}{s} = 1 $$

for all $z \in S_\mu$. By the same argument, along with integration by parts and induction on $N$, for all $z \in S_\mu$ we have

$$ \int_1^\infty \psi(sz) \frac{ds}{s} = P([z])e^{-[z]} $$

where $P$ is a real polynomial of degree $N-1$. Therefore by functional calculus on $\overline{\mathbb{R}(DB^*)}$ we may write

$$ \chi^{+}(DB^{*})D\varphi = \int_{0}^{1} (\psi\chi^{+})(sDB^{*})D\varphi \frac{ds}{s} + P(DB^{*})e^{-DB^{*}}\chi^{+}(DB^{*})D\varphi. $$
By Lemma 6.3 (using $i(\mathbf{p}^{\diamond}) < 2$ and $\theta(\mathbf{p}^{\diamond}) < 0$) we have

$$P(DB^*)e^{-DB^*} \chi^+(DB^*)D\varphi \in E^{\mathbf{p}^{\diamond}},$$

so it suffices to show that

$$\int_0^1 (\psi\chi^+)(sDB^*)D\varphi \frac{ds}{s} \in E^{\mathbf{p}^{\diamond}}.$$

For $f \in L^2(\mathbb{R}^n)$ write

$$G(f) := \int_0^1 (\psi\chi^+)(sDB^*)f \frac{ds}{s};$$

since $\psi\chi^+ \in \Psi_+^\infty$ this is defined for all $f \in L^2(\mathbb{R}^n)$ (not just $f \in \overline{R(DB^*)}$). Note that the family $((\psi\chi^+)(sDB^*))_{s>0}$ satisfies off-diagonal estimates of order $N$ (Theorem 3.8). For $Q, R \in \mathcal{D}_0$ (here $\mathcal{D}_0$ is the set of standard dyadic cubes in $\mathbb{R}^n$ with sidelength 1) with $d(Q, R) \ge 1$ we can estimate

$$\begin{aligned}
\|G(\mathbf{1}_R D\varphi)\|_{L^2(Q)} &= \left( \int_Q \left| \int_0^1 (\psi\chi^+)(sDB^*) \mathbf{1}_R D\varphi(x) \frac{ds}{s} \right|^2 dx \right)^{1/2} \\
&\le \int_0^1 \|(\psi\chi^+)(sDB^*) \mathbf{1}_R D\varphi\|_{L^2(Q)} \frac{ds}{s} \\
&\lesssim \int_0^1 \left( \frac{d(Q, R)}{s} \right)^{-N} \frac{ds}{s} \|\mathbf{1}_R D\varphi\|_2 \\
&\simeq d(Q, R)^{-N} \|\mathbf{1}_R D\varphi\|_2.
\end{aligned}$$

For all other $Q, R \in \mathcal{D}_0$ we have instead

$$\|G(\mathbf{1}_R D\varphi)\|_{L^2(Q)} \lesssim \|\mathbf{1}_R D\varphi\|_2.$$

Therefore we have by the discrete characterisation of slice spaces (Proposition 2.46), writing $R \sim Q$ to mean that $\mathrm{dist}(R, Q) = 0$ and noting that $\mathrm{dist}(R, Q) \ge 1$ if $R \not\sim Q$,

$$\begin{aligned}
\|G(D\varphi)\|_{E^{\mathbf{p}^{\diamond}}} &\approx \left( \sum_{Q \in \mathcal{D}_1} \|G(D\varphi)\|_{L^2(Q)}^{i(\mathbf{p}^{\diamond})} \right)^{1/i(\mathbf{p}^{\diamond})} \\
&\lesssim \left( \sum_{Q \in \mathcal{D}_1} \left( \sum_{R \sim Q} + \sum_{R \not\sim Q} \right) \|G(\mathbf{1}_R D\varphi)\|_{L^2(Q)}^{i(\mathbf{p}^{\diamond})} \right)^{1/i(\mathbf{p}^{\diamond})} \\
&\lesssim \left( \sum_{Q \in \mathcal{D}_1} \|D\varphi\|_{L^2(R)}^{i(\mathbf{p}^{\diamond})} \right)^{1/i(\mathbf{p}^{\diamond})} + \\
&\qquad \left( \sum_{Q \in \mathcal{D}_1} d(Q, R)^{-Ni(\mathbf{p}^{\diamond})} \|D\varphi\|_{L^2(R)}^{i(\mathbf{p}^{\diamond})} \right)^{1/i(\mathbf{p}^{\diamond})} \\
&=: \mathbf{I}_1 + \mathbf{I}_2.
\end{aligned}$$

Since the number of cubes $R \in \mathcal{D}_0$ such that $R \sim Q$ is uniform in $Q$, we have

$$\mathbf{I}_1 \approx \left( \sum_{R \in \mathcal{D}_1} \|D\varphi\|_{L^2(R)}^{i(\mathbf{p}^{\diamond})} \right)^{1/i(\mathbf{p}^{\diamond})} \approx \|D\varphi\|_{E^{\mathbf{p}^{\diamond}}}.$$
To handle $\mathbf{I}_2$ write

$$ (6.27) \quad \mathbf{I}_2 = \left( \sum_{R \in \mathcal{D}_1} \|D\varphi\|_{L^2(R)}^{i(\mathbf{p}^{\diamond})} \sum_{k=1}^{\infty} k^{-Ni(\mathbf{p}^{\diamond})} |\{Q \in \mathcal{D}_1 : d(Q, R) = k\}| \right)^{1/i(\mathbf{p}^{\diamond})} \\ \approx_{N, \mathbf{p}^{\diamond}, n} \|D\varphi\|_{E^{\mathbf{p}^{\diamond}}} $$

using that the innermost sum in (6.27) is independent of $R$ and convergent for $N$ sufficiently large. Therefore

$$ \|G(D\varphi)\|_{E^{\mathbf{p}^{\diamond}}} \lesssim \|D\varphi\|_{E^{\mathbf{p}^{\diamond}}} < \infty $$

which shows that $\chi^+(DB^*)D\varphi \in E^{\mathbf{p}^{\diamond}}$ and completes the proof. $\square$

<figure><img src="image_35.png" /></figure>

**Lemma 6.25.** We have the following stability properties of $\mathbb{D}^{\mathbf{P}}(X)$:

(i) for all $\delta > 0$, $e^{-\delta[B^*D]}\mathbb{D}^{\mathbf{P}}(X) \subset \mathbb{D}^{\mathbf{P}}(X)$,

(ii) $\chi^{\pm}(B^*D)\mathbb{D}^{\mathbf{P}}(X) \subset \mathbb{D}^{\mathbf{P}}(X)$.

**PROOF.** The function $[z \mapsto e^{-\delta[z]}]$ is in $H^{\infty}$ and has a polynomial limit at 0, so $e^{-\delta[B^*D]}\varphi$ may be defined for all $\varphi \in \mathcal{D}(D)$ (not just those in $\overline{\mathbb{R}(B^*D)}$)<fnref n="3" /> as

$$ e^{-\delta[B^*D]}\varphi = e^{-\delta[B^*D]}\mathbb{P}_{B^*D}\varphi + (I - \mathbb{P}_{B^*D})\varphi. $$

For all such $\varphi$, by using the similarity of functional calculi along with the equalities $D(I - \mathbb{P}_{B^*D})\varphi = 0$ and $D\mathbb{P}_{B^*D}\varphi = D\varphi$, we can write

$$ (6.28) \qquad D(e^{-\delta[B^*D]}\varphi) = e^{-\delta[DB^*]}D\varphi. $$

Since $D\varphi$ is in $\mathbb{X}_{DB^*}^{\mathbf{P}^{\diamond}}$, so is $D(e^{-\delta[B^*D]}\varphi)$. To see the slice space containments of spectral projections, write

$$ \chi^{\pm}(DB^*)D(e^{-\delta[B^*D]}\varphi) = e^{-\delta[DB^*]}\chi^{\pm}(DB^*)D\varphi. $$

By assumption $\chi^{\pm}(DB^*)D\varphi$ is in $E^{\mathbf{P}^{\diamond}} \cap \mathbb{X}_{DB^*}^{\mathbf{P}^{\diamond}, \pm} \subset E^{\mathbf{P}^{\diamond}} \cap \mathbb{X}_{DB^*}^{2, \pm}$, and by Corollary 4.39, $e^{-\delta[DB^*]}\chi^{\pm}(DB^*)D\varphi$ is in $E^{\mathbf{P}^{\diamond}}$.

The second part is proven similarly: we have $\chi^{\pm}(B^*D)\mathcal{D}(D) \subset \mathcal{D}(B^*D)$, and by similarity of functional calculi

$$ D\chi^{\pm}(B^*D)\varphi_0 = \chi^{\pm}(DB^*)D\varphi_0 \in \mathbb{X}_{DB^*}^{\mathbf{P}^{\diamond}} $$

and

$$ \chi^{\pm}(DB^*)D\chi^{\pm}(B^*D)\varphi = \chi^{\pm}(DB^*)D\varphi_0 \in E^{\mathbf{P}^{\diamond}} $$

$$ \chi^{\mp}(DB^*)D\chi^{\pm}(B^*D)\varphi = 0 \in E^{\mathbf{P}^{\diamond}}, $$

completing the proof. $\square$

<figure><img src="image_36.png" /></figure>

**Lemma 6.26.** Suppose that $\varphi \in \mathbb{X}_{B^*D}^{\mathbf{P}'} \cap \mathcal{D}(B^*D)$. Then for all $t > 0$, $\chi^{\pm}(DB^*)e^{-t[DB^*]/2}D\varphi$ is defined and contained in $E^{\mathbf{P}'}$. Furthermore, $e^{-t[DB^*]/2}\varphi \in \mathbb{D}^{\mathbf{P}}(X)$.

<footnote n="3">³Although we did not discuss this in Subsection 3.1, this is a standard procedure. The representation (6.28) is all we need.</footnote>
PROOF. Note that $\mathcal{D}(B^*D) = \mathcal{D}(D)$. Since $D\varphi \in \mathbb{X}_{DB^*}^{\mathbb{P}^{\varphi}}$ (Proposition 5.6), by Lemma 6.3 we find that

$$ (6.29) \qquad \chi^{\pm}(DB^{*})e^{-t[DB^{*}]/2}D\varphi = e^{-t[DB^{*}]/2}\chi^{\pm}(DB^{*})D\varphi \in E^{\mathbb{P}^{\varphi}} = E^{\mathbb{P}'} $$

To see that $e^{-t[B^*D]/2}\varphi$ is in $\mathbb{D}^{\mathbb{P}}(X)$, note that

$$ e^{-t[B^*D]/2}\varphi \in \mathcal{D}(B^*D) = \mathcal{D}(D), $$

that

$$ De^{-t[B^*D]/2}\varphi = e^{-t[DB^*]/2}D\varphi \in \mathbb{X}_{DB^*}^{\mathbb{P}^{\varphi}}, $$

and that

$$ \chi^{\pm}(DB^{*})De^{-t[B^*D]/2}\varphi = e^{-t[DB^{*}]/2}\chi^{\pm}(DB^{*})D\varphi \in E^{\mathbb{P}^{\varphi}} $$

by (6.29).

<figure><img src="image_37.png" /></figure>

### Step 2: Verification and application of the initial limiting arguments.

**Lemma 6.27.** *Define the operator*

$$ \tilde{\mathbf{G}}_{t_0} : \varphi \mapsto \operatorname{sgn}(t_0 - t)e^{-(t_0 - t)DB^*} \chi^{\operatorname{sgn}(t_0 - t)}(DB^*)\varphi. $$

Let $K \subset \mathbb{R}_{+,t_0}$ be compact. Then for all $k \in \mathbb{N}$, $\mathbf{1}_{K \times \mathbb{R}^n} \tilde{\mathbf{G}}_{t_0}$ is bounded from $\mathbb{X}_{DB^*}^{\mathbb{P}^{\varphi}}$ to $\mathbb{X}^{\mathbb{P}^{\varphi}+k}$, and this boundedness is uniform in $K$ provided $K_- > t_0 + 1$.

PROOF. We will prove the result for tent spaces; the Z-space result then follows by real interpolation because the assumption on $\mathbf{p}$ is open in $(j(\mathbf{p}), \theta(\mathbf{p}))$.

Suppose $\varphi \in \mathbb{X}_{DB^*}^{\mathbb{P}^{\varphi}}$ and write $K = K_0 \cup K_\infty$, where $K_0 \subset (0, t_0)$ and $K_\infty \subset (t_0, \infty)$. For all $x \in \mathbb{R}^n$,

$$ \begin{aligned} & \mathcal{A}^2(\kappa^{\theta(\mathbf{p})+1-k}\mathbf{1}_{K \times \mathbb{R}^n}\tilde{\mathbf{G}}_{t_0}(\varphi))(x)^2 \\ &= \left( \int_{K_0} \int_{B(x,t)} + \int_{K_\infty} \int_{B(x,t)} \right) |t^{\theta(\mathbf{p})+1-k}\tilde{\mathbf{G}}_{t_0}(\varphi)(t,y)|^2 \frac{dy dt}{t^{1+n}} \\ &=: \mathbf{I}_0 + \mathbf{I}_\infty. \end{aligned} $$

There exists $\alpha > 0$ (depending on $K_0$) such that if $(t_0 - \tau, y) \in (K_0 \times \mathbb{R}^n) \cap \Gamma(x)$, then $(\tau, y) \in \Gamma^\alpha(x)$ (see Figure 7). Thus, using (6.7) and that $t_0 - \tau \simeq_K \tau$ when $t_0 - \tau \in K_0$,

$$ \begin{aligned} \mathbf{I}_0 &\le \iint_{\Gamma^\alpha(x)} \mathbf{1}_{K_0}(t_0 - \tau) |(t_0 - \tau)^{\theta(\mathbf{p})+1-k} \tilde{\mathbf{G}}_{t_0}(\varphi)(t_0 - \tau, y)|^2 \frac{dy d\tau}{(t_0 - \tau)^{1+n}} \\ &\lesssim_{K,k} \iint_{\Gamma^\alpha(x)} |\tau^{\theta(\mathbf{p})+1} e^{-\tau DB^*} \chi^+(DB^*)\varphi(y)|^2 \frac{dy d\tau}{\tau^{1+n}}. \end{aligned} $$

Similarly, there exists $\beta > 0$ such that if $(t_0 + \sigma, y) \in (K_1 \times \mathbb{R}^n) \cap \Gamma(x)$, then $(\sigma, y) \in \Gamma^\beta(x)$, and using $2(\theta(\mathbf{p}) + 1) - n - 1 < 0$ we have

$$ \begin{aligned} \mathbf{I}_\infty &\le \iint_{\Gamma^\beta(x)} \mathbf{1}_{K_\infty}(t_0 + \sigma) |(t_0 + \sigma)^{\theta(\mathbf{p})+1-k} e^{\sigma DB^*} \chi^-(DB^*)\varphi(y)|^2 \frac{dy d\sigma}{(t_0 + \sigma)^{1+n}} \\ &\le (K_\infty)^{-2k} \iint_{\Gamma^\beta(x)} (t_0 + \sigma)^{2(\theta(\mathbf{p})+1)-n-1} |e^{\sigma DB^*} \chi^-(DB^*)\varphi(y)|^2 dy d\sigma \\ &\le (K_\infty)^{-2k} \iint_{\Gamma^\beta(x)} \sigma^{2(\theta(\mathbf{p})+1)-n-1} |e^{\sigma DB^*} \chi^-(DB^*)\varphi(y)|^2 dy d\sigma \end{aligned} $$
<figure><img src="image_38.png" /><figcaption>FIGURE 7. Cones of large aperture, used in Lemma 6.27.</figcaption></figure>

$$= (K_{\infty})_{-}^{-2k} \iint_{\Gamma^{\beta}(x)} |\sigma^{\theta(\mathbf{p})+1} e^{\sigma DB^{*}} \chi^{-}(DB^{*})\varphi(y)|^2 \frac{dy d\sigma}{\sigma^{1+n}}.$$

Therefore we can estimate

$$\begin{align*}
\| \mathbf{1}_{K \times \mathbb{R}^n} \tilde{\mathbf{G}}_{t_0}(\varphi) \|_{T\mathbf{P}^{\heartsuit}+k} & \lesssim C(K, k) \| e^{-tDB^*} \chi^+(DB^*) \varphi \|_{T\mathbf{P}^{\heartsuit}} + (K_{\infty})_{-}^{-2k} \| e^{tDB^*} \chi^-(DB^*) \varphi \|_{T\mathbf{P}^{\heartsuit}} \\
& \lesssim \| \varphi \|_{\mathbb{H}_{DB^*}^{\mathbf{P}^{\heartsuit}}} \\
& < \infty
\end{align*}$$

using the semigroup characterisation of the $\mathbb{H}_{DB^*}^{\mathbf{P}^{\heartsuit}, \pm}$ quasinorm (Theorem 4.34), which is valid since $i(\mathbf{p}^{\heartsuit}) < 2$ and $\theta(\mathbf{p}^{\heartsuit}) < 0$. Note that if $K_- > t_0 + 1$ then $\mathbf{I}_0 = 0$, and that the aperture $\beta$ can remain fixed in this argument, which implies the claimed uniformity in $K$ since $K_-^{-2k}$ is bounded in $K_- > t_0 + 1$. $\square$

**Corollary 6.28.** Let $\varphi \in \mathbb{D}^{\mathbf{P}}(X)$ and $k \in \mathbb{N}$. Then $\mathbf{1}_{K \times \mathbb{R}^n} \mathbf{D}\mathbf{G}_{t_0, \varphi} \in \mathbb{X}^{\mathbf{P}^{\heartsuit}+k}$ for all compact $K \subset \mathbb{R}_{+, t_0}$, with uniform boundedness in $K$ provided $K_- > t_0 + 1$.

**PROOF.** For $\varphi \in \mathbb{D}^{\mathbf{P}}(X)$ we have $D\varphi \in \mathbb{X}_{DB^*}^{\mathbf{P}^{\heartsuit}}$ and $\mathbf{D}\mathbf{G}_{t_0, \varphi} = \tilde{\mathbf{G}}_{t_0}(D\varphi)$, so this follows from Lemma 6.27. $\square$

For $k \in \mathbb{N}$, whenever $F \in \mathbb{X}^{\mathbf{P}^{-k}}$ solves (CR)$_{DB}$ we can invoke Corollary 6.18 when $\varphi \in \mathbb{D}^{\mathbf{P}}(X)$, yielding the equalities (6.11) and (6.12) for sufficiently small $\varepsilon > 0$.

**Corollary 6.29.** Let $\varphi \in \mathbb{D}^{\mathbf{P}}(X)$ and $k \in \mathbb{N}$, and suppose that $F \in \mathbb{X}^{\mathbf{P}^{-k}}$ solves (CR)$_{DB}$. Then

$$(6.30) \quad \lim_{\varepsilon \to 0} \int_{t_0+\varepsilon}^{t_0+2\varepsilon} \int_{\mathbb{R}^n} \langle B^* \mathbf{D}\mathbf{G}_{t_0, \varphi}(t, x), F(t, x) \rangle \, dx \, dt = 0.$$

**PROOF.** For $\varepsilon < 1/2$ we have $t_0 + (2\varepsilon)^{-1} > t_0 + 1$, so by Corollary 6.28 we have

$$\mathbf{1}_{[t_0+(2\varepsilon)^{-1}, t_0+\varepsilon^{-1}]} \times \mathbb{R}^n \mathbf{D}\mathbf{G}_{t_0, \varphi} \in \mathbb{X}^{\mathbf{P}^{\heartsuit}+k+1}$$

with uniformly bounded quasinorms. Since $F \in \mathbb{X}^{\mathbf{P}^{-k}}$, and since $(\mathbf{p} - k)' = \mathbf{p}^{\heartsuit} + k + 1$, absolute convergence of the $X$-space duality integrals implies that
condition (6.9) is satisfied, and also that

$$ \int_{t_0+(2\epsilon)^{-1}}^{t_0+\epsilon^{-1}} \int_{\mathbb{R}^n} |\langle B^* D\mathbf{G}_{t_0,\varphi}(t,x), F(t,x) \rangle| dx dt \lesssim 1 $$

for all $\epsilon < 1/2$. Therefore we can take the limit as $\epsilon \to 0$ of both sides of (6.11) using dominated convergence to conclude that the right hand side vanishes. $\square$

### Step 3: Weak semigroup properties of solutions.

**Lemma 6.30.** Suppose that $F \in X^{\mathbf{P}-k}$ solves $(CR)_{DB}$ for some $k \in \mathbb{N}$. When $t_0 > 0$, $\tau \ge 0$, and $\varphi \in D\mathbf{P}(X)$, we have

$$ (6.31) \qquad \langle B^* D\varphi, F(t_0 + \tau) \rangle_{E^{\mathbf{P}\diamond}} = \langle B^* e^{-\tau D\mathbf{B}^*} \chi^+(DB^*) D\varphi, F(t_0) \rangle_{E^{\mathbf{P}\diamond}}. $$

**PROOF.** We need to rewrite the integrals in (6.30) and (6.12) in terms of duality of slice spaces. By Proposition 6.1, $F(t)$ is in $E^{\mathbf{P}}$ for each $t \in \mathbb{R}_+$. By Lemma 6.3, since $D\varphi \in X_{DB^*}^{\mathbf{P}\diamond}$, we have that $B^* D\mathbf{G}_{t_0,\varphi}(t)$ is in $E^{\mathbf{P}\diamond}$. Hence

$$ \int_{\mathbb{R}^n} \langle B^* D\mathbf{G}_{t_0,\varphi}(t,x), F(t,x) \rangle dx = \langle B^* D\mathbf{G}_{t_0,\varphi}(t), F(t) \rangle_{E^{\mathbf{P}\diamond}} $$

by the slice space duality identification of Proposition 2.42. Therefore (6.30) and (6.12) can be rewritten as

$$ (6.32) \qquad \lim_{\epsilon \to 0} \int_{t_0+\epsilon}^{t_0+2\epsilon} \langle B^* D\mathbf{G}_{t_0,\varphi}(t), F(t) \rangle_{E^{\mathbf{P}\diamond}} dt = 0 $$

and

$$ (6.33) \qquad - \lim_{\epsilon \to 0} \int_{\epsilon}^{2\epsilon} \langle B^* D\mathbf{G}_{t_0,\varphi}(t_0-t), F(t_0-t) \rangle_{E^{\mathbf{P}\diamond}} dt = \lim_{\epsilon \to 0} \int_{\epsilon}^{2\epsilon} \langle B^* D\mathbf{G}_{t_0,\varphi}(t), F(t) \rangle_{E^{\mathbf{P}\diamond}} dt. $$

We evaluate these limits by continuity of the integrands. By Proposition 6.1 we have $F \in C^\infty(\mathbb{R}_+ : E^{\mathbf{P}})$. By the definition of $D\mathbf{P}(X)$, we have that $B^* D\mathbf{G}_{t_0,\varphi}(t) \in E^{\mathbf{P}\diamond}$ for all $t \in \mathbb{R}_{+,t_0}$, with

$$ \lim_{t \searrow t_0} B^* D\mathbf{G}_{t_0,\varphi}(t) = -B^* D\chi^-(B^* D)P_{B^*D}\varphi = -B^* \chi^-(DB^*)D\varphi $$

in $E^{\mathbf{P}\diamond}$ by Corollary 4.39. Therefore (6.32) becomes

$$ (6.34) \qquad \langle B^* \chi^-(DB^*) D\varphi, F(t_0) \rangle_{E^{\mathbf{P}\diamond}} = 0. $$

Next we will prove

$$ (6.35) \qquad \langle B^* e^{-\tau D\mathbf{B}^*} \chi^+(DB^*) D\varphi, F(t_0) \rangle_{E^{\mathbf{P}\diamond}} = \langle B^* \chi^+(DB^*) D\varphi, F(t_0 + \tau) \rangle_{E^{\mathbf{P}\diamond}} $$

by taking the limit of the left hand side of (6.33) and exploiting an algebraic property of the right hand side. Summing (6.34) (at $t_0 + \tau$) and (6.35) will yield (6.31) and complete the proof.

For $\varphi \in D\mathbf{P}(X)$ and $\delta \ge 0$, define

$$ I_{t_0,\varphi}^{\epsilon,\delta} := \int_{\epsilon}^{2\epsilon} \langle B^* D\mathbf{G}_{t_0,\varphi_\delta}(t), F(t) \rangle_{E^{\mathbf{P}\diamond}} dt $$
where $\varphi_\delta := e^{-\delta[B^*D]}\varphi$. By Lemma 6.25, $\varphi_\delta$ is in $\mathbb{D}^p(X)$, and so we can apply (6.33) to get

$$
\begin{align*}
\lim_{\epsilon \to 0} I_{t_0, \varphi}^{\epsilon, \delta} &= - \lim_{\epsilon \to 0} \int_{\epsilon}^{2\epsilon} \langle B^* D \mathbf{G}_{t_0, \varphi_\delta}(t_0 - t), F(t_0 - t) \rangle_{E^p \diamond} dt \\
&= - \lim_{\epsilon \to 0} \int_{\epsilon}^{2\epsilon} \langle B^* e^{-tDB^*} e^{-\delta DB^*} \chi^+(DB^*) D\varphi(t), F(t) \rangle_{E^p \diamond} dt \\
&= - \langle B^* e^{-\delta DB^*} \chi^+(DB^*) D\varphi(t_0), F(t_0) \rangle_{E^p \diamond}
\end{align*}
$$

using the same argument as in the previous paragraph to establish the final equality.
A simple computation shows that $I_{t_0, \varphi}^{\epsilon, \delta} = I_{t_0+\delta, \varphi}^{\epsilon, 0}$, so we can conclude that

$$
\begin{align*}
\langle B^* e^{-\tau DB^*} \chi^+(DB^*) D\varphi, F(t_0) \rangle_{E^p \diamond} &= - \lim_{\epsilon \to 0} I_{t_0, \varphi}^{\epsilon, \tau} \\
&= - \lim_{\epsilon \to 0} I_{t_0 + \tau, \varphi}^{\epsilon, 0} \\
&= \langle B^* \chi^+(DB^*) D\varphi, F(t_0 + \tau) \rangle_{E^p \diamond},
\end{align*}
$$

completing the proof.

□

We can use this lemma, using that $E^P \subset S'$, to see what happens when we test against Schwartz functions.

**Corollary 6.31.** Let $F$, $t_0$, and $\tau$ be as in Lemma 6.30, and suppose $\varphi \in S$.
Then

$$
(6.36) \qquad - \langle \varphi, (\partial_t F)(t_0 + \tau) \rangle_S = \langle B^* e^{-\tau D B^*} \chi^+(DB^*) D\varphi, F(t_0) \rangle_{E^P \diamond}.
$$

PROOF. By Lemma 6.24, $S \subset \mathbb{D}^p(X)$, so we can apply Lemma 6.30 to $\varphi$. Since $F(t_0 + \tau)$ and $(\partial_t F)(t_0 + \tau)$ are in $E^P$, and $\varphi$ and $B^*D\varphi$ are in $E^P \diamond$, we can apply integration by parts in slice spaces (Proposition 2.45) to derive (6.36). $\square$

**Step 4: A reproducing formula for $(\partial_t F)(t_0)$ in terms of higher $t$-derivatives.**

**Lemma 6.32.** Let $t_0 > 0$, $k \in \mathbb{N}_+$, and suppose that $F \in X^P$ solves (CR)$_{DB}$.
Then $(\partial_t^k F)(t_0) \in X_D^P$, with

$$
(6.37) \qquad \|(\partial_t^k F)(t_0)\|_{X_D^P} \lesssim t_0^{-k} \|F\|_{X_P}.
$$

PROOF. Suppose $\varphi \in S$. First note that since $\partial_t^{k-1} F$ solves (CR)$_{DB}$, and is in $X^{P-(k-1)}$ by Proposition 6.1, Corollary 6.31 yields

$$
(6.38) \quad - \langle \varphi, (\partial_t^k F)(t_0/2 + \tau) \rangle_S = \langle B^* e^{-\tau D B^*} \chi^+(DB^*) D\varphi, (\partial_t^{k-1} F)(t_0/2) \rangle_{E^P \diamond}.
$$

Applying this with $\tau = t_0/2$ and using the slice space estimates of Lemma 6.3 and Proposition 6.1, slice space duality, and $\mathbf{p}^{\diamond} \in I(\mathbf{X}, \mathcal{D}\mathbf{B}^*)$, we have

$$
\begin{align*}
\left| \int_{\mathbb{R}^n} \langle \varphi(x), \partial_t^k F(t_0)(x) \rangle dx \right| & \\
&\le \|B^* e^{-t_0 DB^*/2} \chi^+(DB^*) D\varphi\|_{E^P \diamond (t_0/2)} \|(\partial_t^{k-1} F)(t_0/2)\|_{E^{P+1}(t_0/2)} \\
&\lesssim \|D\varphi\|_{X_D^{P \diamond}} t_0^{-k} \|(\partial_t^{k-1} F)(t_0/2)\|_{E^{P-(k-1)}(t_0)} \\
&\lesssim \|D\varphi\|_{X_D^{P \diamond}} t_0^{-k} \|\partial_t^{k-1} F\|_{X^{P-(k-1)}} \\
&\lesssim \|\varphi\|_{X^{P'} t_0^{-k}} \|F\|_{X^P}.
\end{align*}
$$
Since $\varphi$ was arbitrary, this implies that $(\partial_t^k F)(t_0) \in (\mathbf{X}^{\mathbf{P}})' = \mathbf{X}^{\mathbf{P}}$ with the norm estimate (6.37). Furthermore, since $\partial_t^k F$ solves $(\mathrm{CR})_{DB}$, each $(\partial_t^k F)(t_0)$ is in $\overline{\mathcal{R}(DB)} = \overline{\mathcal{R}(D)}$, which implies membership in $\mathbf{X}_D^{\mathbf{P}}$. $\square$

We recall the following elementary lemma.

**Lemma 6.33.** Suppose $k \in \mathbb{N}_+$ and $g \in C^k(\mathbb{R}^+ : \mathbb{C})$, with $t^j g^{(j)}(t) \to 0$ as $t \to \infty$ for all integers $0 \le j \le k-1$. Then for all $t > 0$ we have

$$g(t) = \frac{(-1)^k}{(k-1)!} \int_t^\infty g^{(k)}(\tau)(\tau-t)^{k-1} d\tau.$$

**Corollary 6.34.** Suppose that $F \in \mathbf{X}^{\mathbf{P}}$ solves $(\mathrm{CR})_{DB}$. Then for all $t_0 > 0$ and $\varphi \in S$ we have

$$\langle \varphi, (\partial_t F)(t_0) \rangle_S = \frac{(-1)^k}{(k-1)!} \int_{t_0}^{\infty} \langle \varphi, (\partial_t^{k+1} F)(t) \rangle_{E^{\mathbf{P}'}} (t-t_0)^{k-1} dt.$$

**PROOF.** By Lemma 6.32, the function $t_0 \mapsto (\partial_t F)(t_0)$ is in $C^\infty(\mathbb{R}_+ : \mathbf{X}_D^{\mathbf{P}})$. Therefore for all $\varphi \in S$ the function $g_\varphi$ defined by

$$g_{\varphi}(t_0) := \langle \varphi, (\partial_t F)(t_0) \rangle_S$$

is in $C^\infty(\mathbb{R}_+ : \mathbb{C})$, and for $k \in \mathbb{N}_+$ we have

$$g_{\varphi}^{(k)}(t_0) = \langle \varphi, (\partial_t^{k+1} F)(t_0) \rangle_S = \langle \varphi, (\partial_t^{k+1} F)(t_0) \rangle_{E^{\mathbf{P}'}}.$$

Furthermore, by the same lemma, we have

$$|t_0^k g_{\varphi}(t_0)| = t_0^k |\langle \varphi, (\partial_t F)(t_0) \rangle_{\mathbf{X}_D^{\mathbf{P}'}}| \lesssim_{\varphi,F} t_0^{-1},$$

so the hypotheses of Lemma 6.33 are satisfied, and the result follows. $\square$

**Step 5: Construction of associated 'nice' solutions.**

Since $\mathbf{p}^{\mathcal{P}} \in I(\mathbf{X}, DB^*)$, we have identified $\mathbf{X}_{DB}^{\mathbf{P}} := \mathbf{X}_D^{\mathbf{P}}$ as a completion of $\mathbf{X}_{DB}^{\mathbf{P}}$. In this step of the proof, given a solution $F \in \mathbf{X}^{\mathbf{P}}$ of $(\mathrm{CR})_{DB}$, we will construct distributions modulo polynomials $\tilde{F}(t_0) \in \mathbf{X}_D^{\mathbf{P}}$ which satisfy the properties we want to show for $F(t_0)$. In the remaining steps we will show that $\tilde{F}(t_0) = F(t_0)$, which will complete the proof.

**Lemma 6.35.** Suppose $F \in \mathbf{X}^{\mathbf{P}}$ solves $(\mathrm{CR})_{DB}$. Then for all $t_0 \in [0, \infty)$ and for sufficiently large $N \in \mathbb{N}$ we have

$$\|(t, y) \mapsto t^N (\partial_t^N F)(t_0 + t, y)\|_{\mathbf{X}^{\mathbf{P}}} \lesssim \|F\|_{\mathbf{X}^{\mathbf{P}}}.$$

**PROOF.** This is an immediate corollary of Propositions 2.37 and 6.1. $\square$

Let $F \in \mathbf{X}^{\mathbf{P}}$ solve $(\mathrm{CR})_{DB}$. For $N \in \mathbb{N}$ large enough that Lemma 6.35 applies, define $\zeta \in \Psi_1^\infty$ by

$$\zeta(z) := c_N z e^{-[z]/2}$$

where $c_N = (-1)^{N+1}/N!$. For $k \in \mathbb{N}$ define $\chi_k := \mathbf{1}_{[k-1,k] \times B(0,k)}$, and for all $t_0 \ge 0$ define

$$(6.39) \qquad \tilde{F}_k(t_0) := \mathbb{S}_{\zeta, DB} \left[ t \mapsto \chi_k t^N (\partial_t^N F) \left( t_0 + \frac{t}{2} \right) \right].$$
This is well-defined since the function $t \mapsto \chi_k t^N(\partial_t^N F)(t_0 + (t/2))$ is in $X^2$. Lemma 6.35 and Proposition 4.7 (using $\zeta \in \Psi_1^\infty \subset \Psi_{-\theta(\mathbf{p})+}^{(\theta(\mathbf{p})+n)\frac{1}{2}-j(\mathbf{p})+}$, which requires $\theta(\mathbf{p}) > -1$) imply that each $\tilde{F}_k(t_0)$ is in $X_{DB}^P$, with

$$ \begin{aligned} \|\tilde{F}_k(t_0)\|_{X_{DB}^P} & \lesssim \left\| t \mapsto \chi_k t^N(\partial_t^N F)\left(t_0 + \frac{t}{2}\right) \right\|_{X^P} \\ & \lesssim \|F\|_{X^P}. \end{aligned} $$

The functions $[t \mapsto \chi_k t^N(\partial_t^N F)(t_0 + \frac{t}{2})]$ converge to $[t \mapsto t^N(\partial_t^N F)(t_0 + \frac{t}{2})]$ in $X^P$ as $k \to \infty$, so we can define $\tilde{F}(t_0) \in X_{DB}^P$ by

$$ \tilde{F}(t_0) := S_{\zeta, DB} \left[ t \mapsto t^N(\partial_t^N F)\left(t_0 + \frac{t}{2}\right) \right]. $$

This satisfies the norm estimate

$$ (6.40) \qquad \|\tilde{F}(t_0)\|_{X_{DB}^P} \lesssim \|F\|_{X^P}. $$

**Lemma 6.36.** Let $t_0 \ge 0$. Suppose $F \in X^P$ solves (CR)$_{DB}$ and define $\tilde{F}(t_0) \in X_{DB}^P$ as in the previous paragraphs. Suppose also that $\phi \in X_{B^*D}^{P'} \cap \mathcal{D}(B^*D)$. Then

$$ (6.41) \qquad \langle \phi, \tilde{F}(t_0) \rangle_{X_{B^*D}^{P'}} = -c_N \int_0^\infty \left\langle tB^*e^{-\frac{t}{2}[DB^*]} D\phi, t^N(\partial_t^N F)\left(t_0 + \frac{t}{2}\right) \right\rangle_{E^{P'}} \frac{dt}{t}. $$

**PROOF.** First we show that the $E^{P'}$ duality pairing (6.41) makes sense. Since $\phi \in X_{B^*D}^{P'} \cap \mathcal{D}(B^*D)$, Lemma 6.26 yields $e^{-t[DB^*]/2}D\phi \in E^{P'}$. Since each $tB^*$ is a bounded operator on $E^{P'}$ we have $tB^*e^{-t[DB^*]/2}D\phi \in E^{P'}$. On the other hand, since $t \mapsto (\partial_t^N F)(t_0 + t/2)$ solves (CR)$_{DB}$, by Proposition 6.1 and Lemma 6.35 we have

$$ \|(\partial_t^N F)(t_0 + t/2)\|_{E^{P'-N}(t)} \lesssim \|t \mapsto (\partial_t^N F)(t_0 + t/2)\|_{X^{P'-N}} \lesssim \|F\|_{X^P} $$

for all $t > 0$. Therefore the slice space dual pairing in (6.41) is meaningful.

Now write

$$ \begin{aligned} (6.42) \qquad & \langle \phi, \tilde{F}(t_0) \rangle_{X_{B^*D}^{P'}} \\ &= \lim_{k \to \infty} \langle \phi, \tilde{F}_k(t_0) \rangle_{X_{B^*D}^{P'}} \\ &= \lim_{k \to \infty} \langle \mathbb{Q}_{\zeta, B^*D} \phi, [t \mapsto \chi_k t^N(\partial_t^N F)(t_0 + t/2)] \rangle_{X^P} \\ &= \langle \mathbb{Q}_{\tilde{\zeta}, B^*D} \phi, [t \mapsto t^N(\partial_t^N F)(t_0 + t/2)] \rangle_{X^P} \\ &= -c_N \iint_{\mathbb{R}_+^{1+n}} (t(B^*De^{-t[DB^*]/2}\phi)(x), t^N(\partial_t^N F)(t_0 + t/2, x)) \, dx \frac{dt}{t} \\ &= -c_N \int_0^\infty \langle tB^*e^{-t[DB^*]/2}D\phi, t^N(\partial_t^N F)(t_0 + t/2) \rangle_{E^{P'}} \frac{dt}{t} \end{aligned} $$

using $\tilde{\zeta} = \zeta$ and the slice space containments from the previous paragraph. $\square$

Now we show that the distributions modulo polynomials $(\tilde{F}(t_0))_{t_0 \ge 0}$ are in fact given by the Cauchy operator applied to $\tilde{F}(0)$.
**Proposition 6.37.** Let $F \in X^P$ solve $(CR)_{DB}$ and define $\tilde{F}$ as above. Then for all $t_0 \ge 0$ we have

$$ \tilde{F}(t_0) = e^{-t_0[DB]} \chi^+(DB) \tilde{F}(0). $$

In particular, $\tilde{F}(0) \in X_{DB}^{P,+}$, and so $\tilde{F} = C_{DB}^+(\tilde{F}(0))$.

**PROOF.** Since $\tilde{F}(t_0) \in X_{DB}^P$ and since $X_{B^*D}^{P'} \cap \mathcal{D}(B^*D)$ is dense in $X_{B^*D}^{P'}$ (Corollary 4.9 and density of $\mathcal{D}(B^*D)$ in $X_{B^*D}^2$), it suffices to test against $\phi \in X_{B^*D}^{P'} \cap \mathcal{D}(B^*D)$. For all such $\phi$ write

$$ \begin{aligned} & \langle \phi, e^{-t_0[DB]} \chi^+(DB) \tilde{F}(0) \rangle_{X_{B^*D}^{P'}} \\ &= \langle e^{-t_0[B^*D]} \chi^+(B^*D)\phi, \tilde{F}(0) \rangle_{X_{B^*D}^{P'}} \\ &= -c_N \int_0^\infty \langle tB^*e^{-\frac{t}{2}[DB^*]}D(e^{-t_0[B^*D]}\chi^+(B^*D)\phi), t^N(\partial_t^N F)(t/2) \rangle_{EP'} \frac{dt}{t} \\ &= -c_N \int_0^\infty \langle tB^*e^{-t_0[DB^*]}\chi^+(DB^*)D(e^{-\frac{t}{2}[B^*D]}\phi), t^N(\partial_t^N F)(t/2) \rangle_{EP'} \frac{dt}{t} \\ &= -c_N \int_0^\infty \langle tB^*D(e^{-\frac{t}{2}[B^*D]}\phi), t^N(\partial_t^N F)(t_0 + t/2) \rangle_{EP'} \frac{dt}{t} \\ &= -c_N \int_0^\infty \langle tB^*e^{-\frac{t}{2}[DB^*]}D\phi, t^N(\partial_t^N F)(t_0 + t/2) \rangle_{EP'} \frac{dt}{t} \\ &= \langle \phi, \tilde{F}(t_0) \rangle_{X_{B^*D}^{P'}}. \end{aligned} $$

In the third line we used that $e^{-t_0[B^*D]}\chi^+(B^*D)$ maps $X_{B^*D}^{P'} \cap \mathcal{D}(B^*D)$ into itself, and the representation (6.41). In the fifth line we used Lemma 6.30, which is valid since $e^{-t[BD^*/2]}\phi \in D^P(X)$ (Lemma 6.26) and since $[t \mapsto (\partial_t F)(t/2)] \in X^{P-1}$ solves $(CR)_{DB}$. We use the representation (6.41) once more in the last line. $\square$

This immediately implies the following corollary.

**Corollary 6.38.** Let $F \in X^P$ solve $(CR)_{DB}$. Then $\tilde{F}(0) \in X_D^{P,+}$, $\tilde{F} = C_{DB}^+(\tilde{F}(0))$, and $\tilde{F} \in X^P$.

**PROOF.** All we need to show is that $\tilde{F}$ is in $X^P$. This follows from Theorem 5.26. $\square$

**Step 6: Equality of $\partial_t F$ and $\partial_t \tilde{F}$**.

By Corollary 6.38 and Proposition 4.33, for $F \in X^P$ which solves $(CR)_{DB}$, the function $t_0 \mapsto \tilde{F}(t_0)$ is in $C^\infty(\mathbb{R}_+ : X_D^P)$. Therefore we can consider $(\partial_t \tilde{F})(t_0) \in X_D^P$ as a distribution modulo polynomials.

**Lemma 6.39.** Let $F \in X^P$ solve $(CR)_{DB}$. Then for all $t_0 > 0$ we have $(\partial_t F)(t_0) = (\partial_t \tilde{F})(t_0)$ in $Z'$.

**PROOF.** Fix $\varphi \in Z$. For all $k \in \mathbb{N}$ we have already computed (using that everything is in $L^2$)

$$ (6.43) \qquad \langle \varphi, \tilde{F}_k(t_0) \rangle_Z \\ = -c_N \iint_{\mathbb{R}_+^{1+n}} (t(B^*e^{-t[DB^*]/2}D\varphi)(x), \chi_k t^N(\partial_t^N F)(t_0 + t/2, x)) \, dx \, \frac{dt}{t}. $$
Since $\varphi \in \mathcal{Z}$ we have $D\varphi \in \mathbf{X}_D^{\mathbf{P}'}$, so for each $t > 0$ we may apply the (extended operator) $e^{-t[DB^*]/2}$ to $D\varphi$. We then have

$$ (6.44) \qquad \begin{aligned} & \|t \mapsto tB^*e^{-t[DB^*]/2}D\varphi\|_{X^{\mathbf{P}'}} \\ &= \|t \mapsto B^*e^{-t[DB^*]/2}D\varphi\|_{X^{\mathbf{P}}\mathcal{H}} \\ & \lesssim \|D\varphi\|_{X_D^{\mathbf{P}}\mathcal{H}} \\ & \simeq \|\varphi\|_{X^{\mathbf{P}}'} < \infty, \end{aligned} $$

where (6.44) follows from Proposition 4.4 since $[z \mapsto ze^{-[z]/2}] \in \Psi(X_D^{\mathbf{P}})$ (here we use $i(\mathbf{p}^{\mathcal{H}}) < 2$ and $\theta(\mathbf{p}^{\mathcal{H}}) < 0$). Since $[t \mapsto t^N \partial_t^N F(t_0 + t/2)] \in X^{\mathbf{P}}$ (Lemma 6.35), the integral (6.43) is uniformly bounded in $k$ and so we can take the limit

$$ \langle \varphi, \tilde{F}(t_0) \rangle_{\mathcal{Z}} = \lim_{k \to \infty} \langle \varphi, \tilde{F}_k(t_0) \rangle_{\mathcal{Z}} \\ = -c_N \iint_{\mathbb{R}_+^{1+n}} (t(B^*e^{-t[DB^*]/2}D\varphi)(x), t^N(\partial_t^{N+1}F)(t_0 + t/2, x)) \, dx \, \frac{dt}{t} $$

by dominated convergence. Using dominated convergence again, we can take the derivative:

$$ \begin{aligned} & \langle \varphi, (\partial_t \tilde{F})(t_0) \rangle_{\mathcal{Z}} \\ &= \partial_t \langle \varphi, \tilde{F}(t_0) \rangle_{\mathcal{Z}} \\ &= -c_N \iint_{\mathbb{R}_+^{1+n}} (t(B^*e^{-t[DB^*]/2}D\varphi)(x), t^N(\partial_t^{N+1}F)(t_0 + t/2, x)) \, dx \, \frac{dt}{t} \\ &= -c_N \int_0^\infty \langle tB^*e^{-t[DB^*]/2}D\varphi, t^N(\partial_t^{N+1}F)(t_0 + t/2) \rangle_{E^{\mathbf{P}'}} \frac{dt}{t} \end{aligned} $$

using that $\varphi \in \mathbb{D}^{\mathbf{P}}(X)$ (Lemma 6.24) to conclude that the slice space duality pairing is meaningful as in the proof of Lemma 6.36.

Now we rearrange:

$$ (6.45) \qquad \begin{aligned} & \langle tB^*e^{-t[DB^*]/2}D\varphi, t^N(\partial_t^{N+1}F)(t_0 + t/2) \rangle_{E^{\mathbf{P}'}} \\ &= \langle tB^*D(e^{-t[BD^*]/2}\varphi), t^N(\partial_t^{N+1}F)(t_0 + t/2) \rangle_{E^{\mathbf{P}'}} \end{aligned} $$

$$ (6.46) \qquad \begin{aligned} &= \langle tB^*\chi^+(DB^*)De^{-t[BD^*]/2}\varphi, t^N(\partial_t^{N+1}F)(t_0 + t/2) \rangle_{E^{\mathbf{P}'}} \\ &= t^{N+1}\langle B^*e^{-t[DB^*]/2}\chi^+(DB^*)D\varphi, (\partial_t^{N+1}F)(t_0 + t/2) \rangle_{E^{\mathbf{P}'}} \end{aligned} $$

$$ (6.47) \qquad \begin{aligned} &= t^{N+1} \langle B^* e^{-t[DB^*]/2} \chi^+(DB^*) D\varphi, (\partial_t^{N+1}F)(t_0 + t/2) \rangle_{E^{\mathbf{P}'}} \\ &= t^{N+1} \langle B^* D\varphi, (\partial_t^{N+1}F)(t_0 + t) \rangle_{E^{\mathbf{P}'}} \end{aligned} $$

$$ (6.48) \qquad \begin{aligned} &= t^{N+1} \langle B^* D\varphi, (\partial_t^{N+1}F)(t_0 + t) \rangle_{E^{\mathbf{P}'}} \\ &= -t^{N+1} \langle \varphi, (\partial_t^{N+2}F)(t_0 + t) \rangle_{E^{\mathbf{P}'}} \end{aligned} $$

$$ (6.49) \qquad \begin{aligned} &= -t^{N+1} \langle \varphi, (\partial_t^{N+2}F)(t_0 + t) \rangle_{E^{\mathbf{P}'}} \\ &= -t^{N+1} \langle \varphi, (\partial_t^{N+2}F)(t_0 + t) \rangle_{E^{\mathbf{P}'}} \end{aligned} $$

The first line (6.45) uses that $\varphi \in \mathcal{D}(D) = \mathcal{D}(B^*D)$, (6.46) uses (6.34) and the fact that $e^{-t[BD^*]/2}\varphi$ is in $\mathbb{D}^{\mathbf{P}}(X)$ (Lemma 6.25), (6.47) is just similarity of functional calculi and rearrangement, (6.48) uses the weak semigroup property (6.31), and (6.49) finishes with integration by parts in slice spaces (Proposition 2.45) and $(CR)_{DB}$.

Therefore we have

$$ \begin{aligned} \langle \varphi, (\partial_t \tilde{F})(t_0) \rangle_{\mathcal{Z}} &= c_N \int_0^\infty \langle \varphi, (\partial_t^{N+2}F)(t_0 + t) \rangle_{E^{\mathbf{P}'}} t^{N+1} \frac{dt}{t} \\ &= \frac{(-1)^{N+1}}{N!} \int_{t_0}^\infty \langle \varphi, (\partial_t^{N+2}F)(t) \rangle_{E^{\mathbf{P}'}} (t - t_0)^N dt. \end{aligned} $$
Finally, applying Corollary 6.34 with $k = N + 1$, we get

$$ \langle \varphi, (\partial_t F)(t_0) \rangle_Z = \langle \varphi, (\partial_t \tilde{F})(t_0) \rangle_Z $$

for all $\varphi \in Z$ and all $t_0 > 0$. Therefore we have $(\partial_t F)(t_0) = (\partial_t \tilde{F})(t_0)$ in $Z'$ for all $t_0 > 0$ as claimed. $\square$

### Step 7: Completing the proof.

**Lemma 6.40.** Let $F \in X^P$ solve (CR)$_{DB}$ with $\lim_{t \to \infty} F(t)_{\|} = 0$ in $Z'(\mathbb{R}^n : \mathbb{C}^{nm})$. Then $F = \tilde{F}$.

**PROOF.** By Lemma 6.39 we have $\partial_t F = \partial_t \tilde{F}$ in $Z'$, so there exists $G \in Z'$ such that $F(t_0) = G + \tilde{F}(t_0)$ for all $t_0 \in \mathbb{R}_+$. Since $\lim_{t_0 \to \infty} \tilde{F}(t_0) = 0$ in $X_D^P$ (Proposition 4.33, using the weak-star topology when $p$ is infinite) and hence also in $Z'$, we find that $G_{\|} = 0$. Following the argument of [16, Step 5, page 50], we find that $G = \beta a$ modulo polynomials, where $a$ is invertible in $L^{\infty}$ and $\beta \in \mathbb{C}^m$. To complete the proof it suffices to show that $\beta = 0$.

Note that the constant function $[t \mapsto G = F(t) - \tilde{F}(t)]$ is in $X^P$. If $p$ is finite, then $G \in E^P$ (since $[t \mapsto G]$ solves (CR)$_{DB}$), and this forces $\beta = 0$. If $p$ is infinite, then first we note that if $\beta \neq 0$ then $[t \mapsto G] \notin T_{-1; \tilde{\alpha}}^{\infty}$ for all $\tilde{\alpha} \in [0, 1)$: this follows from estimating

$$ \begin{aligned} \|t \mapsto G\|_{T_{-1; \tilde{\alpha}}^{\infty}} & \ge \frac{1}{R^{\tilde{\alpha}}} \left( \frac{1}{R^n} \int_{B_R} \int_0^R |t^{-1} G(y, t)|^2 \frac{dt}{t} dy \right)^{1/2} \\ & \approx R^{-\tilde{\alpha} - \frac{n}{2}} \left( \int_{B_R} R^{-2} |\beta|^2 |a(y)|^2 dy \right)^{1/2} \\ & \ge R^{-\tilde{\alpha} - 1} |\beta| \|a^{-1}\|_{\infty}^{-1} \end{aligned} $$

for all balls $B_R$ of radius $R > 0$ in $\mathbb{R}^n$, and then taking $R \to 0$. Since $\theta(p) > -1$ we have

$$ G \in X^P \hookrightarrow T_{-1;1+r(p)}^{\infty}, $$

and since $r(p) < 0$ (this is the only time we use this hypothesis), we must have $\beta = 0$. This completes the proof. $\square$

Therefore, by Corollary 6.38, under the assumptions of Theorem 6.9, we have that $F = \tilde{F} = C_{DB}^+(\tilde{F}(0))$, with $\tilde{F}(0) \in X_D^{P,+}$ such that $\|\tilde{F}(0)\|_{X_D^{P,+}} \lesssim \|F\|_{X^P}$ (by (6.40)). Furthermore, if $f \in X_{DB}^{P,+}$ and $F = C_{DB}^+f$, then by Proposition 4.33 we have

$$ f = \lim_{t \to 0} C_{DB}^{+} F(t) = \tilde{F}(0) $$

with limit in $X_{DB}^P$. This completes the proof of Theorem 6.9.

### 6.5. Interpolation of solution spaces

As an interesting aside, we can now show that certain solution spaces for the equation $L_A u = 0$ form interpolation scales. In the context of differential operators with constant coefficients on bounded Lipschitz domains, similar results were proven by Kalton, Mayboroda, and Mitrea [55, Theorem 1.5]. One major difference is that their ‘ambient spaces’ are Triebel-Lizorkin or Besov spaces rather than tent or $Z$-spaces, which is possible since their differential operators have constant (hence very regular) coefficients.
For $\mathbf{p} \in I_{\max}$, define<fnref n="4" />

$$ \widetilde{X^P} := \begin{cases} \{F \in X^P : \lim_{t \to \infty} F(t)\| = 0 \text{ in } \mathcal{Z}'(\mathbb{R}^n)\} & (\theta(\mathbf{p}) < 0), \\ \{F : N_*(F) \in L^i(\mathbf{p})\} & (\theta(\mathbf{p}) = 0). \end{cases} $$

where $N_*(F)$ is defined in (1.4). We set $\|F\|_{\widetilde{X^P}}$ to be $\|F\|_{X^P}$ or $\|N_*F\|_{L^i(\mathbf{p})}$ accordingly, and define

$$ S_{A,X}^{\mathbf{P}} := \{u \in W_{1,\mathrm{loc}}^2(\mathbb{R}^n) : L_A u = 0, \nabla_A u \in \widetilde{X^P}\}, $$

with $\|u\|_{S_{A,X}^{\mathbf{P}}} := \|\nabla_A u\|_{\widetilde{X^P}}$.

Recall that the classification regions $J(\mathbf{X}, \mathcal{DB})$ were defined in Definition 6.10.

**Theorem 6.41 (Interpolation of solution spaces).** Let $\mathbf{p}_0$ and $\mathbf{p}_1$ be exponents, fix $\eta \in (0, 1)$, and set $\mathbf{p}_\eta := [\mathbf{p}_0, \mathbf{p}_1]_\eta$. Let $B = \hat{A}$.

(i) Suppose $\mathbf{p}_0, \mathbf{p}_1 \in J(\mathbf{H}, \mathcal{DB})$, with $\mathbf{p}_0$ finite and $j(\mathbf{p}_1) \ge 0$. Then

$$ [S_{A,T}^{\mathbf{p}_0}, S_{A,T}^{\mathbf{p}_1}]_{\eta} = S_{A,T}^{\mathbf{p}_\eta}. $$

(ii) Suppose $\mathbf{p}_0, \mathbf{p}_1 \in J(\mathbf{B}, \mathcal{DB})$, with $\mathbf{p}_0$ finite. Then

$$ [S_{A,Z}^{\mathbf{p}_0}, S_{A,Z}^{\mathbf{p}_1}]_{\eta} = S_{A,Z}^{\mathbf{p}_\eta}. $$

(iii) Suppose $\mathbf{p}_0, \mathbf{p}_1 \in J(\mathbf{X}, \mathcal{DB})$, with $\theta(\mathbf{p}_0) \neq \theta(\mathbf{p}_1)$. Then

$$ (S_{A,X}^{\mathbf{p}_0}, S_{A,X}^{\mathbf{p}_1})_{\eta, i(\mathbf{p}_\eta)} = S_{A,Z}^{\mathbf{p}_\eta}. $$

The conditions in part (i) exclude the possibility that $S_{A,T}^{\mathbf{p}_\eta}$ corresponds to a BMO-type space.

**PROOF.** For $\mathbf{p} \in J(\mathbf{X}, \mathcal{DB})$, by Theorem 1.5 and Theorems 6.8 and 6.9 (or the corresponding theorems of the second author and Mourgoglou when $\theta(\mathbf{p}) \in \{-1, 0\}$ [16, Theorems 1.1 and 1.3]), and taking into account the decay results of Section 6.2, we have that the operator $\mathbf{Q}_{\mathrm{sgp},\mathcal{DB}}$ is bounded and invertible from $\mathbf{X}_{\mathrm{DB}}^{\mathbf{p}_+}$ into $S_{A,X}^{\mathbf{p}}$. By Theorems 4.34 and 5.26 (see also Remarks 4.29 and 5.28) we have

$$ \|f\|_{\mathbf{X}_{\mathrm{DB}}^{\mathbf{p}_+}} \approx \|\mathbf{Q}_{\mathrm{sgp},\mathcal{DB}} f\|_{X^P} $$

for all $f \in \mathbf{X}_{\mathrm{DB}}^{\mathbf{p}_+}$. Since the regions $J(\mathbf{X}, \mathcal{DB})$ are closed under interpolation given the restrictions on $\mathbf{p}_0$ and $\mathbf{p}_1$ that we have made (this can be shown by combining Theorem 4.32 and Proposition 5.13), the theorem follows from the interpolation theorem for the canonical completions sgp $\mathbf{X}_{\mathrm{DB}}^{\mathbf{p}_+}$ (Theorem 4.28; see Remark 4.29 to justify use of the auxiliary function sgp in this theorem), along with the usual retraction/coretraction interpolation theorem ([75, §1.2.4]). $\square$

We should illustrate this result in a concrete situation, as the current level of abstraction is quite high. Let $\mathbf{p}_0 = (2, 0)$ and $\mathbf{p}_1 = (\infty, s, 0)$, and consider case (i) of the theorem. The corresponding solution space norms are

$$ \|f\|_{S_{A,T}^{\mathbf{p}_0}} = \|N_*(f)\|_{L^2}, $$

$$ \|f\|_{S_{A,T}^{\mathbf{p}_1}} = \|f\|_{T_{s,0}^{\infty}} = \|\mathcal{C}_0(\kappa^{-s}f)\|_{L^{\infty}}, \quad \text{and} $$

$$ \|f\|_{S_{A,T}^{\mathbf{p}_\theta}} = \|f\|_{T_{\theta s}^{\mathbf{p}}} = \|\mathcal{A}(\kappa^{-\theta s}f)\|_{\mathbf{p}}. $$

<footnote n="4">⁴This definition is not totally rigorous, since $F(t)$ need not be defined for all $t > 0$ for arbitrary $F \in X^P$. However we will only consider functions for which $F(t)$ is always defined. In particular the sets $S_{A,X}^{\mathbf{p}}$ defined below are well-defined.</footnote>
with $f = \nabla_A u$ and $p = 2/(1-\theta)$. The interpolation for generic $f$ looks unfamiliar, and is even more surprising when $f$ is the conormal gradient (or equivalently, the gradient) of a solution. This can be understood by seeing that the interpolation theorems for generic function spaces $X^p$ carry over to the solution spaces $S_{A,X}^p$ with exponents in the classification region.

## 6.6. Boundary behaviour of solutions

In this section we establish boundary behaviour of solutions to $L_A u = 0$. What we are looking for is regularity up to the boundary in a classical sense, if possible. Otherwise, we look for almost everywhere non-tangential convergence at the boundary. Since solutions to $L_A u = 0$ need not have pointwise values, we use averages on Whitney regions approaching the boundary. We prove results only for exponents in the classification region of Section 6.3, although this restriction is not always necessary (see for example [60]). This is a natural restriction when using our Cauchy-Riemann approach, and also simplifies part of the argument.

Recall that the spaces $\overline{X}^p$ were defined in the previous section.

**Theorem 6.42.** Let $B = \hat{A}$ and let $\mathbf{p} \in J(\mathbf{X}, DB)$ with $r(\mathbf{p}) \in (-1, 0)$. Let $u$ solve $L_A u = 0$, with $\nabla_A u \in \overline{\mathbf{X}}^p$.

(i) Suppose $\mathbf{p}$ is finite. Then there exists $v \in \mathbf{X}^{p+1}$ such that

$$ (6.50) \qquad \lim_{t \to 0} \iint_{\Omega(t,x)} u(\tau, \xi) \, d\xi \, d\tau = v(x) \quad (\text{a.e. } x \in \mathbb{R}^n) $$
with $\lim_{t \to 0} \nabla_{\|\|} u(t, \cdot) = \nabla_{\|\|} v$ in $\mathcal{Z}'$.

(ii) Suppose $\mathbf{p}$ is infinite. Then $u \in \hat{\Lambda}_{1+r(\mathbf{p})}(\overline{\mathbb{R}}_+^{1+n})$ (i.e. $u$ has a $(1+r(\mathbf{p}))$-Hölder continuous extension to the closed upper half-space $\overline{\mathbb{R}}_+^{1+n}$).

The proof is best presented as a series of lemmas.

**Lemma 6.43.** Suppose that $t > 0$, $\delta \in (0, 1)$, and let $c$ be a Whitney parameter. Then there is a Whitney parameter $\tilde{c}$ such that for all $u$ with $\nabla u \in L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$ and for all $x \in \mathbb{R}^n$ we have

$$ \left| \iint_{\Omega_c(t,x)} u - \iint_{\Omega_c(\delta t, x)} u \right| \lesssim_{\delta} t \left( \iint_{\Omega_{\tilde{c}}(t,x)} |\nabla u|^2 \right)^{1/2} $$

with implicit constant independent of $t$.

**PROOF.** A straightforward computation shows that, writing $c = (c_0, c_1)$, if we define $\tilde{c} := (c_0, c_1/\delta)$ then we have

$$ \Omega_c(t, x) \cup \Omega_c(\delta t, x) \subset \Omega_{\tilde{c}}(t, x) $$

for all $(t, x) \in \mathbb{R}_+^{1+n}$. We then have

$$ \begin{align} \left| \iint_{\Omega_c(t,x)} u - \iint_{\Omega_c(\delta t, x)} u \right| &\le \left| \iint_{\Omega_c(t,x)} u - \iint_{\Omega_{\tilde{c}}(t,x)} u \right| + \left| \iint_{\Omega_{\tilde{c}}(t,x)} u - \iint_{\Omega_c(\delta t, x)} u \right| \nonumber \\ (6.51) \qquad \qquad \qquad \qquad &\lesssim_{\delta} \iint_{\Omega_{\tilde{c}}(t,x)} \left| u - \iint_{\Omega_{\tilde{c}}(t,x)} u \right| \nonumber \\ &\le \left( \iint_{\Omega_{\tilde{c}}(t,x)} \left| u - \iint_{\Omega_{\tilde{c}}(t,x)} u \right|^2 \right)^{1/2} \end{align} $$
$$ (6.52) \qquad \lesssim t \left( \iint_{\Omega_{\tilde{c}}(t,x)} |\nabla u|^2 \right)^{1/2}, $$

using in (6.51) that

$$ \left| \int_A u - \int_B u \right| \le \frac{|B|}{|A|} \left| \int_B u - \int_B u \right| $$

when $A \subseteq B$, and the Poincaré inequality in (6.52). $\square$

Let $c$ be a Whitney parameter. For all $f \in L^0(\mathbb{R}_+^{1+n})$ we define a 'vertical Whitney maximal function' $V_c^p(f)$ on $\mathbb{R}^n$ by

$$ V_c(f)(x) := \sup_{t>0} W_c f(t, x). $$

**Lemma 6.44.** Let $\mathbf{p}$ be a finite exponent and suppose $f \in L^0(\mathbb{R}_+^{1+n})$. Let $c$ be a Whitney parameter. Then

$$ (6.53) \qquad \|V_c(\kappa^{-\theta(\mathbf{p})} f)\|_{L^{i(\mathbf{p})}(\mathbb{R}^n)} \lesssim \|f\|_{X^{\mathbf{p}}}. $$

**PROOF.** First note that $\Gamma_{c_0 c_1}(x) \supset \Omega_c(t, x)$ for all $(t, x) \in \mathbb{R}_+^{1+n}$, so we have

$$ \left( \iint_{\Omega_c(t,x)} |\tau^{-\theta(\mathbf{p})} f(\tau, \xi)|^2 d\tau d\xi \right)^{1/2} \lesssim \left( \iint_{\Gamma_{c_0 c_1}(x)} |\tau^{-\theta(\mathbf{p})} f(\tau, \xi)|^2 \frac{d\tau d\xi}{\tau^{n+1}} \right)^{1/2}. $$

Taking the supremum in $t > 0$ and then integrating in $x$ yields (6.53) with $X = T$. Next, for all $\varepsilon \in (0, 1)$ and for $\tilde{c}_1 \ge c_1 \max(1 + \varepsilon, (1 - \varepsilon)^{-1})$, we have

$$ \Omega_c(t, x) \subset \Omega_{\tilde{c}}(t', x) $$

for all $t' \in [(1 - \varepsilon)t, (1 + \varepsilon)t]$. Therefore

$$ \begin{aligned} & \left( \iint_{\Omega_c(t,x)} |\tau^{-\theta(\mathbf{p})} f(\tau, \xi)|^2 d\tau d\xi \right)^{i(\mathbf{p})/2} \\ & \lesssim \int_{(1-\varepsilon)t}^{(1+\varepsilon)t} \left( \iint_{\Omega_{\tilde{c}}(t',x)} |\tau^{-\theta(\mathbf{p})} f(\tau, \xi)|^2 d\tau d\xi \right)^{i(\mathbf{p})/2} dt' \\ & \lesssim \int_0^\infty \left( \iint_{\Omega_{\tilde{c}}(t',x)} |\tau^{-\theta(\mathbf{p})} f(\tau, \xi)|^2 d\tau d\xi \right)^{i(\mathbf{p})/2} \frac{dt'}{t'}. \end{aligned} $$

Integrating in $x$ then yields (6.53) with $X = Z$. $\square$

<figure><img src="image_39.png" /></figure>

The following lemma is not too difficult to show, so we omit the proof.

**Lemma 6.45.** Suppose $h: (0, \infty) \to \mathbb{R}$ is a function satisfying

$$ |h(t) - h(s)| \lesssim_{\delta} |t - s|^{\alpha} $$

whenever $\delta \in (0, 1)$ and $|t - s| < \delta t$. Then the limit $\lim_{t \to 0} h(t)$ exists, and

$$ (6.54) \qquad |h(t) - \lim_{t \to 0} h(t)| \lesssim |t|^{\alpha}. $$

We can apply the lemmas above to prove the following limit result. Note that we do not need to assume $u$ solves any elliptic equation to establish this.
**Proposition 6.46.** Let **p** be a finite exponent with $\theta(\mathbf{p}) > -1$, and with $\mathbf{p} \in I_{\max}$ if $i(\mathbf{p}) \le 1$. If $u \in L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$ and $\nabla u \in X^{\mathbf{p}}$, then there exists a unique $v \in L^0(\mathbb{R}^n) \cap S'(\mathbb{R}^n)$ such that

$$ (6.55) \qquad \lim_{t \to 0} \iint_{\Omega_c(t,x)} u = v(x) \quad \text{a.e. } x \in \mathbb{R}^n. $$

Moreover,

$$ (6.56) \qquad \lim_{\tau \to 0} \int_{\tau}^{2\tau} u(t, \cdot) dt = v \quad \text{in } S' $$

and likewise

$$ (6.57) \qquad \lim_{\tau \to 0} \int_{\tau}^{2\tau} \nabla_{\|} u(t, \cdot) dt = \nabla_{\|} v \quad \text{in } S'. $$

PROOF. Write

$$ h_x(t) := \iint_{\Omega_c(t,x)} u. $$

By Lemma 6.43, for all $(t, x) \in \mathbb{R}_+^{1+n}$ and for all $\delta \in (0, 1)$ we have

$$ \begin{aligned} |h_x(t) - h_x(\delta t)| &\lesssim_{\delta} t \left( \iint_{\Omega_{\tilde{c}}(t,x)} |\nabla u|^2 \right)^{1/2} \\ &\simeq_{\delta} (t - \delta t)^{1+\theta(\mathbf{p})} \left( \iint_{\Omega_{\tilde{c}}(t,x)} |\tau^{-\theta(\mathbf{p})} \nabla u(\tau, \xi)|^2 d\tau d\xi \right)^{1/2} \\ &\le (t - \delta t)^{1+\theta(\mathbf{p})} g(x) \end{aligned} $$

with $\tilde{c}$ coming from Lemma 6.43 and with $g := V_{\tilde{c}}(\kappa^{-\theta(\mathbf{p})} \nabla u)$. By Lemma 6.44 we find that $g \in L^i(\mathbf{p})(\mathbb{R}^n)$, and therefore $g(x)$ is finite for almost every $x \in \mathbb{R}^n$. Since $1 + \theta(\mathbf{p}) > 0$, for almost every $x$ we can apply Lemma 6.45 to conclude that $\lim_{t \to 0} h_x(t)$ exists. Since the function $x \mapsto h_x(t)$ is measurable for each $t > 0$, the almost-everywhere defined limit

$$ v: x \mapsto \lim_{t \to 0} h_x(t) $$

is measurable, and by construction $v$ is the unique function satisfying (6.55).

Next we will show that $v \in S'(\mathbb{R}^n)$. This will follow from the estimate

$$ (6.58) \qquad |v(x) - v(y)| \lesssim |x - y|^{1+\theta(\mathbf{p})} (g(x) + \bar{g}(x) + g(y)) $$

for all $x, y \in \mathbb{R}^n$ with $\bar{g}$ defined in the same way as $g$ but with a different choice of Whitney parameter. Indeed, assuming (6.58), for all $x \in \mathbb{R}^n$ and all $y \in \mathbb{R}^n$ such that $|v(y)| < \infty$,

$$ |v(x)| \lesssim |v(y)| + |x - y|^{1+\theta(\mathbf{p})} (g(x) + \bar{g}(x) + g(y)), $$

and if $i(\mathbf{p}) > 1$ then for all $\varphi \in S(\mathbb{R}^n)$ we will have

$$ \int_{\mathbb{R}^n} |v(x)||\varphi(x)| dx < \infty $$

since $g, \bar{g} \in L^i(\mathbf{p})(\mathbb{R}^n)$ and $\varphi \in L^i(\mathbf{p})'(\mathbb{R}^n)$. If $i(\mathbf{p}) \le 1$ then since $\mathbf{p} \in I_{\max}$ (by assumption) there exists $\mathbf{q}$ with $\theta(\mathbf{q}) > -1$ and $i(\mathbf{q}) > 1$ such that $\mathbf{p} \to \mathbf{q}$, so that $\nabla u \in X^{\mathbf{q}}$, and we can then argue with $\mathbf{q}$ replacing $\mathbf{p}$.
We now prove (6.58). Let $R = |x - y|$ and let $c' = (c_0 + 1, c_1)$. Then we have $\Omega_c(x, R) \cup \Omega_c(y, R) \subset \Omega_{c'}(x, R)$, and so arguing as in the proof of Lemma 6.43 we have

$$|v(x) - v(y)| \leq |v(x) - \mathcal{I}_{\Omega_c(x, R)} u| + |\mathcal{I}_{\Omega_c(x, R)} u - \mathcal{I}_{\Omega_c(y, R)} u| + |\mathcal{I}_{\Omega_c(y, R)} u - v(y)| \\ \lesssim R^{1+\theta(\mathbf{p})}(g(x) + \bar{g}(x) + g(y))$$

where $\bar{g}$ is defined using the Whitney parameter $c'$ in place of $\tilde{c}$.

It remains to prove the limits (6.56) and (6.57). By the same embedding argument as before, it suffices to prove this for $\mathbf{p}$ such that $i(\mathbf{p}) > 1$. Let $\varphi \in S'(\mathbb{R}^n)$ and $\tau > 0$. Then we have

$$\begin{aligned} & \int_{\tau}^{2\tau} |(u(t, \cdot) - v(\cdot), \varphi)| dt \\ & \leq \int_{\mathbb{R}^n} \int_{\tau}^{2\tau} \int_{B(x, c_0\tau)} |u(t, y) - v(y)| |\varphi(y)| dy dt dx \\ & \leq \int_{\mathbb{R}^n} \int_{\tau}^{2\tau} \int_{B(x, c_0\tau)} |u(t, y) - \mathcal{I}_{\Omega_c(\tau, x)} u(s, z) ds dz| |\varphi(y)| dy dt dx \\ & \quad + \int_{\mathbb{R}^n} \int_{\tau}^{2\tau} \int_{B(x, c_0\tau)} |v(x) - v(y)| |\varphi(y)| dy dt dx \\ & \quad + \int_{\mathbb{R}^n} \int_{\tau}^{2\tau} \int_{B(x, c_0\tau)} |\mathcal{I}_{\Omega_c(\tau, x)} u(s, z) ds dz - v(x)| |\varphi(y)| dy dt dx \\ & =: \mathbf{I}_1 + \mathbf{I}_2 + \mathbf{I}_3. \end{aligned}$$

We estimate these three terms individually. First, for $\tau < 1$, with $c = (c_0, 2)$,

$$\begin{aligned} \mathbf{I}_1 & \lesssim \int_{\mathbb{R}^n} \left( \mathcal{I}_{\Omega_c(\tau, x)} \left| u - \mathcal{I}_{\Omega_c(\tau, x)} u \right| \right) \sup_{y \in B(x, c_0\tau)} |\varphi(y)| dx \\ & \lesssim \tau^{1+\theta(\mathbf{p})} \int_{\mathbb{R}^n} g(x) \sup_{y \in B(x, c_0)} |\varphi(y)| dx \\ & \lesssim_{\varphi} \tau^{1+\theta(\mathbf{p})} \|g\|_{L^i(\mathbf{p})} \end{aligned}$$

since the function $x \mapsto \sup_{y \in B(x, c_0)} |\varphi(y)|$ is in $L^i(\mathbf{p})'$. In the same way we can prove

$$\mathbf{I}_3 \lesssim \tau^{1+\theta(\mathbf{p})} \|g\|_{L^i(\mathbf{p})}$$

for $\tau < 1$ by using (6.54). Finally, using (6.58), for $\tau < 1$ we have

$$\begin{aligned} \mathbf{I}_2 & \leq \int_{\mathbb{R}^n} \int_{B(x, c_0\tau)} |v(x) - v(y)| dy \sup_{z \in B(x, c)} |\varphi(z)| dx \\ & \lesssim \tau^{1+\theta(\mathbf{p})} \int_{\mathbb{R}^n} \int_{B(x, c_0\tau)} g(x) + g(y) dy \sup_{z \in B(x, c)} |\varphi(z)| dx \\ & \lesssim \tau^{1+\theta(\mathbf{p})} \int_{\mathbb{R}^n} (g(x) + Mg(x)) \sup_{z \in B(x, c)} |\varphi(z)| dx \\ & \lesssim_{\varphi} \tau^{1+\theta(\mathbf{p})} \|g\|_{L^i(\mathbf{p})} \end{aligned}$$
where $\mathcal{M}$ is the Hardy–Littlewood maximal operator (which is bounded on $L^i(\mathbf{p}))$. Since $1 + \theta(\mathbf{p}) > 0$ we therefore have

$$ \lim_{\tau \to 0} \int_{\tau}^{2\tau} |(u(t, \cdot) - v(\cdot), \varphi)| dt = 0, $$

which proves (6.56). The limit (6.57) is then proven by taking $\psi \in S(\mathbb{R}^n : \mathbb{C}^n)$ and applying the previous argument to $\varphi = \text{div } \psi$. $\square$

These are all the results we need for the case of finite exponents. We will prove Theorem 6.42 after presenting the results required for infinite exponents (which are simpler).

**Proposition 6.47.** Let $\mathbf{p}$ be infinite. Then for all $u \in L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$ we have

$$ (6.59) \qquad \iint_{\Omega_c(x,R)} |u - \iint_{\Omega_c(x,R)} u|^2 \lesssim R^{2+2r(\mathbf{p})} \|\nabla u\|_{Z_{\mathbf{p}}}^2. $$

Moreover, if $r(\mathbf{p}) \in (-1, 0]$, then for all $x \in \mathbb{R}^n$ the limit

$$ v(x) := \lim_{R \to 0} \iint_{\Omega_c(x,R)} u $$

exists and satisfies

$$ (6.60) \qquad |v(x) - v(y)| \lesssim |x - y|^{1+r(\mathbf{p})}. $$

**PROOF.** The estimate (6.59) follows from the Poincaré inequality and the supremum definition of the $Z^{\mathbf{P}}$ norm (see Corollary 2.27). The remaining statements are then proven exactly as in Proposition 6.46 (using Lemma 6.45, without the additional complication of the function $g$). $\square$

**Remark 6.48.** Proposition 6.47 shows that having a boundary trace in the Hölder space $\dot{\Lambda}^{1+r(\mathbf{p})}(\mathbb{R}^n)$ is a generic property$^5$ of the space $\{u \in L_{\text{loc}}^2 : \nabla u \in Z^{\mathbf{P}}\}$ when $\mathbf{p}$ is infinite and $r(\mathbf{p}) \in (-1, 0]$. This was already observed by Barton and Mayboroda [22, Theorem 6.3].

**PROOF OF THEOREM 6.42.** First suppose $\mathbf{p}$ is finite. Proposition 6.46 furnishes a distribution $v \in L^0 \cap \mathcal{Z}'$ satisfying (6.50). Furthermore, Theorems 1.5, 6.8 and 6.9 imply that there exists $f := \nabla_A u|_{t=0} \in \mathbf{X}_D^{\mathbf{P}} \subset \mathcal{Z}'$ such that $\nabla_A u(t, \cdot) \to \nabla_A u|_{t=0}$ as $t \to 0$ in $\mathbf{X}_{DB}^{\mathbf{P},+} \subset \mathbf{X}_D^{\mathbf{P}}$, hence also in $\mathcal{Z}'$. Along with the limit (6.57), this yields

$$ \nabla_{\|} u|_{t=0} = (\nabla_A u|_{t=0})_{\|} = \nabla_{\|} v $$

which proves that $\nabla_{\|} v \in \mathbf{X}^{\mathbf{P}}$ (and therefore $v \in \mathbf{X}^{\mathbf{P}+1}$), and $\lim_{t \to 0} \nabla_{\|} u(t, \cdot) = \nabla_{\|} v$ in $\mathcal{Z}'$.

Now suppose $\mathbf{p}$ is infinite. We will show that for every cylinder $C(x, a, R)$ in $\mathbb{R}_+^{1+n}$ of the form $(a, a + R) \times B(x, R)$, we have the estimate

$$ (6.61) \qquad \iint_{C(x,a,R)} |u(\tau, y) - \iint_{C(x,a,R)} u|^2 d\tau dy \lesssim R^{2(1+r(\mathbf{p}))}. $$

By Campanato's characterisation of the Hölder space $\dot{\Lambda}_{1+r(\mathbf{p})}(\overline{\mathbb{R}}_+^{1+n})$ [25, Theorem I.2] (which uses that $1 + r(\mathbf{p}) \in (0, 1)$), this will complete the proof of Theorem 6.42 when $\mathbf{p}$ is infinite.

<footnote>$^5$That is, it holds for all $u$ in this space, without requiring that $u$ solves an elliptic equation.</footnote>
By Theorems 1.5 and 6.9, we know that for all $t \ge 0$, $t_0 > 0$ the function $t \mapsto \nabla_A u(t+t_0, \cdot)$ is in $Z^P$ (using the embedding $T^P \hookrightarrow Z^P$ for infinite exponents from Lemma 2.35) with

$$ \|t \mapsto \nabla_A u(t+t_0)\|_{Z^P} \lesssim \|\nabla_A u(t_0)\|_{X_D^P} \lesssim 1. $$

Therefore, using Proposition 6.47 for $u(\cdot + t_0)$ and translating, we get

$$ (6.62) \qquad \iint_{(t_0,0)+\Omega_c(x,R)} \left|u - \iint_{(t_0,0)+\Omega_c(x,R)} u\right|^2 \lesssim R^{2(1+r(\mathbf{p}))}. $$

Now consider a cylinder $C(x, a, R)$ as above. If $a \ge R$, then $C(x, a, R) \subset (R-a, 0) \times \Omega_c(x, R)$, and in this case (6.61) follows from (6.62). To prove (6.61) for $a < R$, it suffices to take $a = 0$. Let $v$ be the boundary trace of $u$ given in Proposition 6.47. Then we have

$$ \begin{aligned} \iint_{C(x,0,R)} \left|u - \iint_{C(x,0,R)} u\right|^2 &\lesssim \iint_{C(x,0,R)} |u-v(x)|^2 + \iint_{C(x,0,R)} \left|v(x) - \iint_{C(x,0,R)} u\right| \\ &\lesssim \iint_{C(x,0,R)} |u-v(x)|^2 + R^{2(1+r(\mathbf{p}))} \end{aligned} $$

using (6.54). To handle the remaining summand, estimate

$$ \begin{aligned} & \iint_{C(x,0,R)} |u(\tau, y) - v(x)|^2 d\tau dy \\ & \lesssim \iint_{C(x,0,2R)} \iint_{\Omega_c(\sigma,z)} |u(\tau, y) - v(x)|^2 d\tau dy d\sigma dz \\ & \lesssim \iint_{C(x,0,2R)} \left( \iint_{\Omega_c(\sigma,z)} |u(\tau, y) - v(z)|^2 d\tau dy \right) + \left( |v(z) - v(x)|^2 \right) d\sigma dz \end{aligned} $$

$$ (6.63) \qquad \lesssim \iint_{C(x,0,2R)} \sigma^{2(1+r(\mathbf{p}))} + |z-x|^{2(1+r(\mathbf{p}))} dz d\sigma $$

$$ (6.64) \qquad \lesssim R^{2(1+r(\mathbf{p}))} $$

using (6.54) and (6.60) in (6.63), and in the last line using that $1+r(\mathbf{p}) \ge 0$. This proves (6.61), and thus completes the proof of Theorem 6.42. $\square$
<figure><img src="image_40.png" /></figure>
# Applications to Boundary Value Problems

## 7.1. Characterisation of well-posedness and corollaries

First let us put the boundary value problems given in the introduction (Subsection 1.1.2) in a more convenient form. Fix $m \in \mathbb{N}$ and let $\mathbf{p}$ be an exponent. By making use of the transversal/tangential splitting $\mathbb{C}^{m(1+n)} = \mathbb{C}^m \oplus \mathbb{C}^{mn}$, we can write

$$ 
\begin{aligned}
\mathbf{X}_D^{\mathbf{P}}(\mathbb{R}^n : \mathbb{C}^{m(1+n)}) &= \mathbb{P}_D(\mathbf{X}^{\mathbf{P}}(\mathbb{R}^n : \mathbb{C}^{m(1+n)})) \\
&= (\mathbf{X}^{\mathbf{P}} \cap D\mathcal{Z}')(\mathbb{R}^n : \mathbb{C}^{m(1+n)}) \\
&= \mathbf{X}^{\mathbf{P}}(\mathbb{R}^n : \mathbb{C}^m) \oplus (\mathbf{X}^{\mathbf{P}}(\mathbb{R}^n : \mathbb{C}^{mn}) \cap \nabla_{\|} \mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{mn})) \\
&=: \mathbf{X}_{\perp}^{\mathbf{P}} \oplus \mathbf{X}_{\|}^{\mathbf{P}}.
\end{aligned}
$$

**Definition 7.1.** For all exponents $\mathbf{p} \in I_{\max}$ we define the *Regularity problem*

$$ (R_{\mathbf{X}})_A^{\mathbf{P}} : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \lim_{t \to 0} \nabla_{\|} u(t, \cdot) = f \in \mathbf{X}_{\|}^{\mathbf{P}}, \\ \|\nabla u\|_{\widetilde{\mathbf{X}}^{\mathbf{P}}} \lesssim \|f\|_{\mathbf{X}^{\mathbf{P}}}, \end{array} \right. $$

and the *Neumann problem*

$$ (N_{\mathbf{X}})_A^{\mathbf{P}} : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \lim_{t \to 0} \partial_{\nu_A} u(t, \cdot) = f \in \mathbf{X}_{\perp}^{\mathbf{P}}, \\ \|\nabla u\|_{\widetilde{\mathbf{X}}^{\mathbf{P}}} \lesssim \|f\|_{\mathbf{X}^{\mathbf{P}}}. \end{array} \right. $$

The spaces $\widetilde{\mathbf{X}}^{\mathbf{P}}$ (which are just the spaces $\mathbf{X}^{\mathbf{P}}$ with an additional decay condition at infinity) were defined in Section 6.5. By $\lim_{t \to 0} \nabla_{\|} u(t, \cdot) = f \in \mathbf{X}_{\|}^{\mathbf{P}}$ we mean that $f \in \mathbf{X}_{\|}^{\mathbf{P}}$ and that the limit is in the $\mathbf{X}_{\|}^{\mathbf{P}}$ topology, and likewise for the limit in the Neumann problem. These limits are imposed in the weak-star topology when $\mathbf{p}$ is infinite. We say that such a problem is *well-posed* if for all boundary data $f$ there exists a unique $u$ (up to additive constant) satisfying the conditions of the problem.

We denote these problems simultaneously by $(P_{\mathbf{X}})_A^{\mathbf{P}}$, with $P$ standing for either $R$ or $N$. We define the *well-posedness region*

$$ \mathrm{WP}(P_{\mathbf{X}})_A := \{\mathbf{p} \in I_{\max} : (P_{\mathbf{X}})_A^{\mathbf{P}} \text{ is well-posed}\}. $$

**Remark 7.2.** The problems $(R_{\mathbf{X}})_A^{\mathbf{P}}$ and $(N_{\mathbf{X}})_A^{\mathbf{P}}$ include all Regularity and Neumann problems introduced in Subsection 1.1.2. The definition above is much more concise (but, initially, much less clear).

**Remark 7.3.** We stress that in the case of Hardy-Sobolev spaces, there is no a priori connection between the interior control and the boundary condition. In