$\mathcal{E}_{-\Delta}(u_0)$, which proves (6.4) and also implies that $u(t)$ converges to $u_0$ in $\mathcal{S}'$ as $t \to 0$. Moreover, Theorem 1.1 (ii) says that $u_0$ has the properties asserted in (i) and (ii). This completes the proof. $\square$

## 7. HOMOGENEOUS CAUCHY PROBLEM

This section is devoted to investigating existence of global weak solutions to the homogeneous Cauchy problem (HC), where $v_0$ lies in $\dot{H}^{2\beta+1,p}$ with $-1 < \beta < 0$, as asserted in Theorem 1.3.

Before stating our results, let us first briefly revisit the heat equation. For any $v_0 \in \mathcal{S}'$, the heat extension of $v_0$, $\mathcal{E}_{-\Delta}(v_0) \in C^\infty(\mathbb{R}_+^{1+n})$ is a global weak (hence classical) solution to the heat equation with initial data $v_0$. We established a series of estimates on boundedness of the heat extension on $\dot{H}^{s,p}$, cf. Proposition 3.1, Theorem 3.2, and Corollary 3.4, with the correspondence $s = 2\beta + 1$. For general $L$, we cannot start from tempered distributions, and our weak solutions will be constructed by extending $\mathcal{E}_L$ (as defined in (1.3) on $L^2$) to $\dot{H}^{2\beta+1,p}$. When $L = -\Delta$, both approaches are consistent.

### 7.1. Main properties of the semigroup solution map.
Let us state the main theorem on extension of $\mathcal{E}_L$ to $\dot{H}^{2\beta+1,p}$, of which the existence of the global weak solution asserted in Theorem 1.3 is a direct consequence.

**Theorem 7.1** (Extension of $\mathcal{E}_L$ to $\dot{H}^{2\beta+1,p}$). Let $-1 < \beta < 0$ and $\tilde{p}_L(\beta) < p \le \infty$. Then $\mathcal{E}_L$ extends to an operator from $\dot{H}^{2\beta+1,p}$ to $L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$, also denoted by $\mathcal{E}_L$, so that the following properties hold for any $v_0 \in \dot{H}^{2\beta+1,p}$ and $v := \mathcal{E}_L(v_0)$.

(a) (Regularity) $\nabla v$ belongs to $T_{\beta+1/2}^p$ with equivalence

$$ (7.1) \qquad \|\nabla v\|_{T_{\beta+1/2}^p} \stackrel{\sim}{=} \|v_0\|_{\dot{H}^{2\beta+1,p}}. \qquad \text{Moreover, if } -1 < \beta < -1/2, \text{ then } v \text{ lies in } T_{\beta+1}^p \text{ with} $$

$$ (7.2) \qquad \|\nabla v\|_{T_{\beta+1/2}^p} \stackrel{\sim}{=} \|v\|_{T_{\beta+1}^p} \stackrel{\sim}{=} \|v_0\|_{\dot{H}^{2\beta+1,p}}. $$

(b) (Explicit formulae) It holds that

$$ (7.3) \qquad v = \mathcal{E}_{-\Delta}(v_0) - \mathcal{R}_{1/2}^L((A - \mathbb{I})\nabla \mathcal{E}_{-\Delta}(v_0)) $$

$$ (7.4) \qquad = \mathcal{E}_{-\Delta}(v_0) + \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla v), $$

where $\mathcal{E}_{-\Delta}$ is the heat extension given in Section 3.1, and $\mathcal{R}_{1/2}^L, \mathcal{R}_{1/2}^{-\Delta}$ are given by Theorem 5.1.

(c) (Continuity) $v$ belongs to $C([0, \infty); \mathcal{S}')$ with $v(0) = v_0$.

(d) (Strong continuity) When $p_-((2\beta+1, L)) < p < p_+((2\beta+1, L))$, $v$ also belongs to $C^\infty((0, \infty); \dot{H}^{2\beta+1,p}) \cap C_0([0, \infty); \dot{H}^{2\beta+1,p})$ with

$$ \sup_{t>0} \|v(t)\|_{\dot{H}^{2\beta+1,p}} \stackrel{\sim}{=} \|v_0\|_{\dot{H}^{2\beta+1,p}}. $$

(e) $v$ is a global weak solution to (HC) with initial data $v_0$.

The proof is deferred to Section 7.3. Let us give some remarks.

**Remark 7.2.** When $\beta > -1/2$, the equivalence (7.2) fails: [AH23b, Theorem 4.3] shows that any weak solution $v \in T_{\beta+1}^p$ to the equation $\partial_t v - \text{div}(A\nabla v) = 0$ must be zero when $p_L^b(\beta) < p \le \infty$, where $p_L^b(\beta)$ is given by (1.10).
$\mathcal{E}_{-\Delta}(u_0)$, which proves (6.4) and also implies that $u(t)$ converges to $u_0$ in $\mathcal{S}'$ as $t \to 0$. Moreover, Theorem 1.1 (ii) says that $u_0$ has the properties asserted in (i) and (ii). This completes the proof. $\square$

## 7. HOMOGENEOUS CAUCHY PROBLEM

This section is devoted to investigating existence of global weak solutions to the homogeneous Cauchy problem (HC), where $v_0$ lies in $\dot{H}^{2\beta+1,p}$ with $-1 < \beta < 0$, as asserted in Theorem 1.3.

Before stating our results, let us first briefly revisit the heat equation. For any $v_0 \in \mathcal{S}'$, the heat extension of $v_0$, $\mathcal{E}_{-\Delta}(v_0) \in C^\infty(\mathbb{R}_+^{1+n})$ is a global weak (hence classical) solution to the heat equation with initial data $v_0$. We established a series of estimates on boundedness of the heat extension on $\dot{H}^{s,p}$, cf. Proposition 3.1, Theorem 3.2, and Corollary 3.4, with the correspondence $s = 2\beta + 1$. For general $L$, we cannot start from tempered distributions, and our weak solutions will be constructed by extending $\mathcal{E}_L$ (as defined in (1.3) on $L^2$) to $\dot{H}^{2\beta+1,p}$. When $L = -\Delta$, both approaches are consistent.

### 7.1. Main properties of the semigroup solution map.
Let us state the main theorem on extension of $\mathcal{E}_L$ to $\dot{H}^{2\beta+1,p}$, of which the existence of the global weak solution asserted in Theorem 1.3 is a direct consequence.

**Theorem 7.1** (Extension of $\mathcal{E}_L$ to $\dot{H}^{2\beta+1,p}$). Let $-1 < \beta < 0$ and $\tilde{p}_L(\beta) < p \le \infty$. Then $\mathcal{E}_L$ extends to an operator from $\dot{H}^{2\beta+1,p}$ to $L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$, also denoted by $\mathcal{E}_L$, so that the following properties hold for any $v_0 \in \dot{H}^{2\beta+1,p}$ and $v := \mathcal{E}_L(v_0)$.

(a) (Regularity) $\nabla v$ belongs to $T_{\beta+1/2}^p$ with equivalence

$$ (7.1) \qquad \|\nabla v\|_{T_{\beta+1/2}^p} \stackrel{\sim}{=} \|v_0\|_{\dot{H}^{2\beta+1,p}}. \qquad \text{Moreover, if } -1 < \beta < -1/2, \text{ then } v \text{ lies in } T_{\beta+1}^p \text{ with} $$

$$ (7.2) \qquad \|\nabla v\|_{T_{\beta+1/2}^p} \stackrel{\sim}{=} \|v\|_{T_{\beta+1}^p} \stackrel{\sim}{=} \|v_0\|_{\dot{H}^{2\beta+1,p}}. $$

(b) (Explicit formulae) It holds that

$$ (7.3) \qquad v = \mathcal{E}_{-\Delta}(v_0) - \mathcal{R}_{1/2}^L((A - \mathbb{I})\nabla \mathcal{E}_{-\Delta}(v_0)) $$

$$ (7.4) \qquad = \mathcal{E}_{-\Delta}(v_0) + \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla v), $$

where $\mathcal{E}_{-\Delta}$ is the heat extension given in Section 3.1, and $\mathcal{R}_{1/2}^L, \mathcal{R}_{1/2}^{-\Delta}$ are given by Theorem 5.1.

(c) (Continuity) $v$ belongs to $C([0, \infty); \mathcal{S}')$ with $v(0) = v_0$.

(d) (Strong continuity) When $p_-((2\beta+1, L)) < p < p_+((2\beta+1, L))$, $v$ also belongs to $C^\infty((0, \infty); \dot{H}^{2\beta+1,p}) \cap C_0([0, \infty); \dot{H}^{2\beta+1,p})$ with

$$ \sup_{t>0} \|v(t)\|_{\dot{H}^{2\beta+1,p}} \stackrel{\sim}{=} \|v_0\|_{\dot{H}^{2\beta+1,p}}. $$

(e) $v$ is a global weak solution to (HC) with initial data $v_0$.

The proof is deferred to Section 7.3. Let us give some remarks.

**Remark 7.2.** When $\beta > -1/2$, the equivalence (7.2) fails: [AH23b, Theorem 4.3] shows that any weak solution $v \in T_{\beta+1}^p$ to the equation $\partial_t v - \text{div}(A\nabla v) = 0$ must be zero when $p_L^b(\beta) < p \le \infty$, where $p_L^b(\beta)$ is given by (1.10).
For $\beta = -1/2$, it also fails. An alternative of the equivalence (7.2) can be achieved with the Kenig-Pipher space $X^p$ as

$$ \left\{ \begin{array}{ll} \|v\|_{X^p} \approx \|\nabla v\|_{T_0^p} \approx \|v_0\|_{L^p} & \text{if } p_-(L) = p_-(0, L) < p < \infty \\ \|v\|_{X^\infty} \approx \|v_0\|_{L^\infty}, & \text{if } p = \infty \\ \|\nabla v\|_{T_0^\infty} \approx \|v_0\|_{\text{BMO}} & \text{if } p = \infty \end{array} \right. . $$

See [AMP19, Theorem 7.6, Corollary 5.5 & 5.10] and [Zat20, Theorem 7.6].

**7.2. L-adapted Hardy-Sobolev spaces.** Our main tool is L-adapted Hardy-Sobolev spaces. We give a brief review, and the reader can refer to [AA18, Chapter 4] and [AE23, §8.2] for details. See also [HvNVW17, Chapter 10] for definitions and basic facts of sectorial operators and $H^\infty$-calculus.

Denote by $S_\mu^+$ the set $\{z \in \mathbb{C} \setminus \{0\} : |\arg(z)| < \mu\}$ for $0 < \mu < \pi$. The class $\Psi_\infty^\infty(S_\mu^+)$ consists of holomorphic functions $\psi : S_\mu^+ \to \mathbb{C}$ fulfilling that for any $\sigma \in \mathbb{R}$, there exists a constant $C > 0$ so that

$$ |\psi(z)| \le C|z|^\sigma. $$

The class $H^\infty(S_\mu^+)$ consists of bounded holomorphic functions on $S_\mu^+$.

It is well-known that $L$ is a sectorial operator of angle $\omega$ for some $\omega \in [0, \pi/2)$ and has bounded $H^\infty(S_\mu^+)$-calculus for any $\mu \in (\omega, \pi)$. For any $\psi \in H^\infty(S_\mu^+)$, define the operator $Q_{\psi,L} : L^2 \to L^\infty((0, \infty); L^2)$ by

$$ Q_{\psi,L}(f)(t) := \psi(tL)f. $$

Note that for $\psi(z) = e^{-z}$, $Q_{\psi,L}$ identifies with $\mathcal{E}_L$.

**Definition 7.3 (L-adapted Hardy-Sobolev spaces).** Let $s \in \mathbb{R}$, $0 < p < \infty$, and $\omega < \mu < \pi$. Let $\psi$ be a non-zero function in $H^\infty(S_\mu^+)$. The L-adapted Hardy-Sobolev space $H_{L,\psi}^{s,p}$ consists of $f \in L^2$ so that $Q_{\psi,L}(f)$ lies in $T_{(s+1)/2}^p \cap L_{1/2}^2(\mathbb{R}_+^{1+n})$, equipped with the (quasi-)norm

$$ \|f\|_{H_{L,\psi}^{s,p}} := \|Q_{\psi,L}(f)\|_{T_{(s+1)/2}^p}. $$

A priori, this space depends on the choice of $\psi$. For each $(s, p)$, there is a subclass of bounded holomorphic functions, called *admissible functions for $H_{L,\psi}^{s,p}$*, for which the spaces $H_{L,\psi}^{s,p}$ agree as sets with mutually equivalent (quasi-)norms. We denote that space by $H_L^{s,p}$. All non-zero $\psi \in \Psi_\infty^\infty(S_\mu^+)$ are admissible functions for $H_L^{s,p}$. See [AE23, §8.2] for some explicit classes of admissible functions for $H_L^{s,p}$. For example, $\psi(z) = e^{-z}$ is admissible for $p \le 2$ and $s < 0$.

The spaces $H_L^{s,p}$ are not complete for the defining (quasi-)norms. Aside from abstract completion, we are interested in realizing the completion as a subspace of tempered distributions. The next result provides a range of $(s, p)$, called the identification range, within which a completion in fact equals to $\dot{H}^{s,p}$. When $0 \le s \le 1$, sharpness of this range is shown in [AE23, §19.1]. We describe the range for $-1 \le s \le 0$, and also exhibit a possible extra range where a completion exists in $\dot{H}^{s,p}$ without knowing equality. This fact will be useful when we come to uniqueness.

We call *identification range* of $L$ the set

$$ \mathcal{I}_L := \{(s, p) \in [-1, 1] \times (0, \infty) : p_-(s, L) < p < p_+(s, L)\}. $$

**Proposition 7.4 (Completion of $H_L^{s,p}$).** Let $-1 \le s \le 1$ and $0 < p < \infty$.
For $\beta = -1/2$, it also fails. An alternative of the equivalence (7.2) can be achieved with the Kenig-Pipher space $X^p$ as

$$ \left\{ \begin{array}{ll} \|v\|_{X^p} \approx \|\nabla v\|_{T_0^p} \approx \|v_0\|_{L^p} & \text{if } p_-(L) = p_-(0, L) < p < \infty \\ \|v\|_{X^\infty} \approx \|v_0\|_{L^\infty}, & \text{if } p = \infty \\ \|\nabla v\|_{T_0^\infty} \approx \|v_0\|_{\text{BMO}} & \text{if } p = \infty \end{array} \right. . $$

See [AMP19, Theorem 7.6, Corollary 5.5 & 5.10] and [Zat20, Theorem 7.6].

**7.2. L-adapted Hardy-Sobolev spaces.** Our main tool is L-adapted Hardy-Sobolev spaces. We give a brief review, and the reader can refer to [AA18, Chapter 4] and [AE23, §8.2] for details. See also [HvNVW17, Chapter 10] for definitions and basic facts of sectorial operators and $H^\infty$-calculus.

Denote by $S_\mu^+$ the set $\{z \in \mathbb{C} \setminus \{0\} : |\arg(z)| < \mu\}$ for $0 < \mu < \pi$. The class $\Psi_\infty^\infty(S_\mu^+)$ consists of holomorphic functions $\psi : S_\mu^+ \to \mathbb{C}$ fulfilling that for any $\sigma \in \mathbb{R}$, there exists a constant $C > 0$ so that

$$ |\psi(z)| \le C|z|^\sigma. $$

The class $H^\infty(S_\mu^+)$ consists of bounded holomorphic functions on $S_\mu^+$.

It is well-known that $L$ is a sectorial operator of angle $\omega$ for some $\omega \in [0, \pi/2)$ and has bounded $H^\infty(S_\mu^+)$-calculus for any $\mu \in (\omega, \pi)$. For any $\psi \in H^\infty(S_\mu^+)$, define the operator $Q_{\psi,L} : L^2 \to L^\infty((0, \infty); L^2)$ by

$$ Q_{\psi,L}(f)(t) := \psi(tL)f. $$

Note that for $\psi(z) = e^{-z}$, $Q_{\psi,L}$ identifies with $\mathcal{E}_L$.

**Definition 7.3 (L-adapted Hardy-Sobolev spaces).** Let $s \in \mathbb{R}$, $0 < p < \infty$, and $\omega < \mu < \pi$. Let $\psi$ be a non-zero function in $H^\infty(S_\mu^+)$. The L-adapted Hardy-Sobolev space $H_{L,\psi}^{s,p}$ consists of $f \in L^2$ so that $Q_{\psi,L}(f)$ lies in $T_{(s+1)/2}^p \cap L_{1/2}^2(\mathbb{R}_+^{1+n})$, equipped with the (quasi-)norm

$$ \|f\|_{H_{L,\psi}^{s,p}} := \|Q_{\psi,L}(f)\|_{T_{(s+1)/2}^p}. $$

A priori, this space depends on the choice of $\psi$. For each $(s, p)$, there is a subclass of bounded holomorphic functions, called *admissible functions for $H_{L,\psi}^{s,p}$*, for which the spaces $H_{L,\psi}^{s,p}$ agree as sets with mutually equivalent (quasi-)norms. We denote that space by $H_L^{s,p}$. All non-zero $\psi \in \Psi_\infty^\infty(S_\mu^+)$ are admissible functions for $H_L^{s,p}$. See [AE23, §8.2] for some explicit classes of admissible functions for $H_L^{s,p}$. For example, $\psi(z) = e^{-z}$ is admissible for $p \le 2$ and $s < 0$.

The spaces $H_L^{s,p}$ are not complete for the defining (quasi-)norms. Aside from abstract completion, we are interested in realizing the completion as a subspace of tempered distributions. The next result provides a range of $(s, p)$, called the identification range, within which a completion in fact equals to $\dot{H}^{s,p}$. When $0 \le s \le 1$, sharpness of this range is shown in [AE23, §19.1]. We describe the range for $-1 \le s \le 0$, and also exhibit a possible extra range where a completion exists in $\dot{H}^{s,p}$ without knowing equality. This fact will be useful when we come to uniqueness.

We call *identification range* of $L$ the set

$$ \mathcal{I}_L := \{(s, p) \in [-1, 1] \times (0, \infty) : p_-(s, L) < p < p_+(s, L)\}. $$

**Proposition 7.4 (Completion of $H_L^{s,p}$).** Let $-1 \le s \le 1$ and $0 < p < \infty$.
(i) (Identification) If $(s, p) \in \mathcal{I}_L$, then $\mathbb{H}_L^{s,p}$ agrees with $\dot{H}^{s,p} \cap L^2$ with equiv-alent (quasi-)norms

$$ \|f\|_{\mathbb{H}_L^{s,p}} \approx \|f\|_{\dot{H}^{s,p}}. $$

In particular, $\dot{H}^{s,p}$ is a completion of $\mathbb{H}_L^{s,p}$.

(ii) (Extra embedding) Suppose $p_-(L) < 1$. Let $-1 < s < 0$ and $1 < p < 2$. Then $\mathbb{H}_L^{s,p} \subset \dot{H}^{s,p}$ with

$$ \|f\|_{\dot{H}^{s,p}} \lesssim \|f\|_{\mathbb{H}_L^{s,p}}. $$

If, moreover,

$$ (7.5) \qquad s > -n \left( \frac{1}{p_-(L)} - 1 \right) - \frac{q_+(L^*)}{p'} \left( 1 - n \left( \frac{1}{p_-(L)} - 1 \right) \right). $$

then the closure of $\mathbb{H}_L^{s,p}$ in $\dot{H}^{s,p}$ with respect to the norm $\| \cdot \|_{\mathbb{H}_L^{s,p}}$ is a completion of $\mathbb{H}_L^{s,p}$.

The proof is deferred to Appendix A. Of course, the extra embedding range is only interesting when $(s, p) \notin \mathcal{I}_L$. In fact, setting $s = 2\beta + 1$, when $p_-(L) < 1$ and $\beta < \beta(L)$, the condition (7.5) exactly means $p > \tilde{p}_L(\beta)$, so we observe from Figure 1 that such range can be non-empty.

**Lemma 7.5** (Semigroup extension of $\mathcal{E}_L$). Let $s \in \mathbb{R}$ and $0 < p \le \infty$. Then $(e^{-tL})$ is a continuous bounded semigroup on $\mathbb{H}_L^{s,p}$. In particular, for any $f \in \mathbb{H}_L^{s,p}$, $\mathcal{E}_L(f)$ lies in $C^\infty((0, \infty); \mathbb{H}_L^{s,p}) \cap C_0([0, \infty); \mathbb{H}_L^{s,p})$ with

$$ \sup_{t \ge 0} \|\mathcal{E}_L(f)(t)\|_{\mathbb{H}_L^{s,p}} = \sup_{t \ge 0} \|e^{-tL}f\|_{\mathbb{H}_L^{s,p}} \approx \|f\|_{\mathbb{H}_L^{s,p}}. $$

Restricting to the identification range $(s, p) \in \mathcal{I}_L$, $(e^{-tL})$ extends to a continuous bounded semigroup on $\dot{H}^{s,p}$, and for any $f \in \dot{H}^{s,p}$,

$$ \mathcal{E}_L(f) \in C^\infty((0, \infty); \dot{H}^{s,p}) \cap C_0([0, \infty); \dot{H}^{s,p}) $$

with

$$ \sup_{t>0} \|\mathcal{E}_L(f)(t)\|_{\dot{H}^{s,p}} \approx \|f\|_{\dot{H}^{s,p}}. $$

*Proof.* The first point is a general result. See again [AE23, §8.2] applied to $T = L$ and in particular, [AE23, Proposition 8.3] replacing $\sqrt{T}$ by $T$. The second point follows from the first by Proposition 7.4 (i). □

7.3. **Proof of Theorem 7.1.** We use the formula of $\mathcal{E}_L$ shown in Corollary 4.5 (ii) to construct its extension. Define the operator $\mathcal{T}_{1/2}$ from $L^2$ to $L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$ by

$$ (7.6) \qquad \mathcal{T}_{1/2}(f) = -\mathcal{R}_{1/2}^L((A - \mathbb{I})\nabla \mathcal{E}_{-\Delta}(f)). $$

We first extend $\mathcal{T}_{1/2}$ to $\dot{H}^{2\beta+1,p}$.

**Lemma 7.6** (Extension of $\mathcal{T}_{1/2}$). Let $-1 < \beta < 0$ and $\tilde{p}_L(\beta) < p \le \infty$. Then for any $f \in \dot{H}^{2\beta+1,p} \cap L^2$,

$$ \|\mathcal{T}_{1/2}(f)\|_{\mathcal{T}_{\beta+1}^p} + \|\nabla \mathcal{T}_{1/2}(f)\|_{\mathcal{T}_{\beta+1/2}^p} \lesssim \|f\|_{\dot{H}^{2\beta+1,p}}. $$

<footnote n="4">⁴For $p = \infty$, $\mathbb{H}_L^{s,\infty}$ is contained in the dual of $\mathbb{H}_L^{-s,1}$ by $L^2(\mathbb{R}^n)$-inner product. Here, it is equipped with the induced weak*-topology.</footnote>
(i) (Identification) If $(s,p) \in \mathcal{I}_L$, then $\mathbb{H}_L^{s,p}$ agrees with $\dot{H}^{s,p} \cap L^2$ with equiv-alent (quasi-)norms

$$ \|f\|_{\mathbb{H}_L^{s,p}} \approx \|f\|_{\dot{H}^{s,p}}. $$

In particular, $\dot{H}^{s,p}$ is a completion of $\mathbb{H}_L^{s,p}$.

(ii) (Extra embedding) Suppose $p_-(L) < 1$. Let $-1 < s < 0$ and $1 < p < 2$. Then $\mathbb{H}_L^{s,p} \subset \dot{H}^{s,p}$ with

$$ \|f\|_{\dot{H}^{s,p}} \lesssim \|f\|_{\mathbb{H}_L^{s,p}}. $$

If, moreover,

$$ (7.5) \qquad s > -n \left( \frac{1}{p_-(L)} - 1 \right) - \frac{q_+(L^*)}{p'} \left( 1 - n \left( \frac{1}{p_-(L)} - 1 \right) \right). $$

then the closure of $\mathbb{H}_L^{s,p}$ in $\dot{H}^{s,p}$ with respect to the norm $\| \cdot \|_{\mathbb{H}_L^{s,p}}$ is a completion of $\mathbb{H}_L^{s,p}$.

The proof is deferred to Appendix A. Of course, the extra embedding range is only interesting when $(s,p) \notin \mathcal{I}_L$. In fact, setting $s = 2\beta + 1$, when $p_-(L) < 1$ and $\beta < \beta(L)$, the condition (7.5) exactly means $p > \tilde{p}_L(\beta)$, so we observe from Figure 1 that such range can be non-empty.

**Lemma 7.5** (Semigroup extension of $\mathcal{E}_L$). Let $s \in \mathbb{R}$ and $0 < p \le \infty$. Then $(e^{-tL})$ is a continuous bounded semigroup on $\mathbb{H}_L^{s,p}$. In particular, for any $f \in \mathbb{H}_L^{s,p}$, $\mathcal{E}_L(f)$ lies in $C^\infty((0, \infty); \mathbb{H}_L^{s,p}) \cap C_0([0, \infty); \mathbb{H}_L^{s,p})$ with

$$ \sup_{t \ge 0} \|\mathcal{E}_L(f)(t)\|_{\mathbb{H}_L^{s,p}} = \sup_{t \ge 0} \|e^{-tL}f\|_{\mathbb{H}_L^{s,p}} \approx \|f\|_{\mathbb{H}_L^{s,p}}. $$

Restricting to the identification range $(s,p) \in \mathcal{I}_L$, $(e^{-tL})$ extends to a continuous bounded semigroup on $\dot{H}^{s,p}$, and for any $f \in \dot{H}^{s,p}$,

$$ \mathcal{E}_L(f) \in C^\infty((0, \infty); \dot{H}^{s,p}) \cap C_0([0, \infty); \dot{H}^{s,p}) $$

with

$$ \sup_{t>0} \|\mathcal{E}_L(f)(t)\|_{\dot{H}^{s,p}} \approx \|f\|_{\dot{H}^{s,p}}. $$

*Proof.* The first point is a general result. See again [AE23, §8.2] applied to $T = L$ and in particular, [AE23, Proposition 8.3] replacing $\sqrt{T}$ by $T$. The second point follows from the first by Proposition 7.4 (i). $\square$

7.3. **Proof of Theorem 7.1.** We use the formula of $\mathcal{E}_L$ shown in Corollary 4.5 (ii) to construct its extension. Define the operator $\mathcal{T}_{1/2}$ from $L^2$ to $L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$ by

$$ (7.6) \qquad \mathcal{T}_{1/2}(f) = -\mathcal{R}_{1/2}^L((A - \mathbb{I})\nabla \mathcal{E}_{-\Delta}(f)). $$

We first extend $\mathcal{T}_{1/2}$ to $\dot{H}^{2\beta+1,p}$.

**Lemma 7.6** (Extension of $\mathcal{T}_{1/2}$). Let $-1 < \beta < 0$ and $\tilde{p}_L(\beta) < p \le \infty$. Then for any $f \in \dot{H}^{2\beta+1,p} \cap L^2$,

$$ \|\mathcal{T}_{1/2}(f)\|_{\mathcal{T}_{\beta+1}^p} + \|\nabla \mathcal{T}_{1/2}(f)\|_{\mathcal{T}_{\beta+1/2}^p} \lesssim \|f\|_{\dot{H}^{2\beta+1,p}}. $$

<footnote n="4">⁴For $p = \infty$, $\mathbb{H}_L^{s,\infty}$ is contained in the dual of $\mathbb{H}_L^{-s,1}$ by $L^2(\mathbb{R}^n)$-inner product. Here, it is equipped with the induced weak*-topology.</footnote>
Hence, $\mathcal{T}_{1/2}$ extends by density (or weak*-density if $p = \infty$) to a bounded operator from $\dot{H}^{2\beta+1,p}$ to $T_{\beta+1}^p$ with the above estimate. We use the same notation for the extension.

**Proof.** We just need to show the inequality. To this end, fix $f \in \dot{H}^{2\beta+1,p} \cap L^2$. The formula (7.6), together with estimates from Theorem 5.1 (a) applied to $\mathcal{R}_{1/2}^L$, yields that

$$ \|\mathcal{T}_{1/2}(f)\|_{T_{\beta+1}^p} + \|\nabla \mathcal{T}_{1/2}(f)\|_{T_{\beta+1/2}^p} \lesssim \|\nabla \mathcal{E}_{-\Delta}(f)\|_{T_{\beta+1/2}^p} \lesssim \|f\|_{\dot{H}^{2\beta+1,p}}. $$

The last inequality follows from Corollary 3.4 (i). □

<figure><img src="image_5.png" /></figure>

**Proof of Theorem 7.1.** Let $-1 < \beta < 0$ and $\tilde{p}_L(\beta) < p \le \infty$. We define the extension $\mathcal{E}_L$ by

$$ (7.7) \qquad \mathcal{E}_L(v_0) := \mathcal{E}_{-\Delta}(v_0) + \mathcal{T}_{1/2}(v_0), \quad \forall v_0 \in \dot{H}^{2\beta+1,p}, $$

where $\mathcal{T}_{1/2}$ is given by Lemma 7.6.

Let us verify the properties for $v := \mathcal{E}_L(v_0)$. As $\dot{H}^{2\beta+1,p}$ is contained in $\mathcal{S}'$, we get $\mathcal{E}_{-\Delta}(v_0) \in C^\infty(\mathbb{R}_+^{1+n})$. Also, Lemma 7.6 says $\mathcal{T}_{1/2}(v_0) \in T_{\beta+1}^p$ and $\nabla \mathcal{T}_{1/2}(v_0) \in T_{\beta+1/2}^p$. In particular, it implies $\mathcal{T}_{1/2}(v_0) \in L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$, and so does $v$.

First consider (a). The inequality "$\lesssim$" in (7.1) follows directly from Corollary 3.4 (i) and Lemma 7.6 as

$$ \|\nabla v\|_{T_{\beta+1/2}^p} \le \|\nabla \mathcal{E}_{-\Delta}(v_0)\|_{T_{\beta+1/2}^p} + \|\nabla \mathcal{T}_{1/2}(v_0)\|_{T_{\beta+1/2}^p} \lesssim \|v_0\|_{\dot{H}^{2\beta+1,p}}. $$

We postpone the proof of the converse inequality "$\gtrsim$" in (7.1) and (7.2) to the end of the proof.

Next, the formulae in (b) hold when $v_0 \in \dot{H}^{2\beta+1,p} \cap L^2$ by Corollary 4.5, and the above bounds of the terms involved allow us to use density (or weak*-density) to extend them to all $v_0 \in \dot{H}^{s,p}$, valued in $L_{\text{loc}}^2(\mathbb{R}_+^{n+1})$.

To prove (c), we use the formula (7.4) and prove that each term has the desired regularity. As $\dot{H}^{2\beta+1,p}$ is contained in $\mathcal{S}'$, we have $\mathcal{E}_{-\Delta}(v_0) \in C([0, \infty); \mathcal{S}')$. Moreover, as $\tilde{p}_{-\Delta}(\beta) \le \tilde{p}_L(\beta) < p$ and $\nabla v \in T_{\beta+1/2}^p$, we deduce from Theorem 5.1 (d) that $\mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla v) \in C([0, \infty); \mathcal{S}')$ as wanted.

Property (d) is a direct consequence of uniqueness of extensions by density: the extension defined by (7.7) agrees with the one given by Lemma 7.5, which has the desired regularity properties.

For (e), note that $\mathcal{E}_{-\Delta}(v_0)$ is a global weak solution to the heat equation, and Theorem 5.1 yields that $w := \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla v)$ is a global weak solution to $\partial_t w - \Delta w = \text{div}((A - \mathbb{I})\nabla v)$. Therefore, we infer from formula (7.4) that $v$ is a global weak solution to $\partial_t v - \Delta v = \text{div}((A - \mathbb{I})\nabla v)$, that is $\partial_t v - \text{div}(A\nabla v) = 0$. Meanwhile, (c) shows that $v(t)$ converges to $v_0$ in $\mathcal{S}'$ as $t \to 0$. We hence conclude that $v$ is a global weak solution to (HC) with initial data $v_0$.

Let us now prove the rest of (a). We begin with the inequality "$\gtrsim$" in (7.1). We just proved in (e) that $v$ is a global weak solution to the equation $\partial_t v - \text{div}(A\nabla v) = 0$ with $\nabla v \in T_{\beta+1/2}^p$, and that $v(t)$ converges to $v_0$ in $\mathcal{S}'$ as $t \to 0$, so $v_0$ agrees with the trace $g$ of $v$ given by Proposition 6.2 (ii) in $\dot{H}^{2\beta+1,p}$. In particular, we get $\|v_0\|_{\dot{H}^{2\beta+1,p}} \lesssim \|\nabla v\|_{T_{\beta+1/2}^p}$ as desired. This proves (7.1).

Next, to see (7.2), we only need to show that $\|v\|_{T_{\beta+1}^p} \lesssim \|v_0\|_{\dot{H}^{2\beta+1,p}}$, thanks to (7.1) and Corollary 4.3. This inequality is a direct consequence of (7.4), Theorem
Hence, $\mathcal{T}_{1/2}$ extends by density (or weak*-density if $p = \infty$) to a bounded operator from $\dot{H}^{2\beta+1,p}$ to $T_{\beta+1}^p$ with the above estimate. We use the same notation for the extension.

*Proof.* We just need to show the inequality. To this end, fix $f \in \dot{H}^{2\beta+1,p} \cap L^2$. The formula (7.6), together with estimates from Theorem 5.1 (a) applied to $\mathcal{R}_{1/2}^L$, yields that

$$ \|\mathcal{T}_{1/2}(f)\|_{T_{\beta+1}^p} + \|\nabla \mathcal{T}_{1/2}(f)\|_{T_{\beta+1/2}^p} \lesssim \|\nabla \mathcal{E}_{-\Delta}(f)\|_{T_{\beta+1/2}^p} \lesssim \|f\|_{\dot{H}^{2\beta+1,p}}. $$

The last inequality follows from Corollary 3.4 (i). □

<figure><img src="image_5.png" /></figure>

*Proof of Theorem 7.1.* Let $-1 < \beta < 0$ and $\tilde{p}_L(\beta) < p \le \infty$. We define the extension $\mathcal{E}_L$ by

$$ (7.7) \qquad \mathcal{E}_L(v_0) := \mathcal{E}_{-\Delta}(v_0) + \mathcal{T}_{1/2}(v_0), \quad \forall v_0 \in \dot{H}^{2\beta+1,p}, $$

where $\mathcal{T}_{1/2}$ is given by Lemma 7.6.

Let us verify the properties for $v := \mathcal{E}_L(v_0)$. As $\dot{H}^{2\beta+1,p}$ is contained in $\mathcal{S}'$, we get $\mathcal{E}_{-\Delta}(v_0) \in C^\infty(\mathbb{R}_+^{1+n})$. Also, Lemma 7.6 says $\mathcal{T}_{1/2}(v_0) \in T_{\beta+1}^p$ and $\nabla \mathcal{T}_{1/2}(v_0) \in T_{\beta+1/2}^p$. In particular, it implies $\mathcal{T}_{1/2}(v_0) \in L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$, and so does $v$.

First consider (a). The inequality "$\lesssim$" in (7.1) follows directly from Corollary 3.4 (i) and Lemma 7.6 as

$$ \|\nabla v\|_{T_{\beta+1/2}^p} \le \|\nabla \mathcal{E}_{-\Delta}(v_0)\|_{T_{\beta+1/2}^p} + \|\nabla \mathcal{T}_{1/2}(v_0)\|_{T_{\beta+1/2}^p} \lesssim \|v_0\|_{\dot{H}^{2\beta+1,p}}. $$

We postpone the proof of the converse inequality "$\gtrsim$" in (7.1) and (7.2) to the end of the proof.

Next, the formulae in (b) hold when $v_0 \in \dot{H}^{2\beta+1,p} \cap L^2$ by Corollary 4.5, and the above bounds of the terms involved allow us to use density (or weak*-density) to extend them to all $v_0 \in \dot{H}^{s,p}$, valued in $L_{\text{loc}}^2(\mathbb{R}_+^{n+1})$.

To prove (c), we use the formula (7.4) and prove that each term has the desired regularity. As $\dot{H}^{2\beta+1,p}$ is contained in $\mathcal{S}'$, we have $\mathcal{E}_{-\Delta}(v_0) \in C([0, \infty); \mathcal{S}')$. Moreover, as $\tilde{p}_{-\Delta}(\beta) \le \tilde{p}_L(\beta) < p$ and $\nabla v \in T_{\beta+1/2}^p$, we deduce from Theorem 5.1 (d) that $\mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla v) \in C([0, \infty); \mathcal{S}')$ as wanted.

Property (d) is a direct consequence of uniqueness of extensions by density: the extension defined by (7.7) agrees with the one given by Lemma 7.5, which has the desired regularity properties.

For (e), note that $\mathcal{E}_{-\Delta}(v_0)$ is a global weak solution to the heat equation, and Theorem 5.1 yields that $w := \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla v)$ is a global weak solution to $\partial_t w - \Delta w = \text{div}((A - \mathbb{I})\nabla v)$. Therefore, we infer from formula (7.4) that $v$ is a global weak solution to $\partial_t v - \Delta v = \text{div}((A - \mathbb{I})\nabla v)$, that is $\partial_t v - \text{div}(A\nabla v) = 0$. Meanwhile, (c) shows that $v(t)$ converges to $v_0$ in $\mathcal{S}'$ as $t \to 0$. We hence conclude that $v$ is a global weak solution to (HC) with initial data $v_0$.

Let us now prove the rest of (a). We begin with the inequality "$\gtrsim$" in (7.1). We just proved in (e) that $v$ is a global weak solution to the equation $\partial_t v - \text{div}(A\nabla v) = 0$ with $\nabla v \in T_{\beta+1/2}^p$, and that $v(t)$ converges to $v_0$ in $\mathcal{S}'$ as $t \to 0$, so $v_0$ agrees with the trace $g$ of $v$ given by Proposition 6.2 (ii) in $\dot{H}^{2\beta+1,p}$. In particular, we get $\|v_0\|_{\dot{H}^{2\beta+1,p}} \lesssim \|\nabla v\|_{T_{\beta+1/2}^p}$ as desired. This proves (7.1).

Next, to see (7.2), we only need to show that $\|v\|_{T_{\beta+1}^p} \lesssim \|v_0\|_{\dot{H}^{2\beta+1,p}}$, thanks to (7.1) and Corollary 4.3. This inequality is a direct consequence of (7.4), Theorem
3.2 (i), and (7.1) (for the Laplacian) as

$$
\begin{align*}
\|v\|_{T_{\beta+1}^p} &\lesssim \|\mathcal{E}_{-\Delta}(v_0)\|_{T_{\beta+1}^p} + \|\mathcal{R}_{1/2}^{-\Delta}((A-I)\nabla v)\|_{T_{\beta+1}^p} \\
&\lesssim \|\mathcal{E}_{-\Delta}(v_0)\|_{T_{\beta+1}^p} + \|\nabla v\|_{T_{\beta+1/2}^p} \lesssim \|v_0\|_{\dot{H}^{2\beta+1,p}}.
\end{align*}
$$

This shows (a) and the proof is complete.

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

8. UNIQUENESS AND REPRESENTATION

In this section, we prove uniqueness of weak solutions to the homogeneous
Cauchy problem and Lions' equation as asserted in Theorems 1.3 and 1.6. We
also provide the proof of Theorem 1.4.

8.1. Uniqueness. The main theorem on uniqueness is as follows.

**Theorem 8.1** (Uniqueness). Let $\beta > -1$ and $\tilde{p}_L(\beta) < p \le \infty$. Let $u$ be a global weak solution to the initial value problem

$$
\left\{
\begin{array}{l}
\partial_t u - \operatorname{div}(A \nabla u) = 0, \\
u(0) = 0
\end{array}
\right.
,
$$

with $\nabla u \in T_{\beta+1/2}^p$. Then $u=0$.

Remark 8.2. Let us mention that [AH23b, Theorem 4.3] proves uniqueness for $\beta > -1/2$ and $p_L^\beta(\beta) < p \le \infty$ (where $p_L^\beta(\beta)$ is defined in (1.10) and equals $\tilde{p}_L(\beta)$ when $p_-(L) \ge 1$), requiring that $u$ lies in $T_{\beta+1}^p$ (but with no initial data needed). However, recall that Corollary 4.3 asserts the inequality $\|\nabla v\|_{T_{\beta+1/2}^p} \lesssim \|v\|_{T_{\beta+1}^p}$ holds for any global weak solution $v$ to the equation $\partial_t v - \operatorname{div}(A \nabla v) = 0$. Therefore, we improve the class of uniqueness in this range of $\beta$ with more values of $p$.

Proof of Theorem 8.1. Let $u$ be a global weak solution to $\partial_t u - \operatorname{div}(A\nabla u) = 0$ with $\nabla u \in T_{\beta+1/2}^p$ and null initial data. Thanks to Lemma 6.1, we may assume $p > \max\{\tilde{p}_L(\beta), 1\}$ in what follows. Let us begin by some important common facts. First, as $p > \tilde{p}_L(\beta) \ge \frac{n}{n+2\beta+2}$ and $u(0) = 0$, Proposition 6.2 yields

$$
(8.1) \qquad u = \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u).
$$

Hence, Theorem 5.1 for $\mathcal{R}_{1/2}^{-\Delta}$ applies: $u \in T_{\beta+1}^p$ and $u(s)$ tends to 0 as $s \to 0$ in $\mathcal{S}'$ and in various finer topology depending on $\beta$ and $p$.

Second, Lemma 3.7 gives local $L^2$-norm estimates of $u$ allowing us to invoke
[AMP19, Theorem 5.1] to get an interior representation of $u$ as follows: For $0 <$
$s < t < \infty$ and $h \in C_c^\infty(\mathbb{R}^n)$, it holds that

$$
(8.2) \quad \int_{\mathbb{R}^n} u(t, x) \bar{h}(x) dx = \int_{\mathbb{R}^n} u(s, x) \overline{e^{-(t-s)L^*} h}(x) dx.
$$

Now, fix $t > 0$ and $h \in C_c^\infty(\mathbb{R}^n)$. We wish to let $s \to 0$, yielding $u(t) = 0$ in $\mathcal{S}'(\mathbb{R}^n)$. To take advantage of the different modes of convergence established in Theorem 5.1 (d), we split the argument in two cases.

Case 1: $\beta > -1/2$. As $p > \max\{\tilde{p}_L(\beta), 1\} \ge p_L^\beta(\beta)$ and $u \in T_{\beta+1}^p$, we can use
[AH23b, Theorem 4.3] to conclude $u=0$ (the proof there uses (8.2) as well).
3.2 (i), and (7.1) (for the Laplacian) as

$$
\begin{align*}
\|v\|_{T_{\beta+1}^p} &\lesssim \|\mathcal{E}_{-\Delta}(v_0)\|_{T_{\beta+1}^p} + \|\mathcal{R}_{1/2}^{-\Delta}((A-I)\nabla v)\|_{T_{\beta+1}^p} \\
&\lesssim \|\mathcal{E}_{-\Delta}(v_0)\|_{T_{\beta+1}^p} + \|\nabla v\|_{T_{\beta+1/2}^p} \lesssim \|v_0\|_{\dot{H}^{2\beta+1,p}}.
\end{align*}
$$

This shows (a) and the proof is complete.

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

8. UNIQUENESS AND REPRESENTATION

In this section, we prove uniqueness of weak solutions to the homogeneous
Cauchy problem and Lions' equation as asserted in Theorems 1.3 and 1.6. We
also provide the proof of Theorem 1.4.

8.1. **Uniqueness.** The main theorem on uniqueness is as follows.

**Theorem 8.1** (Uniqueness). Let $\beta > -1$ and $\tilde{p}_L(\beta) < p \le \infty$. Let $u$ be a global weak solution to the initial value problem

$$
\left\{
\begin{array}{l}
\partial_t u - \operatorname{div}(A \nabla u) = 0, \\
u(0) = 0
\end{array}
\right.
,
$$

with $\nabla u \in T_{\beta+1/2}^p$. Then $u=0$.

Remark 8.2. Let us mention that [AH23b, Theorem 4.3] proves uniqueness for $\beta > -1/2$ and $p_L^\beta(\beta) < p \le \infty$ (where $p_L^\beta(\beta)$ is defined in (1.10) and equals $\tilde{p}_L(\beta)$ when $p_-(L) \ge 1$), requiring that $u$ lies in $T_{\beta+1}^p$ (but with no initial data needed). However, recall that Corollary 4.3 asserts the inequality $\|\nabla v\|_{T_{\beta+1/2}^p} \lesssim \|v\|_{T_{\beta+1}^p}$ holds for any global weak solution $v$ to the equation $\partial_t v - \operatorname{div}(A \nabla v) = 0$. Therefore, we improve the class of uniqueness in this range of $\beta$ with more values of $p$.

Proof of Theorem 8.1. Let $u$ be a global weak solution to $\partial_t u - \operatorname{div}(A\nabla u) = 0$ with $\nabla u \in T_{\beta+1/2}^p$ and null initial data. Thanks to Lemma 6.1, we may assume $p > \max\{\tilde{p}_L(\beta), 1\}$ in what follows. Let us begin by some important common facts. First, as $p > \tilde{p}_L(\beta) \ge \frac{n}{n+2\beta+2}$ and $u(0) = 0$, Proposition 6.2 yields

$$
(8.1) \qquad u = \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u).
$$

Hence, Theorem 5.1 for $\mathcal{R}_{1/2}^{-\Delta}$ applies: $u \in T_{\beta+1}^p$ and $u(s)$ tends to 0 as $s \to 0$ in $\mathcal{S}'$ and in various finer topology depending on $\beta$ and $p$.

Second, Lemma 3.7 gives local $L^2$-norm estimates of $u$ allowing us to invoke
[AMP19, Theorem 5.1] to get an interior representation of $u$ as follows: For $0 <$
$s < t < \infty$ and $h \in C_c^\infty(\mathbb{R}^n)$, it holds that

$$
(8.2) \quad \int_{\mathbb{R}^n} u(t, x) \bar{h}(x) dx = \int_{\mathbb{R}^n} u(s, x) \overline{e^{-(t-s)L^*} h(x)} dx.
$$

Now, fix $t > 0$ and $h \in C_c^\infty(\mathbb{R}^n)$. We wish to let $s \to 0$, yielding $u(t) = 0$ in $\mathcal{S}'(\mathbb{R}^n)$. To take advantage of the different modes of convergence established in Theorem 5.1 (d), we split the argument in two cases.

Case 1: $\beta > -1/2$. As $p > \max\{\tilde{p}_L(\beta), 1\} \ge p_L^\beta(\beta)$ and $u \in T_{\beta+1}^p$, we can use
[AH23b, Theorem 4.3] to conclude $u=0$ (the proof there uses (8.2) as well).
Case 2: $-1 < \beta \le -1/2$. We begin with $p \ge 2$ and then $\max\{\tilde{p}_L(\beta), 1\} < p \le 2$, which is further split into two sub-cases.

Case 2(a): $2 \le p \le \infty$. Pick $\delta > 0$. We use the slice spaces $E_{\delta}^{-1,p}$ and $E_{\delta}^{1,p'}$ introduced in Section 2.2. Recall that the $L^2(\mathbb{R}^n)$-inner product realizes $E_{\delta}^{-1,p}$ as the dual of $E_{\delta}^{1,p'}$. The formula (8.1) and Theorem 5.1 (d) for $\mathcal{R}_{1/2}^{-\Delta}$ imply

$$ \lim_{s \to 0} u(s) = 0 \quad \text{in } E_{\delta}^{-1,p}. $$

Moreover, we claim that

$$ (8.3) \qquad \lim_{s \to 0} e^{-(t-s)L^*} h = e^{-tL^*} h \quad \text{in } E_{\delta}^{1,p'}. $$

Then taking limit of the right-hand integral in (8.2) yields $u(t) = 0$.

To prove (8.3), pick $\lambda \in (0,1)$ and $R > \delta > 0$ so that $\supp(h) \subset B := B(0, R)$ and for any $|x| > R$, $D(x) := \text{dist}(B(x, \delta), \text{supp}(h)) \ge \lambda|x|$. If $|x| \le R$, we have

$$ \|\mathbb{1}_{B(x,\delta)} \nabla e^{-tL^*} h\|_2 \le \|\nabla e^{-tL^*} h\|_2 \lesssim t^{-1/2} \|h\|_2. $$

If $|x| > R$, the exponential $L^2 - L^2$ off-diagonal estimates of $(t^{1/2}\nabla e^{-tL^*})$ yield there exists $\gamma > 0$ so that

$$ \|\mathbb{1}_{B(x,\delta)} \nabla e^{-tL^*} h\|_2 \lesssim t^{-1/2} e^{-\frac{\gamma D(x)}{t}} \|h\|_2 \lesssim t^{-1/2} e^{-\frac{\gamma \lambda^2 |x|^2}{t}} \|h\|_2. $$

We hence get

$$ \sup_{0 \le s \le t/2} \|\mathbb{1}_{B(x,\delta)} \nabla e^{-(t-s)L^*} h\|_2 \lesssim t^{-1/2} (\mathbb{1}_B(x) + \mathbb{1}_{B^c}(x) e^{-\frac{\gamma \lambda^2 |x|^2}{t}}) \|h\|_2. $$

Note that the function on the right-hand side lies in $L^{p'}$, so $(\nabla e^{-(t-s)L^*} h)_{0 \le s \le t/2}$ form a uniformly bounded set in $E_{\delta}^{p'}$. Moreover, Lebesgue's dominated convergence theorem and continuity of $(\nabla e^{-tL^*})$ on $L^2$ imply that $\nabla e^{-(t-s)L^*} h$ converges to $\nabla e^{-tL^*} h$ in $E_{\delta}^{p'}$ as $s \to 0$. Hence equivalently, $e^{-(t-s)L^*} h$ converges to $e^{-tL^*} h$ in $E_{\delta}^{1,p'}$ as $s \to 0$. This proves (8.3).

Case 2(b): $p_-(2\beta + 1, L) < p < 2$. Recall that we also assume $p > 1$, and we know that $\tilde{p}_{-\Delta}(\beta) \le p_-(2\beta + 1, L)$, so in particular, we have

$$ \max\{\tilde{p}_{-\Delta}(\beta), 1\} \le \max\{p_-(2\beta + 1, L), 1\} < p < 2. $$

Using the formula (8.1) and Theorem 5.1 (d) for $\mathcal{R}_{1/2}^{-\Delta}$, we get

$$ \lim_{s \to 0} u(s) = 0 \quad \text{in } \dot{H}^{2\beta+1,p}. $$

On the other hand, as $h \in \dot{H}^{-(2\beta+1),p'} \cap L^2$ and

$$ 2 < p' < (\max\{p_-(2\beta + 1, L), 1\})' = p_+(-(2\beta + 1), L^*), $$

we infer from Theorem 7.1 (d) that

$$ \lim_{s \to 0} e^{-(t-s)L^*} h = \lim_{s \to 0} \mathcal{E}_{L^*}(h)(t-s) = e^{-tL^*} h \quad \text{in } \dot{H}^{-(2\beta+1),p'}. $$

Realizing the right-hand integral of (8.2) in the sense of duality for $\dot{H}^{2\beta+1,p}$ and $\dot{H}^{-(2\beta+1),p'}$, we obtain that it tends to 0 as $s \to 0$.
Case 2: $-1 < \beta \le -1/2$. We begin with $p \ge 2$ and then $\max\{\tilde{p}_L(\beta), 1\} < p \le 2$, which is further split into two sub-cases.

Case 2(a): $2 \le p \le \infty$. Pick $\delta > 0$. We use the slice spaces $E_{\delta}^{-1,p}$ and $E_{\delta}^{1,p'}$ introduced in Section 2.2. Recall that the $L^2(\mathbb{R}^n)$-inner product realizes $E_{\delta}^{-1,p}$ as the dual of $E_{\delta}^{1,p'}$. The formula (8.1) and Theorem 5.1 (d) for $\mathcal{R}_{1/2}^{-\Delta}$ imply

$$ \lim_{s \to 0} u(s) = 0 \quad \text{in } E_{\delta}^{-1,p}. $$

Moreover, we claim that

$$ (8.3) \qquad \lim_{s \to 0} e^{-(t-s)L^*} h = e^{-tL^*} h \quad \text{in } E_{\delta}^{1,p'}. $$

Then taking limit of the right-hand integral in (8.2) yields $u(t) = 0$.

To prove (8.3), pick $\lambda \in (0,1)$ and $R > \delta > 0$ so that $\supp(h) \subset B := B(0, R)$ and for any $|x| > R$, $D(x) := \text{dist}(B(x, \delta), \text{supp}(h)) \ge \lambda|x|$. If $|x| \le R$, we have

$$ \|\mathbb{1}_{B(x,\delta)} \nabla e^{-tL^*} h\|_2 \le \|\nabla e^{-tL^*} h\|_2 \lesssim t^{-1/2} \|h\|_2. $$

If $|x| > R$, the exponential $L^2 - L^2$ off-diagonal estimates of $(t^{1/2}\nabla e^{-tL^*})$ yield there exists $\gamma > 0$ so that

$$ \|\mathbb{1}_{B(x,\delta)} \nabla e^{-tL^*} h\|_2 \lesssim t^{-1/2} e^{-\frac{\gamma D(x)}{t}} \|h\|_2 \lesssim t^{-1/2} e^{-\frac{\gamma \lambda^2 |x|^2}{t}} \|h\|_2. $$

We hence get

$$ \sup_{0 \le s \le t/2} \|\mathbb{1}_{B(x,\delta)} \nabla e^{-(t-s)L^*} h\|_2 \lesssim t^{-1/2} (\mathbb{1}_B(x) + \mathbb{1}_{B^c}(x) e^{-\frac{\gamma \lambda^2 |x|^2}{t}}) \|h\|_2. $$

Note that the function on the right-hand side lies in $L^{p'}$, so $(\nabla e^{-(t-s)L^*} h)_{0 \le s \le t/2}$ form a uniformly bounded set in $E_{\delta}^{p'}$. Moreover, Lebesgue's dominated convergence theorem and continuity of $(\nabla e^{-tL^*})$ on $L^2$ imply that $\nabla e^{-(t-s)L^*} h$ converges to $\nabla e^{-tL^*} h$ in $E_{\delta}^{p'}$ as $s \to 0$. Hence equivalently, $e^{-(t-s)L^*} h$ converges to $e^{-tL^*} h$ in $E_{\delta}^{1,p'}$ as $s \to 0$. This proves (8.3).

Case 2(b): $p_-(2\beta + 1, L) < p < 2$. Recall that we also assume $p > 1$, and we know that $\tilde{p}_{-\Delta}(\beta) \le p_-(2\beta + 1, L)$, so in particular, we have

$$ \max\{\tilde{p}_{-\Delta}(\beta), 1\} \le \max\{p_-(2\beta + 1, L), 1\} < p < 2. $$

Using the formula (8.1) and Theorem 5.1 (d) for $\mathcal{R}_{1/2}^{-\Delta}$, we get

$$ \lim_{s \to 0} u(s) = 0 \quad \text{in } \dot{H}^{2\beta+1,p}. $$

On the other hand, as $h \in \dot{H}^{-(2\beta+1),p'} \cap L^2$ and

$$ 2 < p' < (\max\{p_-(2\beta + 1, L), 1\})' = p_+(-(2\beta + 1), L^*), $$

we infer from Theorem 7.1 (d) that

$$ \lim_{s \to 0} e^{-(t-s)L^*} h = \lim_{s \to 0} \mathcal{E}_{L^*}(h)(t-s) = e^{-tL^*} h \quad \text{in } \dot{H}^{-(2\beta+1),p'}. $$

Realizing the right-hand integral of (8.2) in the sense of duality for $\dot{H}^{2\beta+1,p}$ and $\dot{H}^{-(2\beta+1),p'}$, we obtain that it tends to 0 as $s \to 0$.
Case 2(c): $\tilde{p}_L(\beta) < p \le p_-(2\beta + 1, L)$. This case only occurs when $p_-(L) < 1$. Theorem 5.1 (d) still gives us a limit $u(s) \to 0$ in $\dot{H}^{2\beta+1,p}$ but now we do not know whether $e^{-(t-s)L^*}h$ tends to $e^{-tL^*}h$ in the dual space. However, the extra embedding in Proposition 7.4 (ii) allows us to enhance the convergence of $u(s)$.

**Lemma 8.3** (Null limit of $u(s)$ in $H_L^{2\beta+1,p}$). In the range determined by this case, for all $t > 0$, $u(t)$ lies in $H_L^{2\beta+1,p}$ with a uniform bound. Moreover,

$$ \lim_{t \to 0} u(t) = 0 \quad \text{in } H_L^{2\beta+1,p}. $$

The proof is presented right after. Admitting this lemma, let us show $u = 0$. Since $u(s) \in H_L^{2\beta+1,p}$ for all $s > 0$, we have $u(s) \in L^2$ and we get from (8.2) that $u(t) = e^{-(t-s)L}u(s)$ for $t \ge s$. By Lemma 7.5, we get

$$ \sup_{t \ge s} \|u(t)\|_{H_L^{2\beta+1,p}} \lesssim \|u(s)\|_{H_L^{2\beta+1,p}}, $$

with a uniform bound in $s$. As the right-hand side tends to 0 as $s \to 0$, we conclude that $\sup_{t>0} \|u(t)\|_{H_L^{2\beta+1,p}} = 0$, so $u = 0$. $\square$

*Proof of Lemma 8.3.* The proof is divided into 4 steps.

**Step 1:** Regularity of $u$. We prove $u \in C^\infty((0, \infty); L^p \cap L^2)$. Since $1 < p < 2$, as proved in [AH23b, (4.19)], for a.e. $s > 0$, we have $u(s) \in L^p$ with

$$ \|u(s)\|_p \lesssim s^{\beta+1/2} \|u\|_{T_{\beta+1}^p}. $$

For such an $s$, $L^p$-boundedness of $(e^{-tL})$ yields $e^{-(t-s)L}u(s) \in L^p$ for all $t \ge s$, and it follows from (8.2) that it equals to $u(t)$. Thus we obtain $u(t) = e^{-(t-s)L}u(s)$ for any $t \ge s > 0$. Applying analyticity of $(e^{-tL})$ on $L^p$, we get $u \in C^\infty((0, \infty); L^p)$. As $e^{-tL}$ also maps $L^p$ to $L^2$ for any $t > 0$, we have $u \in C^\infty((0, \infty); L^2)$ as desired.

**Step 2:** A key estimate. For any $k \ge 1$, we show that

$$ (8.4) \qquad \|t^k(\partial_t^k u)(t)\|_{T_{\beta+1}^p} \lesssim_k \|u\|_{T_{\beta+1}^p}. $$

We prove it for $k = 1$, and iteration concludes the argument. Observe that by Step 1, we have that for any $t > 0$ and $\tau > 0$,

$$ (8.5) \qquad (\partial_t u)(t + \tau) = \partial_\tau (u(t + \tau)) = \partial_\tau (e^{-\tau L} u(t)) = -Le^{-\tau L} u(t). $$

In particular, pick $\tau = t$, we get $(\partial_t u)(2t) = -Le^{-tL}u(t)$, and hence,

$$ \begin{aligned} \|t \partial_t u\|_{T_{\beta+1}^p} &\approx \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x,(2s)^{1/2})} |s^{-(\beta+1)}s(\partial_t u)(2s,y)|^2 ds dy \right)^{p/2} dx \right)^{1/p} \\ &= \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x,(2s)^{1/2})} |s^{-(\beta+1)}(sLe^{-sL}u(s))(y)|^2 ds dy \right)^{p/2} dx \right)^{1/p}. \end{aligned} $$

Write $B := B(x, (2s)^{1/2})$, $C_0 := 2B$, and $C_j := 2^{j+1}B \setminus 2^j B$ for $j \ge 1$. The $L^2 - L^2$ off-diagonal estimates of $(sLe^{-sL})$ yield that

$$ \|\mathbb{1}_B s Le^{-sL} u(s)\|_2 \le \sum_{j \ge 0} \|\mathbb{1}_B s Le^{-sL} \mathbb{1}_{C_j} u(s)\|_2 \lesssim \sum_{j \ge 0} e^{-c2^{2j}} \|\mathbb{1}_{C_j} u(s)\|_2, $$
Case 2(c): $\tilde{p}_L(\beta) < p \le p_-(2\beta + 1, L)$. This case only occurs when $p_-(L) < 1$. Theorem 5.1 (d) still gives us a limit $u(s) \to 0$ in $\dot{H}^{2\beta+1,p}$ but now we do not know whether $e^{-(t-s)L^*}h$ tends to $e^{-tL^*}h$ in the dual space. However, the extra embedding in Proposition 7.4 (ii) allows us to enhance the convergence of $u(s)$.

**Lemma 8.3** (Null limit of $u(s)$ in $H_L^{2\beta+1,p}$). In the range determined by this case, for all $t > 0$, $u(t)$ lies in $H_L^{2\beta+1,p}$ with a uniform bound. Moreover,

$$ \lim_{t \to 0} u(t) = 0 \quad \text{in } H_L^{2\beta+1,p}. $$

The proof is presented right after. Admitting this lemma, let us show $u = 0$. Since $u(s) \in H_L^{2\beta+1,p}$ for all $s > 0$, we have $u(s) \in L^2$ and we get from (8.2) that $u(t) = e^{-(t-s)L}u(s)$ for $t \ge s$. By Lemma 7.5, we get

$$ \sup_{t \ge s} \|u(t)\|_{H_L^{2\beta+1,p}} \lesssim \|u(s)\|_{H_L^{2\beta+1,p}}, $$

with a uniform bound in $s$. As the right-hand side tends to 0 as $s \to 0$, we conclude that $\sup_{t>0} \|u(t)\|_{H_L^{2\beta+1,p}} = 0$, so $u = 0$. $\square$

*Proof of Lemma 8.3.* The proof is divided into 4 steps.

**Step 1:** Regularity of $u$. We prove $u \in C^\infty((0, \infty); L^p \cap L^2)$. Since $1 < p < 2$, as proved in [AH23b, (4.19)], for a.e. $s > 0$, we have $u(s) \in L^p$ with

$$ \|u(s)\|_p \lesssim s^{\beta+1/2} \|u\|_{T_{\beta+1}^p}. $$

For such an $s$, $L^p$-boundedness of $(e^{-tL})$ yields $e^{-(t-s)L}u(s) \in L^p$ for all $t \ge s$, and it follows from (8.2) that it equals to $u(t)$. Thus we obtain $u(t) = e^{-(t-s)L}u(s)$ for any $t \ge s > 0$. Applying analyticity of $(e^{-tL})$ on $L^p$, we get $u \in C^\infty((0, \infty); L^p)$. As $e^{-tL}$ also maps $L^p$ to $L^2$ for any $t > 0$, we have $u \in C^\infty((0, \infty); L^2)$ as desired.

**Step 2:** A key estimate. For any $k \ge 1$, we show that

$$ (8.4) \qquad \|t^k (\partial_t^k u)(t)\|_{T_{\beta+1}^p} \lesssim_k \|u\|_{T_{\beta+1}^p}. $$

We prove it for $k = 1$, and iteration concludes the argument. Observe that by Step 1, we have that for any $t > 0$ and $\tau > 0$,

$$ (8.5) \qquad (\partial_t u)(t + \tau) = \partial_\tau (u(t + \tau)) = \partial_\tau (e^{-\tau L} u(t)) = -Le^{-\tau L} u(t). $$

In particular, pick $\tau = t$, we get $(\partial_t u)(2t) = -Le^{-tL}u(t)$, and hence,

$$ \begin{aligned} \|t \partial_t u\|_{T_{\beta+1}^p} &\approx \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x,(2s)^{1/2})} |s^{-(\beta+1)} s (\partial_t u)(2s, y)|^2 ds dy \right)^{p/2} dx \right)^{1/p} \\ &= \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x,(2s)^{1/2})} |s^{-(\beta+1)} (sLe^{-sL}u(s))(y)|^2 ds dy \right)^{p/2} dx \right)^{1/p}. \end{aligned} $$

Write $B := B(x, (2s)^{1/2})$, $C_0 := 2B$, and $C_j := 2^{j+1}B \setminus 2^j B$ for $j \ge 1$. The $L^2 - L^2$ off-diagonal estimates of $(sLe^{-sL})$ yield that

$$ \|\mathbf{1}_B s Le^{-sL} u(s)\|_2 \le \sum_{j \ge 0} \|\mathbf{1}_B s Le^{-sL} \mathbf{1}_{C_j} u(s)\|_2 \lesssim \sum_{j \ge 0} e^{-c2^{2j}} \|\mathbf{1}_{C_j} u(s)\|_2, $$
where $c > 0$ is independent of $s$. Applying this on the above computation, we get

$$
\begin{align*}
\|t \partial_t u\|_{T_{\beta+1}^p} &\lesssim \sum_{j \ge 0} e^{-c2^{2j}} \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{C_j} s^{-\frac{n}{2}} |s^{-(\beta+1)} u(s, y)|^2 ds dy \right)^{p/2} dx \right)^{1/p} \\
&\lesssim \sum_{j \ge 0} 2^{j \frac{n}{2}} e^{-c2^{2j}} \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x, 2^{j+1}(2s)^{1/2})} |s^{-(\beta+1)} u(s, y)|^2 ds dy \right)^{p/2} dx \right)^{1/p} \\
&\lesssim \sum_{j \ge 0} 2^{j(\frac{n}{2} + \frac{n}{p})} e^{-c2^{2j}} \|u\|_{T_{\beta+1}^p} \lesssim \|u\|_{T_{\beta+1}^p}.
\end{align*}
$$

The factor $2^{j \frac{n}{p}}$ in the third inequality comes from change of angle for tent space norms, see [Aus11, Theorem 1.1].

Step 3: Boundedness of $u(t)$ in $H_L^{2\beta+1,p}$. Now we prove that for any $t > 0$, $u(t)$ belongs to $H_L^{2\beta+1,p}$ with

$$ (8.6) \qquad \sup_{t>0} \|u(t)\|_{H_L^{2\beta+1,p}} \lesssim \|u\|_{T_{\beta+1}^p}. $$

By Step 1, $u(t)$ lies in $L^2$ for any $t > 0$, so we just need to show the norm estimate. To this end, pick an integer $k \ge 1$ with $(k - \beta - 1)p - \frac{n}{2} > 0$. From [AE23, §8.2], $\psi(z) := z^k e^{-z}$ is a valid admissible function for $H_L^{2\beta+1,p}$. So we use (8.4), (8.5) in Step 2, and the identity $\partial_\tau^k(u(t+\tau)) = (\partial_t^k u)(t+\tau)$ to deduce that for any $t > 0$,

$$
\begin{align*}
\|u(t)\|_{H_L^{2\beta+1,p}} &\approx \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x, \tau^{1/2})} |\tau^{-(\beta+1)} ((\tau L)^k e^{-\tau L} u(t))(y)|^2 dy d\tau \right)^{p/2} dx \right)^{1/p} \\
&\approx \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x, \tau^{1/2})} |\tau^{-(\beta+1)} \tau^k \partial_\tau^k (u(t+\tau, y))|^2 dy d\tau \right)^{p/2} dx \right)^{1/p} \\
&\approx \left( \int_{\mathbb{R}^n} \left( \int_t^\infty \int_{B(x, (\sigma-t)^{1/2})} (\sigma-t)^{(k-\beta-1)p-\frac{n}{2}} |(\partial_t^k u)(\sigma, y)|^2 dy d\sigma \right)^{p/2} dx \right)^{1/p} \\
&\lesssim \|\sigma^k (\partial_t^k u)(\sigma)\|_{T_{\beta+1}^p} \lesssim \|u\|_{T_{\beta+1}^p}.
\end{align*}
$$

The first inequality with implicit constant independent of $t$ uses $(k-\beta-1)p - \frac{n}{2} > 0$. This proves (8.6).

Step 4: Limit at $t = 0$. We finish by showing that $u(t)$ tends to 0 in $H_L^{2\beta+1,p}$ as $t \to 0$. Suppose $(k - \beta - 1)p - \frac{n}{2} > 0$. Remark that examination of the argument in Step 2 allows time truncation. More precisely, if $0 < t \le \delta$, then one finds

$$ \|1_{\{\tau < 2\delta\}} \partial_{\tau}^{k}(u(t + \tau))\|_{T_{\beta+1-k}^{p}} \lesssim \|1_{\{\tau < \delta\}} u\|_{T_{\beta+1}^{p}}, $$
where $c > 0$ is independent of $s$. Applying this on the above computation, we get

$$ \begin{aligned} \|t \partial_t u\|_{T_{\beta+1}^p} &\lesssim \sum_{j \ge 0} e^{-c2^{2j}} \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{C_j} s^{-\frac{n}{2}} |s^{-(\beta+1)} u(s, y)|^2 ds dy \right)^{p/2} dx \right)^{1/p} \\ &\lesssim \sum_{j \ge 0} 2^{j \frac{n}{2}} e^{-c2^{2j}} \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x, 2^{j+1}(2s)^{1/2})} |s^{-(\beta+1)} u(s, y)|^2 ds dy \right)^{p/2} dx \right)^{1/p} \\ &\lesssim \sum_{j \ge 0} 2^{j(\frac{n}{2} + \frac{n}{p})} e^{-c2^{2j}} \|u\|_{T_{\beta+1}^p} \lesssim \|u\|_{T_{\beta+1}^p}. \end{aligned} $$

The factor $2^{j \frac{n}{p}}$ in the third inequality comes from change of angle for tent space norms, see [Aus11, Theorem 1.1].

Step 3: Boundedness of $u(t)$ in $H_L^{2\beta+1,p}$. Now we prove that for any $t > 0$, $u(t)$ belongs to $H_L^{2\beta+1,p}$ with

$$ (8.6) \qquad \sup_{t>0} \|u(t)\|_{H_L^{2\beta+1,p}} \lesssim \|u\|_{T_{\beta+1}^p}. $$

By Step 1, $u(t)$ lies in $L^2$ for any $t > 0$, so we just need to show the norm estimate. To this end, pick an integer $k \ge 1$ with $(k - \beta - 1)p - \frac{n}{2} > 0$. From [AE23, §8.2], $\psi(z) := z^k e^{-z}$ is a valid admissible function for $H_L^{2\beta+1,p}$. So we use (8.4), (8.5) in Step 2, and the identity $\partial_\tau^k(u(t+\tau)) = (\partial_t^k u)(t+\tau)$ to deduce that for any $t > 0$,

$$ \begin{aligned} \|u(t)\|_{H_L^{2\beta+1,p}} &\lesssim \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x, \tau^{1/2})} |\tau^{-(\beta+1)} ((\tau L)^k e^{-\tau L} u(t))(y)|^2 dy d\tau \right)^{p/2} dx \right)^{1/p} \\ &\lesssim \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x, \tau^{1/2})} |\tau^{-(\beta+1)} \tau^k \partial_\tau^k (u(t+\tau, y))|^2 dy d\tau \right)^{p/2} dx \right)^{1/p} \\ &\lesssim \left( \int_{\mathbb{R}^n} \left( \int_t^\infty \int_{B(x, (\sigma-t)^{1/2})} (\sigma-t)^{(k-\beta-1)p-\frac{n}{2}} |(\partial_t^k u)(\sigma, y)|^2 dy d\sigma \right)^{p/2} dx \right)^{1/p} \\ &\lesssim \|\sigma^k (\partial_t^k u)(\sigma)\|_{T_{\beta+1}^p} \lesssim \|u\|_{T_{\beta+1}^p}. \end{aligned} $$

The first inequality with implicit constant independent of $t$ uses $(k-\beta-1)p - \frac{n}{2} > 0$. This proves (8.6).

Step 4: Limit at $t = 0$. We finish by showing that $u(t)$ tends to 0 in $H_L^{2\beta+1,p}$ as $t \to 0$. Suppose $(k - \beta - 1)p - \frac{n}{2} > 0$. Remark that examination of the argument in Step 2 allows time truncation. More precisely, if $0 < t \le \delta$, then one finds

$$ \|1_{\{\tau < 2\delta\}} \partial_{\tau}^{k}(u(t + \tau))\|_{T_{\beta+1-k}^{p}} \lesssim \|1_{\{\tau < \delta\}} u\|_{T_{\beta+1}^{p}}, $$
where the implicit constant does not depend on $t, \delta$. Using Newton-Leibniz formula for $L^2$-valued functions, we get

$$ \tau^k \partial_{\tau}^k (u(t+\tau)) - \tau^k \partial_{\tau}^k (u(t'+\tau)) = \frac{1}{\tau} \int_{t'}^t \tau^{k+1} \partial_{\tau}^{k+1} (u(h+\tau)) dh. $$

Also, following the argument of Step 2 with the help of Minkowski's integral inequality, we find for $t, t' \in (0, \delta]$,

$$ \|1_{\{\tau<2\delta\}} (\partial_{\tau}^{k}(u(t+\tau)) - \partial_{\tau}^{k}(u(t+\tau)))\|_{T_{\beta+1-k}^{p}} \lesssim \frac{|t-t'|}{\delta} \|u\|_{T_{\beta+1}^{p}}, $$

again with implicit constant independent of $t, t', \delta$. This implies that $((\tau, x) \mapsto \partial_{\tau}^k(u(t+\tau, x)))_{t>0}$ is Cauchy in $T_{\beta+1-k}^p$ when $t \to 0$. We deduce as in Step 3 that $(u(t))_{t>0}$ is Cauchy in $H_L^{2\beta+1,p}$ when $t \to 0$. As mentioned in the first paragraph of Case 2(c), $u(t)$ tends to 0 in $\dot{H}^{2\beta+1,p}$. The conditions on $\beta, p$ allow us to apply Proposition 7.4 (ii), so we conclude that the limit of $u(t)$ exists and must be zero in $H_L^{2\beta+1,p}$. This completes the proof. $\square$

## 8.2. Representation.
Let us demonstrate Theorem 1.4.

*Proof of Theorem 1.4.* Let $u$ be a global weak solution to $\partial_t u - \text{div}(A\nabla u) = 0$ with $\nabla u \in T_{\beta+1/2}^p$. As $p > \tilde{p}_L(\beta) \ge \frac{n}{n+2\beta+2}$, Proposition 6.2 asserts that there exists $u_0 \in \mathcal{S}'$ so that

$$ u = \mathcal{E}_{-\Delta}(u_0) + \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u). $$

In case (i), as $\beta \ge 0$ and $\frac{n}{n+2\beta} \le p \le \infty$, Proposition 6.2 (i) says $u_0$ is a constant. Hence, $w := u - \mathcal{E}_{-\Delta}(u_0)$ is a global weak solution to the Cauchy problem

$$ \begin{cases} \partial_t w - \text{div}(A \nabla w) = 0, \\ w(0) = 0 \end{cases} $$

with $\nabla w \in T_{\beta+1/2}^p$. As $\tilde{p}_L(\beta) < p \le \infty$, we invoke Theorem 8.1 to get $w = 0$, so $u = \mathcal{E}_{-\Delta}(u_0)$ is a constant as desired.

In case (ii), Proposition 6.2 (ii) shows that there exist $g \in \dot{H}^{2\beta+1,p}$ and $c \in \mathbb{C}$ so that $u_0 = g + c$. Then we apply Theorem 7.1 to $g$ to get

$$ u = \mathcal{E}_{-\Delta}(g) + \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u) + c = \mathcal{E}_L(g) + c. $$

This completes the proof. $\square$

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

# 9. RESULTS FOR HOMOGENEOUS BESOV SPACES

One can also study the case where the initial data are taken in homogeneous Besov spaces $\dot{B}_{p,p}^s$. The definition of $\dot{B}_{p,p}^s$ can be analogously adapted from Definition 2.1, as “realization” of homogeneous Besov spaces defined on $\mathcal{S}'/\mathcal{P}$. In fact, one can also take the definition of $\dot{B}_{p,p}^s$ as in [BCD11, Definition 2.15], which does not contain polynomials for any $s$ and $p$. See [BCD11, Remark 2.26] for more detailed discussion.

The counterparts of tent spaces are $Z$-spaces, introduced by [BM16] (with a different notation). For any $p \in (0, \infty)$ and $\beta \in \mathbb{R}$, the (parabolic) $Z$-space $Z_{\beta}^p$
where the implicit constant does not depend on $t, \delta$. Using Newton-Leibniz formula for $L^2$-valued functions, we get

$$ \tau^k \partial_{\tau}^k(u(t+\tau)) - \tau^k \partial_{\tau}^k(u(t'+\tau)) = \frac{1}{\tau} \int_{t'}^t \tau^{k+1} \partial_{\tau}^{k+1}(u(h+\tau)) dh. $$

Also, following the argument of Step 2 with the help of Minkowski's integral inequality, we find for $t, t' \in (0, \delta]$,

$$ \|1_{\{\tau<2\delta\}} (\partial_{\tau}^{k}(u(t+\tau)) - \partial_{\tau}^{k}(u(t+\tau)))\|_{T_{\beta+1-k}^{p}} \lesssim \frac{|t-t'|}{\delta} \|u\|_{T_{\beta+1}^{p}}, $$

again with implicit constant independent of $t, t', \delta$. This implies that $((\tau, x) \mapsto \partial_{\tau}^k(u(t+\tau, x)))_{t>0}$ is Cauchy in $T_{\beta+1-k}^p$ when $t \to 0$. We deduce as in Step 3 that $(u(t))_{t>0}$ is Cauchy in $H_L^{2\beta+1,p}$ when $t \to 0$. As mentioned in the first paragraph of Case 2(c), $u(t)$ tends to 0 in $\dot{H}^{2\beta+1,p}$. The conditions on $\beta, p$ allow us to apply Proposition 7.4 (ii), so we conclude that the limit of $u(t)$ exists and must be zero in $H_L^{2\beta+1,p}$. This completes the proof. $\square$

## 8.2. Representation. Let us demonstrate Theorem 1.4.

*Proof of Theorem 1.4.* Let $u$ be a global weak solution to $\partial_t u - \text{div}(A\nabla u) = 0$ with $\nabla u \in T_{\beta+1/2}^p$. As $p > \tilde{p}_L(\beta) \ge \frac{n}{n+2\beta+2}$, Proposition 6.2 asserts that there exists $u_0 \in \mathcal{S}'$ so that

$$ u = \mathcal{E}_{-\Delta}(u_0) + \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u). $$

In case (i), as $\beta \ge 0$ and $\frac{n}{n+2\beta} \le p \le \infty$, Proposition 6.2 (i) says $u_0$ is a constant. Hence, $w := u - \mathcal{E}_{-\Delta}(u_0)$ is a global weak solution to the Cauchy problem

$$ \begin{cases} \partial_t w - \text{div}(A \nabla w) = 0, \\ w(0) = 0 \end{cases} $$

with $\nabla w \in T_{\beta+1/2}^p$. As $\tilde{p}_L(\beta) < p \le \infty$, we invoke Theorem 8.1 to get $w = 0$, so $u = \mathcal{E}_{-\Delta}(u_0)$ is a constant as desired.

In case (ii), Proposition 6.2 (ii) shows that there exist $g \in \dot{H}^{2\beta+1,p}$ and $c \in \mathbb{C}$ so that $u_0 = g + c$. Then we apply Theorem 7.1 to $g$ to get

$$ u = \mathcal{E}_{-\Delta}(g) + \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u) + c = \mathcal{E}_L(g) + c. $$

This completes the proof. $\square$

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

## 9. RESULTS FOR HOMOGENEOUS BESOV SPACES

One can also study the case where the initial data are taken in homogeneous Besov spaces $\dot{B}_{p,p}^s$. The definition of $\dot{B}_{p,p}^s$ can be analogously adapted from Definition 2.1, as “realization” of homogeneous Besov spaces defined on $\mathcal{S}'/\mathcal{P}$. In fact, one can also take the definition of $\dot{B}_{p,p}^s$ as in [BCD11, Definition 2.15], which does not contain polynomials for any $s$ and $p$. See [BCD11, Remark 2.26] for more detailed discussion.

The counterparts of tent spaces are $Z$-spaces, introduced by [BM16] (with a different notation). For any $p \in (0, \infty)$ and $\beta \in \mathbb{R}$, the (parabolic) $Z$-space $Z_{\beta}^p$
consists of measurable functions $F$ on $\mathbb{R}_+^{1+n}$ for which the (quasi-)norm

$$ \|F\|_{Z_{\beta}^{p}} := \left( \int_{\mathbb{R}_{+}^{1+n}} \left( \int_{t/2}^{t} \int_{B(x,t^{1/2})} |s^{-\beta} F(s, y)|^2 ds dy \right)^{p/2} \frac{dt}{t} dx \right)^{1/p} < \infty. $$

For $p = \infty$, we set

$$ \|F\|_{Z_{\beta}^{\infty}} := \sup_{t>0, x \in \mathbb{R}^n} \left( \int_{t/2}^{t} \int_{B(x,t^{1/2})} |s^{-\beta} F(s,y)|^2 ds dy \right)^{1/2}. $$

The relation between $\dot{B}_{p,p}^s$ (resp. $Z_{\beta}^p$) and $\dot{H}^{s,p}$ (resp. $T_{\beta}^p$) is given by real interpolation, see [AA18, Theorem 2.30] and [AE23, §2.6]. Let $0 < p_0, p_1 \le \infty$ and $\theta \in (0, 1)$. Let $s_0, s_1, \beta_0, \beta_1 \in \mathbb{R}$ with $s_0 \ne s_1$ and $\beta_0 \ne \beta_1$. Define $s, \beta \in \mathbb{R}$ and $p \in (0, \infty]$ so that $s := (1-\theta)s_0 + \theta s_1$, $\beta := (1-\theta)\beta_0 + \theta \beta_1$, and $\frac{1}{p} := \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$. Then we have

$$ (9.1) \qquad (\dot{H}^{s_0, p_0}, \dot{H}^{s_1, p_1})_{\theta, p} = \dot{B}_{p, p}^{s}, \quad (T_{\beta_0}^{p_0}, T_{\beta_1}^{p_1})_{\theta, p} = Z_{\beta}^{p}. $$

Moreover, it has been shown in [AA18, Theorem 2.34] that

$$ (9.2) \qquad T_{\beta_0}^{p_0} \hookrightarrow Z_{\beta_1}^{p_1} \hookrightarrow T_{\beta_2}^{p_2}, \\ \text{if } 0 < p_0 < p_1 < p_2 \le \infty \text{ and } 2\beta_0 - \frac{n}{p_0} = 2\beta_1 - \frac{n}{p_1} = 2\beta_2 - \frac{n}{p_2}. $$

Let us first consider a special case for $\beta = 0$ (or $s = 1$) and $p = \infty$. Recall that we say a distribution $g$ belongs to the homogeneous Sobolev space $\dot{W}^{1,\infty}$ if $\nabla g$ is bounded, and $\|g\|_{\dot{W}^{1,\infty}} := \|\nabla g\|_{\infty}$. Rademacher's theorem asserts $\dot{W}^{1,\infty}$ coincides with the set of Lipschitz continuous functions (up to almost everywhere equality).

**Proposition 9.1.** (i) Let $g \in \dot{W}^{1,\infty}$ be a Lipschitz function. Then the function $(t, x) \mapsto \nabla e^{t\Delta}g(x)$ belongs to $Z_{1/2}^{\infty}$ with

$$ \|\nabla e^{t\Delta} g\|_{Z_{1/2}^{\infty}} \asymp \|g\|_{\dot{W}^{1,\infty}}. $$

(ii) Let $u$ be a distributional solution to the heat equation on $\mathbb{R}_+^{1+n}$ with $\nabla u \in Z_{1/2}^{\infty}$. Then there exists $u_0 \in \dot{W}^{1,\infty}$ such that $u(t) = e^{t\Delta}u_0$ for all $t > 0$.

*Proof.* Observe that the space $Z_{1/2}^{\infty}$ coincides with the parabolic version of the Kenig-Pipher space $X^{\infty}$ introduced by [KP93]. Then we invoke [AMP19, Theorem 5.4] to get (i) as

$$ \|\nabla e^{t\Delta} g\|_{Z_{1/2}^{\infty}} = \|e^{t\Delta} \nabla g\|_{X^{\infty}} \asymp \|\nabla g\|_{L^{\infty}} = \|g\|_{\dot{W}^{1,\infty}}. $$

Moreover, it also asserts that any weak solution $G \in X^{\infty}$ to the heat equation has a trace $g \in L^{\infty}$ so that $G(t) = e^{t\Delta}g$ for all $t > 0$.

To prove (ii), we claim that such $u$ also has a distributional limit $u_0 \in \mathcal{S}'$ as $t \to 0$, and $u(t) = e^{t\Delta}u_0$ for any $t > 0$. Then applying the above assertion to $\nabla u \in X^{\infty}$ yields $\nabla u_0 = g$, so $u_0$ must be (equal almost everywhere to) a Lipschitz function as desired.

The claim follows by a verbatim adaptation of the proof of Proposition 3.5. We just list the main modifications here. In this case, the estimate in Lemma 3.7 that yields the size condition becomes: For $0 < a < b < \infty$ and $R > 1$,

$$ \int_a^b \int_{B(0,R)} |u|^2 \lesssim_{a,b} R^{3n+2} \left( \|\nabla u\|_{Z_{1/2}^{\infty}}^2 + \|u\|_{L^2((a,b) \times B(0,1))}^2 \right). $$
consists of measurable functions $F$ on $\mathbb{R}_+^{1+n}$ for which the (quasi-)norm

$$ \|F\|_{Z_{\beta}^{p}} := \left( \int_{\mathbb{R}_{+}^{1+n}} \left( \int_{t/2}^{t} \int_{B(x,t^{1/2})} |s^{-\beta} F(s,y)|^2 ds dy \right)^{p/2} \frac{dt}{t} dx \right)^{1/p} < \infty. $$

For $p = \infty$, we set

$$ \|F\|_{Z_{\beta}^{\infty}} := \sup_{t>0, x \in \mathbb{R}^n} \left( \int_{t/2}^{t} \int_{B(x,t^{1/2})} |s^{-\beta} F(s,y)|^2 ds dy \right)^{1/2}. $$

The relation between $\dot{B}_{p,p}^s$ (resp. $Z_{\beta}^p$) and $\dot{H}^{s,p}$ (resp. $T_{\beta}^p$) is given by real interpolation, see [AA18, Theorem 2.30] and [AE23, §2.6]. Let $0 < p_0, p_1 \le \infty$ and $\theta \in (0, 1)$. Let $s_0, s_1, \beta_0, \beta_1 \in \mathbb{R}$ with $s_0 \ne s_1$ and $\beta_0 \ne \beta_1$. Define $s, \beta \in \mathbb{R}$ and $p \in (0, \infty]$ so that $s := (1-\theta)s_0 + \theta s_1$, $\beta := (1-\theta)\beta_0 + \theta \beta_1$, and $\frac{1}{p} := \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$. Then we have

$$ (9.1) \qquad (\dot{H}^{s_0, p_0}, \dot{H}^{s_1, p_1})_{\theta, p} = \dot{B}_{p, p}^{s}, \quad (T_{\beta_0}^{p_0}, T_{\beta_1}^{p_1})_{\theta, p} = Z_{\beta}^{p}. $$

Moreover, it has been shown in [AA18, Theorem 2.34] that

$$ (9.2) \qquad T_{\beta_0}^{p_0} \hookrightarrow Z_{\beta_1}^{p_1} \hookrightarrow T_{\beta_2}^{p_2}, $$ 
 if $0 < p_0 < p_1 < p_2 \le \infty$ and $2\beta_0 - \frac{n}{p_0} = 2\beta_1 - \frac{n}{p_1} = 2\beta_2 - \frac{n}{p_2}$. $$

Let us first consider a special case for $\beta = 0$ (or $s = 1$) and $p = \infty$. Recall that we say a distribution $g$ belongs to the homogeneous Sobolev space $\dot{W}^{1,\infty}$ if $\nabla g$ is bounded, and $\|g\|_{\dot{W}^{1,\infty}} := \|\nabla g\|_{\infty}$. Rademacher's theorem asserts $\dot{W}^{1,\infty}$ coincides with the set of Lipschitz continuous functions (up to almost everywhere equality).

**Proposition 9.1.** (i) Let $g \in \dot{W}^{1,\infty}$ be a Lipschitz function. Then the function $(t, x) \mapsto \nabla e^{t\Delta}g(x)$ belongs to $Z_{1/2}^{\infty}$ with

$$ \|\nabla e^{t\Delta} g\|_{Z_{1/2}^{\infty}} \approx \|g\|_{\dot{W}^{1,\infty}}. $$

(ii) Let $u$ be a distributional solution to the heat equation on $\mathbb{R}_+^{1+n}$ with $\nabla u \in Z_{1/2}^{\infty}$. Then there exists $u_0 \in \dot{W}^{1,\infty}$ such that $u(t) = e^{t\Delta}u_0$ for all $t > 0$.

*Proof.* Observe that the space $Z_{1/2}^{\infty}$ coincides with the parabolic version of the Kenig-Pipher space $X^{\infty}$ introduced by [KP93]. Then we invoke [AMP19, Theorem 5.4] to get (i) as

$$ \|\nabla e^{t\Delta} g\|_{Z_{1/2}^{\infty}} = \|e^{t\Delta} \nabla g\|_{X^{\infty}} \approx \|\nabla g\|_{L^{\infty}} = \|g\|_{\dot{W}^{1,\infty}}. $$

Moreover, it also asserts that any weak solution $G \in X^{\infty}$ to the heat equation has a trace $g \in L^{\infty}$ so that $G(t) = e^{t\Delta}g$ for all $t > 0$.

To prove (ii), we claim that such $u$ also has a distributional limit $u_0 \in \mathcal{S}'$ as $t \to 0$, and $u(t) = e^{t\Delta}u_0$ for any $t > 0$. Then applying the above assertion to $\nabla u \in X^{\infty}$ yields $\nabla u_0 = g$, so $u_0$ must be (equal almost everywhere to) a Lipschitz function as desired.

The claim follows by a verbatim adaptation of the proof of Proposition 3.5. We just list the main modifications here. In this case, the estimate in Lemma 3.7 that yields the size condition becomes: For $0 < a < b < \infty$ and $R > 1$,

$$ \int_a^b \int_{B(0,R)} |u|^2 \lesssim_{a,b} R^{3n+2} \left( \|\nabla u\|_{Z_{1/2}^\infty}^2 + \|u\|_{L^2((a,b) \times B(0,1))}^2 \right). $$
For the uniform control condition (3.8), it follows from (3.9), which still holds for $N > n + 1$ since

$$ \int_0^2 \int_{\mathbb{R}^n} |\nabla u||\nabla \phi| \lesssim \|\nabla u\|_{Z_{1/2}^{\infty}} \|\mathbf{1}_{(0,2)}\phi\|_{Z_{-1/2}^1} \lesssim \|\nabla u\|_{Z_{1/2}^{\infty}} \mathcal{P}_N(\phi). $$

Therefore, applying [AH23a, Theorem 1.1] again provides the trace $u_0 \in \mathcal{S}'$ as wanted. This completes the proof. □

**Theorem 9.2** (Well-posedness of Cauchy problems of type (1.1) for Besov spaces). Let $\beta > -1$, $s \in \mathbb{R}$, and $0 < p \le \infty$. With modifications for $\beta = 0$ (or $s = 1$) and $p = \infty$ as in Proposition 9.1 (i.e., the initial data $u_0$ lies in $\dot{W}^{1,\infty}$), Theorems 1.1, 1.3, 1.4, 1.6, and 1.7 are all valid in the same range of exponents, when replacing homogeneous Hardy–Sobolev spaces (resp. tent spaces) by homogeneous Besov spaces (resp. Z-spaces) with the same exponents.

*Proof.* We just provide some key ingredients of the proof. For convenience, we still use the same label of the theorems for their variants in Besov spaces. Interpolation (9.1) allows one to prove Theorem 1.1 (i) and the needed boundedness for existence in Theorems 1.3, 1.6, and 1.7. For $\beta = -1/2$ (or $s = 0$) and $p = \infty$, one uses Proposition 9.1. To prove uniqueness, for $p \ne \infty$, it follows from embedding into tent spaces in (9.2). For $p = \infty$, following the proof of Theorem 8.1, we may suppose $u = \mathcal{R}_{1/2}^{-\Delta}((A - \mathbf{I})\nabla u)$ (cf. (8.1)), so it suffices to show that for any $\tilde{F} \in Z_{\beta+1/2}^{\infty}$, $\mathcal{R}_{1/2}^{-\Delta}(\tilde{F})(t)$ tends to 0 as $t \to 0$ in $E_{\delta}^{-1,\infty}$ for any $\delta > 0$. Using Theorem 5.1 (b) and [AH23b, Corollary 5.6], we have

$$ \|\mathcal{R}_{1/2}^{-\Delta}(\tilde{F})(t)\|_{E_{\delta}^{-1,\infty}} \lesssim \|\mathcal{L}_1^{-\Delta}(\tilde{F})(t)\|_{E_{\delta}^{\infty}} \lesssim t^{\beta+1} \|\tilde{F}\|_{T_{\beta+1/2}^{\infty}}. $$

Then interpolation (9.1) yields that $\|\mathcal{R}_{1/2}^{-\Delta}(\tilde{F})(t)\|_{E_{\delta}^{-1,\infty}} \lesssim t^{\beta+1} \|\tilde{F}\|_{Z_{\beta+1/2}^{\infty}}$ holds for $\beta > -1$, which implies the desired zero limit when $t \to 0$.

Next, we prove Theorem 1.1 (ii), from which Theorem 1.4 also follows, using the same argument in Section 8.2. Existence of the trace again follows from a verbatim adaptation of the proof of Proposition 3.5, as is shown in Proposition 9.1 (ii). Then we show the properties of the trace. For (1), when $s \ge 0$, $\frac{n}{n+s} \le p \le \infty$ but $(s,p) \ne (0, \infty)$, one can imitate the proof of Corollary 3.4 (ii), using interpolation and direct computation, to get that for $0 < a < 1$ and $\phi \in C_c^{\infty}(\mathbb{R}^n)$,

$$ \int_a^{2a} |\langle \nabla e^{t\Delta} u_0, \phi \rangle| dt \lesssim_{\phi} \|\mathbf{1}_{(a,2a)} \nabla \mathcal{E}_{-\Delta}(u_0)\|_{Z_{s/2}^p}, $$

which tends to 0 as $a \to 0$, so $\nabla u_0 = 0$ and $u_0$ is a constant. For $(s,p) = (0, \infty)$, this is Proposition 9.1 (ii). This hence proves (1).

For (2), when $s < 1$ and $\frac{n}{n+s+1} < p \le \infty$, we apply interpolation for the map $u \mapsto u_0$ to obtain $u_0 \in \dot{B}_{p,p}^s$ with $\|u_0\|_{\dot{B}_{p,p}^s} \lesssim \|u\|_{Z_{s/2}^p}$. This completes the proof. □

10. AN ENDPOINT CASE $\beta = -1$

In this section, we show existence of global weak solutions to the homogeneous Cauchy problem for $\beta = -1$.

**Proposition 10.1.** Let $p_-(-1, L) = q_+(L^*)' < p \le \infty$. For any $v_0 \in \dot{H}^{-1,p}$, there is a global weak solution $v \in C([0, \infty); \mathcal{S}')$ to (HC) with initial data $v_0$ so that

$$ (10.1) \qquad \|\nabla v\|_{T_{-1/2}^p} \lesssim \|v\|_{T_0^p} \lesssim \|v_0\|_{\dot{H}^{-1,p}}. $$
For the uniform control condition (3.8), it follows from (3.9), which still holds for $N > n + 1$ since

$$ \int_0^2 \int_{\mathbb{R}^n} |\nabla u| |\nabla \phi| \lesssim \|\nabla u\|_{Z_{1/2}^{\infty}} \|\mathbf{1}_{(0,2)}\phi\|_{Z_{-1/2}^1} \lesssim \|\nabla u\|_{Z_{1/2}^{\infty}} \mathcal{P}_N(\phi). $$

Therefore, applying [AH23a, Theorem 1.1] again provides the trace $u_0 \in \mathcal{S}'$ as wanted. This completes the proof. □

**Theorem 9.2** (Well-posedness of Cauchy problems of type (1.1) for Besov spaces). Let $\beta > -1$, $s \in \mathbb{R}$, and $0 < p \le \infty$. With modifications for $\beta = 0$ (or $s = 1$) and $p = \infty$ as in Proposition 9.1 (i.e., the initial data $u_0$ lies in $\dot{W}^{1,\infty}$), Theorems 1.1, 1.3, 1.4, 1.6, and 1.7 are all valid in the same range of exponents, when replacing homogeneous Hardy–Sobolev spaces (resp. tent spaces) by homogeneous Besov spaces (resp. Z-spaces) with the same exponents.

*Proof.* We just provide some key ingredients of the proof. For convenience, we still use the same label of the theorems for their variants in Besov spaces. Interpolation (9.1) allows one to prove Theorem 1.1 (i) and the needed boundedness for existence in Theorems 1.3, 1.6, and 1.7. For $\beta = -1/2$ (or $s = 0$) and $p = \infty$, one uses Proposition 9.1. To prove uniqueness, for $p \ne \infty$, it follows from embedding into tent spaces in (9.2). For $p = \infty$, following the proof of Theorem 8.1, we may suppose $u = \mathcal{R}_{1/2}^{-\Delta}((A - \mathbf{I})\nabla u)$ (cf. (8.1)), so it suffices to show that for any $\tilde{F} \in Z_{\beta+1/2}^{\infty}$, $\mathcal{R}_{1/2}^{-\Delta}(\tilde{F})(t)$ tends to 0 as $t \to 0$ in $E_{\delta}^{-1,\infty}$ for any $\delta > 0$. Using Theorem 5.1 (b) and [AH23b, Corollary 5.6], we have

$$ \|\mathcal{R}_{1/2}^{-\Delta}(\tilde{F})(t)\|_{E_{\delta}^{-1,\infty}} \lesssim \|\mathcal{L}_1^{-\Delta}(\tilde{F})(t)\|_{E_{\delta}^{\infty}} \lesssim t^{\beta+1} \|\tilde{F}\|_{T_{\beta+1/2}^{\infty}}. $$

Then interpolation (9.1) yields that $\|\mathcal{R}_{1/2}^{-\Delta}(\tilde{F})(t)\|_{E_{\delta}^{-1,\infty}} \lesssim t^{\beta+1} \|\tilde{F}\|_{Z_{\beta+1/2}^{\infty}}$ holds for $\beta > -1$, which implies the desired zero limit when $t \to 0$.

Next, we prove Theorem 1.1 (ii), from which Theorem 1.4 also follows, using the same argument in Section 8.2. Existence of the trace again follows from a verbatim adaptation of the proof of Proposition 3.5, as is shown in Proposition 9.1 (ii). Then we show the properties of the trace. For (1), when $s \ge 0$, $\frac{n}{n+s} \le p \le \infty$ but $(s,p) \ne (0, \infty)$, one can imitate the proof of Corollary 3.4 (ii), using interpolation and direct computation, to get that for $0 < a < 1$ and $\phi \in C_c^{\infty}(\mathbb{R}^n)$,

$$ \int_a^{2a} |\langle \nabla e^{t\Delta} u_0, \phi \rangle| dt \lesssim_{\phi} \|\mathbf{1}_{(a,2a)} \nabla \mathcal{E}_{-\Delta}(u_0)\|_{Z_{s/2}^p}, $$

which tends to 0 as $a \to 0$, so $\nabla u_0 = 0$ and $u_0$ is a constant. For $(s,p) = (0, \infty)$, this is Proposition 9.1 (ii). This hence proves (1).

For (2), when $s < 1$ and $\frac{n}{n+s+1} < p \le \infty$, we apply interpolation for the map $u \mapsto u_0$ to obtain $u_0 \in \dot{B}_{p,p}^s$ with $\|u_0\|_{\dot{B}_{p,p}^s} \lesssim \|u\|_{Z_{s/2}^p}$. This completes the proof. □

10. AN ENDPOINT CASE $\beta = -1$

In this section, we show existence of global weak solutions to the homogeneous Cauchy problem for $\beta = -1$.

**Proposition 10.1.** Let $p_-(-1, L) = q_+(L^*)' < p \le \infty$. For any $v_0 \in \dot{H}^{-1,p}$, there is a global weak solution $v \in C([0, \infty); \mathcal{S}')$ to (HC) with initial data $v_0$ so that

$$ (10.1) \qquad \|\nabla v\|_{T_{-1/2}^p} \lesssim \|v\|_{T_0^p} \lesssim \|v_0\|_{\dot{H}^{-1,p}}. $$
Moreover, if $p_{-}(-1, L) < p < p_{+}(-1, L)$, then $v \in C([0, \infty); \dot{H}^{-1, p})$.

**Remark 10.2.** For $\beta = -1$, neither $\mathcal{R}_{1/2}^L$ nor $\mathcal{R}_{1/2}^{-\Delta}$ is defined on $T_{-1/2}^p$, so Lemma 7.6 fails, and we do not have methods to prove uniqueness or representation for general parabolic equations. However, it is possible for the heat equation; for example, a representation result is presented in [AH23a, Theorem 3.1] for $v \in T_0^\infty$ with trace $v_0 \in \dot{H}^{-1, \infty} \simeq \text{BMO}^{-1}$. Similarly, we do not know the converse inequality in (10.1).

Our main strategy here is to define the operator $\mathcal{E}_L$ on $\dot{H}^{-1, p}$ by an extension of $(e^{-tL} \text{div})_{t>0}$ acting on $\dot{H}^{0,p}$, since any $\dot{H}^{-1,p}$-distributions can be written as the divergence of $\dot{H}^{0,p}$-functions. This fact follows from boundedness of Riesz transforms on $L^p$ when $1 < p < \infty$ and on BMO when $p = \infty$. Here and in the sequel, we omit to specify that the operator applies to $C^n$-valued functions.

**Lemma 10.3.** Let $q_+(L^*)' < p \le \infty$ and $t > 0$.

(i) The operator $e^{-tL} \text{div} : W^{1,2} \to W^{1,2}$ extends to a bounded operator from $\dot{H}^{0,p}$ to $W_{\text{loc}}^{1,2}$, denoted by $\mathcal{G}_t$.

(ii) For any $f \in \dot{H}^{0,p}$, $\mathcal{G}_t(f)$ converges to $\text{div} f$ in $\mathcal{S}'$ as $t \to 0$.

**Proof.** The extension in (i) is constructed by duality. Let $B$ be a ball in $\mathbb{R}^n$ and $\phi \in L^2(B)$. When $q_+(L^*)' < p \le 2$, $L^2 - L^{p'}$ estimates of $(t^{1/2}\nabla e^{-tL^*})$ yields

$$ \|\nabla e^{-tL^*} \phi\|_{p'} \lesssim t^{-\frac{1}{2}-\frac{n}{2}[p',2]} \|\phi\|_2. $$

When $p = \infty$, standard molecular estimates combining $L^2 - L^2$ off-diagonal estimates of $(t^{1/2}\nabla e^{-tL^*})$ and the fact that $\nabla e^{-tL^*} \phi$ has mean value 0 imply

$$ \|\nabla e^{-tL^*} \phi\|_{H^1} \lesssim (t^{-1/2}|B|^{1/2} + t^{\frac{n}{4}-\frac{1}{2}})\|\phi\|_2. $$

By interpolation, we have $\|\nabla e^{-tL^*} \phi\|_{\dot{H}^{0,p'}} \lesssim_{t,B} \|\phi\|_2$ for $q_+(L^*)' < p \le \infty$. Thus, for any $t > 0$, we define $\mathcal{G}_t: \dot{H}^{0,p} \to L_{\text{loc}}^2$ by the pairing

$$ \langle \mathcal{G}_t(f), \phi \rangle := \langle f, \nabla e^{-tL^*} \phi \rangle, \quad \forall \phi \in L_c^2(\mathbb{R}^n). $$

Meanwhile, note that $(t\nabla e^{-tL^*} \text{div})$ also has $L^2 - L^{p'}$ off-diagonal estimates for $q_+(L^*)' < p \le 2$, considering the decomposition

$$ (10.2) \qquad t \nabla e^{-tL^*} \text{div} = (t^{1/2} \nabla e^{-t/2L^*}) (t^{1/2} e^{-t/2L^*} \text{div}). $$

So repeating the above argument for $(t\nabla e^{-tL^*} \text{div})$ yields $\nabla \mathcal{G}_t(f)$ also lies in $L_{\text{loc}}^2$, hence $\mathcal{G}_t(f) \in W_{\text{loc}}^{1,2}$. This proves (i).

Next, we proceed with (ii). Pick $\phi \in \mathcal{S}$. It suffices to prove $\nabla e^{-tL^*} \phi$ tends to $\nabla \phi$ in $\dot{H}^{0,p'}$ when $t \to 0$. For $q_+(L^*)' < p \le 2$, as $2 \le p' < q_+(L^*) = p_+(1, L^*)$, it follows by continuity of the semigroup $(e^{-tL^*})$ on $H^{1,p'}$, see Lemma 7.5. For $p = \infty$, we first assert the following

**Lemma 10.4.** Let $\psi \in C_c^\infty(\mathbb{R}^n)$ and $B$ be a ball in $\mathbb{R}^n$ with $\text{supp}(\psi) \subset B$. Then

$$ \|\nabla e^{-tL^*} \psi - \nabla \psi\|_{H^1} \lesssim |B|^{1/2} \|\nabla e^{-tL^*} \psi - \nabla \psi\|_2 + t^{n/4} \|\nabla \psi\|_2. $$

The proof is provided right below. Admitting it, let us show that for any $\phi \in \mathcal{S}$, $\nabla e^{-tL^*} \phi$ tends to $\nabla \phi$ in $H^1$ as $t \to 0$. Indeed, write $B := B(0, 1)$, $C_0 := 4B$, and
Moreover, if $p_{-}(-1, L) < p < p_{+}(-1, L)$, then $v \in C([0, \infty); \dot{H}^{-1, p})$.

**Remark 10.2.** For $\beta = -1$, neither $\mathcal{R}_{1/2}^L$ nor $\mathcal{R}_{1/2}^{-\Delta}$ is defined on $T_{-1/2}^p$, so Lemma 7.6 fails, and we do not have methods to prove uniqueness or representation for general parabolic equations. However, it is possible for the heat equation; for example, a representation result is presented in [AH23a, Theorem 3.1] for $v \in T_0^\infty$ with trace $v_0 \in \dot{H}^{-1, \infty} \simeq \text{BMO}^{-1}$. Similarly, we do not know the converse inequality in (10.1).

Our main strategy here is to define the operator $\mathcal{E}_L$ on $\dot{H}^{-1, p}$ by an extension of $(e^{-tL} \text{div})_{t>0}$ acting on $\dot{H}^{0,p}$, since any $\dot{H}^{-1,p}$-distributions can be written as the divergence of $\dot{H}^{0,p}$-functions. This fact follows from boundedness of Riesz transforms on $L^p$ when $1 < p < \infty$ and on BMO when $p = \infty$. Here and in the sequel, we omit to specify that the operator applies to $C^n$-valued functions.

**Lemma 10.3.** Let $q_+(L^*)' < p \le \infty$ and $t > 0$.

(i) The operator $e^{-tL} \text{div} : W^{1,2} \to W^{1,2}$ extends to a bounded operator from $\dot{H}^{0,p}$ to $W_{\text{loc}}^{1,2}$, denoted by $\mathcal{G}_t$.

(ii) For any $f \in \dot{H}^{0,p}$, $\mathcal{G}_t(f)$ converges to $\text{div} f$ in $\mathcal{S}'$ as $t \to 0$.

**Proof.** The extension in (i) is constructed by duality. Let $B$ be a ball in $\mathbb{R}^n$ and $\phi \in L^2(B)$. When $q_+(L^*)' < p \le 2$, $L^2 - L^{p'}$ estimates of $(t^{1/2} \nabla e^{-tL^*})$ yields

$$ \|\nabla e^{-tL^*} \phi\|_{p'} \lesssim t^{-\frac{1}{2}-\frac{n}{2}[p',2]} \|\phi\|_2. $$

When $p = \infty$, standard molecular estimates combining $L^2 - L^2$ off-diagonal estimates of $(t^{1/2} \nabla e^{-tL^*})$ and the fact that $\nabla e^{-tL^*} \phi$ has mean value 0 imply

$$ \|\nabla e^{-tL^*} \phi\|_{H^1} \lesssim (t^{-1/2}|B|^{1/2} + t^{\frac{n}{4}-\frac{1}{2}})\|\phi\|_2. $$

By interpolation, we have $\|\nabla e^{-tL^*} \phi\|_{\dot{H}^{0,p'}} \lesssim_{t,B} \|\phi\|_2$ for $q_+(L^*)' < p \le \infty$. Thus, for any $t > 0$, we define $\mathcal{G}_t : \dot{H}^{0,p} \to L_{\text{loc}}^2$ by the pairing

$$ \langle \mathcal{G}_t(f), \phi \rangle := \langle f, \nabla e^{-tL^*} \phi \rangle, \quad \forall \phi \in L_c^2(\mathbb{R}^n). $$

Meanwhile, note that $(t \nabla e^{-tL^*} \text{div})$ also has $L^2 - L^{p'}$ off-diagonal estimates for $q_+(L^*)' < p \le 2$, considering the decomposition

$$ (10.2) \qquad t \nabla e^{-tL^*} \text{div} = (t^{1/2} \nabla e^{-tL^*}) (t^{1/2} e^{-tL^*} \text{div}). $$

So repeating the above argument for $(t \nabla e^{-tL^*} \text{div})$ yields $\nabla \mathcal{G}_t(f)$ also lies in $L_{\text{loc}}^2$, hence $\mathcal{G}_t(f) \in W_{\text{loc}}^{1,2}$. This proves (i).

Next, we proceed with (ii). Pick $\phi \in \mathcal{S}$. It suffices to prove $\nabla e^{-tL^*} \phi$ tends to $\nabla \phi$ in $\dot{H}^{0,p'}$ when $t \to 0$. For $q_+(L^*)' < p \le 2$, as $2 \le p' < q_+(L^*) = p_+(1, L^*)$, it follows by continuity of the semigroup $(e^{-tL^*})$ on $H^{1,p'}$, see Lemma 7.5. For $p = \infty$, we first assert the following

**Lemma 10.4.** Let $\psi \in C_c^\infty(\mathbb{R}^n)$ and $B$ be a ball in $\mathbb{R}^n$ with $\text{supp}(\psi) \subset B$. Then

$$ \|\nabla e^{-tL^*} \psi - \nabla \psi\|_{H^1} \lesssim |B|^{1/2} \|\nabla e^{-tL^*} \psi - \nabla \psi\|_2 + t^{n/4} \|\nabla \psi\|_2. $$

The proof is provided right below. Admitting it, let us show that for any $\phi \in \mathcal{S}$, $\nabla e^{-tL^*} \phi$ tends to $\nabla \phi$ in $H^1$ as $t \to 0$. Indeed, write $B := B(0, 1)$, $C_0 := 4B$, and
$C_j := 2^{j+2}B \setminus 2^{j-2}B$ for $j \ge 1$. Let $(\chi_j)$ be a smooth partition of unity so that $\text{supp}(\chi_j) \subset C_j$ with $\|\chi_j\|_{\infty} \le 1$ and $\|\nabla\chi_j\|_{\infty} \lesssim 1$. Define $\phi_j := \phi\chi_j$. Note that (10.3)

$$ \begin{aligned} \|\nabla \phi_j\|_2 &\lesssim \left( \int_{C_j} |y|^{-2N} |\langle y \rangle^N \nabla \phi(y)|^2 dy \right)^{1/2} + \left( \int_{C_j} |y|^{-2N} |\langle y \rangle^N \phi(y)|^2 dy \right)^{1/2} \\ &\lesssim 2^{-j(N-\frac{n}{2})} \mathcal{P}_{N+1}(\phi), \end{aligned} $$

where $\mathcal{P}_{N+1}$ is the semi-norm on $\mathcal{S}$ defined in (3.5). Then applying Lemma 10.4 on $\psi = \phi_j$, we get

$$ \begin{aligned} \|\nabla e^{-tL^*} \phi - \nabla \phi\|_{H^1} &\lesssim \sum_{j \ge 0} \|\nabla e^{-tL^*} \phi_j - \nabla \phi_j\|_{H^1} \\ &\lesssim 2^{jn/2} \|\nabla e^{-tL^*} \phi_j - \nabla \phi_j\|_2 + t^{n/4} \|\nabla \phi_j\|_2. \end{aligned} \quad (10.4) $$

Using (10.3), we have

$$ \sum_{j \ge 0} 2^{jn/2} \|\nabla e^{-tL^*} \phi_j - \nabla \phi_j\|_2 + t^{n/4} \|\nabla \phi_j\|_2 \lesssim \sum_{j \ge 0} 2^{-j(N-n)} \mathcal{P}_{N+1}(\phi), $$

which converges when $N > n$. Therefore, Lebesgue's dominated convergence theorem implies the right-hand side in (10.4) tends to 0 as $t \to 0$, noting that the first term tends to 0 by continuity of $(e^{-tL^*})$ on $\dot{H}^{1,2}$. This completes the proof. $\square$

**Proof of Lemma 10.4.** The conservation property of the semigroup yields $\nabla e^{-tL^*} \psi = \nabla e^{-tL^*} (\psi - \int_B \psi)$, so the desired inequality again follows from applying the molecular decomposition and Poincaré's inequality. $\square$

Let us finish by proving Proposition 10.1.

**Proof of Proposition 10.1.** Let $q_+(L^*)' < p \le \infty$. Define the operator $\mathcal{G}: \dot{H}^{0,p} \to L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$ by

$$ (10.5) \qquad \mathcal{G}(f)(t) := \mathcal{G}_t(f). $$

We first show that $v := \mathcal{G}(f)$ satisfies

$$ (10.6) \qquad \|v\|_{T_0^p} \lesssim \|f\|_{\dot{H}^{0,p}} $$

When $q_+(L^*)' < p \le 2$, we know from [AE23, §8.2] that $\psi(z) := z^{1/2}e^{-z}$ is an admissible function for $s \le 0$ and $p \le 2$. We hence infer from $L^p$-boundedness of $L^{-1/2}$ div (see e.g., [Aus07, Theorem 4.1]) that

$$ (10.7) \qquad \|\mathcal{G}(f)\|_{T_0^p} = \|(tL)^{1/2}e^{-tL}L^{-1/2} \text{div } f\|_{T_{1/2}^p} \lesssim \|L^{-1/2} \text{div } f\|_p \lesssim \|f\|_p. $$

Next, when $p = \infty$, we use a classical argument on the relation between Carleson measures and BMO-functions. Let $f$ be in $\dot{H}^{0,\infty} \simeq \text{BMO}$. Recall that

$$ \|\mathcal{G}(f)\|_{T_0^\infty} = \sup_B \left( \frac{1}{|B|} \int_0^{r(B)^2} \int_B |\mathcal{G}_t(f)(y)|^2 dt dy \right)^{1/2}. $$

Let $B$ be a ball in $\mathbb{R}^n$. Write $C_0 := 2B$ and $C_j := 2^{j+1}B \setminus 2^j B$ for any $j \ge 1$. Consider the decomposition

$$ f = \langle f \rangle_{C_0} + \sum_{j \ge 0} (f - \langle f \rangle_{C_0}) \mathbb{1}_{C_j} := \langle f \rangle_{C_0} + \sum_{j \ge 0} f_j \quad \text{in } \mathcal{D}', $$
$C_j := 2^{j+2}B \setminus 2^{j-2}B$ for $j \ge 1$. Let $(\chi_j)$ be a smooth partition of unity so that $\text{supp}(\chi_j) \subset C_j$ with $\|\chi_j\|_{\infty} \le 1$ and $\|\nabla\chi_j\|_{\infty} \lesssim 1$. Define $\phi_j := \phi\chi_j$. Note that (10.3)

$$ \begin{aligned} \|\nabla \phi_j\|_2 &\lesssim \left( \int_{C_j} |y|^{-2N} |\langle y \rangle^N \nabla \phi(y)|^2 dy \right)^{1/2} + \left( \int_{C_j} |y|^{-2N} |\langle y \rangle^N \phi(y)|^2 dy \right)^{1/2} \\ &\lesssim 2^{-j(N-\frac{n}{2})} \mathcal{P}_{N+1}(\phi), \end{aligned} $$

where $\mathcal{P}_{N+1}$ is the semi-norm on $\mathcal{S}$ defined in (3.5). Then applying Lemma 10.4 on $\psi = \phi_j$, we get

$$ \begin{aligned} \|\nabla e^{-tL^*} \phi - \nabla \phi\|_{H^1} &\lesssim \sum_{j \ge 0} \|\nabla e^{-tL^*} \phi_j - \nabla \phi_j\|_{H^1} \\ &\lesssim \sum_{j \ge 0} 2^{jn/2} \|\nabla e^{-tL^*} \phi_j - \nabla \phi_j\|_2 + t^{n/4} \|\nabla \phi_j\|_2. \end{aligned} \quad (10.4) $$

Using (10.3), we have

$$ \sum_{j \ge 0} 2^{jn/2} \|\nabla e^{-tL^*} \phi_j - \nabla \phi_j\|_2 + t^{n/4} \|\nabla \phi_j\|_2 \lesssim \sum_{j \ge 0} 2^{-j(N-n)} \mathcal{P}_{N+1}(\phi), $$

which converges when $N > n$. Therefore, Lebesgue's dominated convergence theorem implies the right-hand side in (10.4) tends to 0 as $t \to 0$, noting that the first term tends to 0 by continuity of $(e^{-tL^*})$ on $\dot{H}^{1,2}$. This completes the proof. $\square$

**Proof of Lemma 10.4.** The conservation property of the semigroup yields $\nabla e^{-tL^*} \psi = \nabla e^{-tL^*} (\psi - \int_B \psi)$, so the desired inequality again follows from applying the molecular decomposition and Poincaré's inequality. $\square$

Let us finish by proving Proposition 10.1.

**Proof of Proposition 10.1.** Let $q_+(L^*)' < p \le \infty$. Define the operator $\mathcal{G}: \dot{H}^{0,p} \to L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$ by

$$ (10.5) \qquad \mathcal{G}(f)(t) := \mathcal{G}_t(f). $$

We first show that $v := \mathcal{G}(f)$ satisfies

$$ (10.6) \qquad \|v\|_{T_0^p} \lesssim \|f\|_{\dot{H}^{0,p}}. $$

When $q_+(L^*)' < p \le 2$, we know from [AE23, §8.2] that $\psi(z) := z^{1/2}e^{-z}$ is an admissible function for $s \le 0$ and $p \le 2$. We hence infer from $L^p$-boundedness of $L^{-1/2}$ div (see e.g., [Aus07, Theorem 4.1]) that

$$ (10.7) \qquad \|\mathcal{G}(f)\|_{T_0^p} = \|(tL)^{1/2}e^{-tL}L^{-1/2} \operatorname{div} f\|_{T_{1/2}^p} \lesssim \|L^{-1/2} \operatorname{div} f\|_p \lesssim \|f\|_p. $$

Next, when $p = \infty$, we use a classical argument on the relation between Carleson measures and BMO-functions. Let $f$ be in $\dot{H}^{0,\infty} \simeq \text{BMO}$. Recall that

$$ \|\mathcal{G}(f)\|_{T_0^\infty} = \sup_B \left( \frac{1}{|B|} \int_0^{r(B)^2} \int_B |\mathcal{G}_t(f)(y)|^2 dt dy \right)^{1/2}. $$

Let $B$ be a ball in $\mathbb{R}^n$. Write $C_0 := 2B$ and $C_j := 2^{j+1}B \setminus 2^j B$ for any $j \ge 1$. Consider the decomposition

$$ f = \langle f \rangle_{C_0} + \sum_{j \ge 0} (f - \langle f \rangle_{C_0}) \mathbb{1}_{C_j} := \langle f \rangle_{C_0} + \sum_{j \ge 0} f_j \quad \text{in } \mathcal{D}', $$
where $\langle f \rangle_{C_0} := f_{C_0} f$. As $f_j \in L^2$, by definition, one finds that for any $t > 0$, $G(f)(t) = \sum_{j \ge 0} G(f_j)(t) = \sum_{j \ge 0} e^{-tL} \text{div } f_j$ in $\mathcal{D}'$, so we get

$$ \|G(f)\|_{L^2((0,r(B)^2) \times B)} \lesssim \sum_{j \ge 0} \|e^{-tL} \text{div } f_j\|_{L^2((0,r(B)^2) \times B)} =: I_j. $$

For $I_0$, this follows from (10.7) as

$$ I_0 \le \|t^{1/2}e^{-tL} \text{div } f_0\|_{L^2_{1/2}(\mathbb{R}^{1+n}_+)} \lesssim \|f_0\|_2 = \|f - \langle f \rangle_{C_0}\|_{L^2(C_0)} \lesssim |B|^{1/2} \|f\|_{\text{BMO}}. $$

For $j \ge 1$, we infer from $L^2 - L^2$ off-diagonal estimates of $(t^{1/2}e^{-tL} \text{div})$ that

$$ \begin{aligned} I_j &\lesssim \left( \int_0^{r(B)^2} e^{-2c \frac{(2^j r(B))^2}{t}} \frac{dt}{t} \int_{C_j} |f - \langle f \rangle_{C_0}|^2 \right)^{1/2} \\ &\lesssim \left( \int_0^{r(B)^2} |2^{j+1} B| e^{-2c \frac{(2^j r(B))^2}{t}} \frac{dt}{t} \int_{2^{j+1} B} |f - \langle f \rangle_{2^{j+1} B}|^2 \right)^{1/2} \\ &\quad + \left( \int_0^{r(B)^2} |2^{j+1} B| e^{-2c \frac{(2^j r(B))^2}{t}} \frac{dt}{t} \right)^{1/2} |\langle f \rangle_{2^{j+1} B} - \langle f \rangle_{2B}| \\ &\lesssim 2^{2jn} \log(1+j) e^{-c2^{2j}} |B|^{1/2} \|f\|_{\text{BMO}}. \end{aligned} $$

Gathering the estimates, we obtain $\|G(f)\|_{L^2((0,r(B)^2) \times B)} \lesssim |B|^{1/2} \|f\|_{\text{BMO}}$. Note that the controlling constant is independent of the ball $B$, so by taking supremum over all balls $B$ in $\mathbb{R}^n$, we have $\|G(f)\|_{T_0^\infty} \lesssim \|f\|_{\text{BMO}}$ as desired. The rest for $2 \le p < \infty$ hence follows from interpolation. This proves (10.6).

Meanwhile, observe that when $f \in \mathcal{S}_\infty$, $v$ is clearly a global weak solution to $\partial_t v - \text{div } A \nabla v = 0$ on $\mathbb{R}_+^{1+n}$, so by Corollary 4.3, we also have $\|\nabla v\|_{T_{-1/2}^p} \lesssim \|v\|_{T_0^p}$.

Now, for any $v_0 \in \dot{H}^{-1,p}$, pick $V_0 \in \dot{H}^{0,p}$ so that $v_0 = \text{div } V_0$ with $\|v_0\|_{\dot{H}^{-1,p}} \lesssim \|V_0\|_{\dot{H}^{0,p}}$. Using the above bounds for $f = V_0$, we obtain existence of the weak solution and the estimates (10.1) by a standard density argument.

To prove continuity, recall that Lemma 10.3 (ii) says $v(t)$ converges to $v_0$ in $\mathcal{S}'$ as $t \to 0$. In fact, a similar argument yields $v \in C((0, \infty); \mathcal{S}')$. When $p_-(1, L) < p < p_+(1, L)$, the desired stronger continuity follows from Lemma 7.5, due to uniqueness of extensions by density. This completes the proof. $\square$

## APPENDIX A. PROOF OF PROPOSITION 7.4

### A.1. Proof of Proposition 7.4 (i).
Let us first establish an appropriate atomic decomposition for distributions in $\dot{H}^{1,p}$ with $\frac{n}{n+2} < p \le \frac{n}{n+1}$. For $\frac{n}{n+1} < p \le 1$, it was done by [AE23, Proposition 8.31].

**Definition A.1** ($\dot{H}^{1,p}$-atom). Let $\frac{n}{n+2} < p \le \frac{n}{n+1}$. A function $a \in L^2$ is called an $\dot{H}^{1,p}$-atom if

(i) there is a ball $B \subset \mathbb{R}^n$ so that $\text{supp}(a) \subset B$, which is called associated to $a$;

(ii) $a$ is of mean zero, i.e., $\int_{\mathbb{R}^n} a = 0$;

(iii) $\|\nabla a\|_2 \le |B|^{[2,p]}$.

Compared with the case for $\frac{n}{n+1} < p \le 1$ (see [AE23, Definition 8.30]), the only extra condition we impose is (ii).
where $\langle f \rangle_{C_0} := f_{C_0} f$. As $f_j \in L^2$, by definition, one finds that for any $t > 0$, $G(f)(t) = \sum_{j \ge 0} G(f_j)(t) = \sum_{j \ge 0} e^{-tL} \text{div } f_j$ in $\mathcal{D}'$, so we get

$$ \|G(f)\|_{L^2((0,r(B)^2) \times B)} \lesssim \sum_{j \ge 0} \|e^{-tL} \text{div } f_j\|_{L^2((0,r(B)^2) \times B)} =: I_j. $$

For $I_0$, this follows from (10.7) as

$$ I_0 \le \|t^{1/2}e^{-tL} \text{div } f_0\|_{L^2_{1/2}(\mathbb{R}^{1+n}_+)} \lesssim \|f_0\|_2 = \|f - \langle f \rangle_{C_0}\|_{L^2(C_0)} \lesssim |B|^{1/2} \|f\|_{\text{BMO}}. $$

For $j \ge 1$, we infer from $L^2 - L^2$ off-diagonal estimates of $(t^{1/2}e^{-tL} \text{div})$ that

$$ \begin{aligned} I_j &\lesssim \left( \int_0^{r(B)^2} e^{-2c \frac{(2^j r(B))^2}{t}} \frac{dt}{t} \int_{C_j} |f - \langle f \rangle_{C_0}|^2 \right)^{1/2} \\ &\lesssim \left( \int_0^{r(B)^2} |2^{j+1} B| e^{-2c \frac{(2^j r(B))^2}{t}} \frac{dt}{t} \int_{2^{j+1} B} |f - \langle f \rangle_{2^{j+1} B}|^2 \right)^{1/2} \\ &\quad + \left( \int_0^{r(B)^2} |2^{j+1} B| e^{-2c \frac{(2^j r(B))^2}{t}} \frac{dt}{t} \right)^{1/2} |\langle f \rangle_{2^{j+1} B} - \langle f \rangle_{2B}| \\ &\lesssim 2^{2jn} \log(1+j) e^{-c2^{2j}} |B|^{1/2} \|f\|_{\text{BMO}}. \end{aligned} $$

Gathering the estimates, we obtain $\|G(f)\|_{L^2((0,r(B)^2) \times B)} \lesssim |B|^{1/2} \|f\|_{\text{BMO}}$. Note that the controlling constant is independent of the ball $B$, so by taking supremum over all balls $B$ in $\mathbb{R}^n$, we have $\|G(f)\|_{T_0^\infty} \lesssim \|f\|_{\text{BMO}}$ as desired. The rest for $2 \le p < \infty$ hence follows from interpolation. This proves (10.6).

Meanwhile, observe that when $f \in \mathcal{S}_\infty$, $v$ is clearly a global weak solution to $\partial_t v - \text{div } A \nabla v = 0$ on $\mathbb{R}_+^{1+n}$, so by Corollary 4.3, we also have $\|\nabla v\|_{T_{-1/2}^p} \lesssim \|v\|_{T_0^p}$.

Now, for any $v_0 \in \dot{H}^{-1,p}$, pick $V_0 \in \dot{H}^{0,p}$ so that $v_0 = \text{div } V_0$ with $\|v_0\|_{\dot{H}^{-1,p}} \lesssim \|V_0\|_{\dot{H}^{0,p}}$. Using the above bounds for $f = V_0$, we obtain existence of the weak solution and the estimates (10.1) by a standard density argument.

To prove continuity, recall that Lemma 10.3 (ii) says $v(t)$ converges to $v_0$ in $\mathcal{S}'$ as $t \to 0$. In fact, a similar argument yields $v \in C((0, \infty); \mathcal{S}')$. When $p_-(1, L) < p < p_+(1, L)$, the desired stronger continuity follows from Lemma 7.5, due to uniqueness of extensions by density. This completes the proof. $\square$

## APPENDIX A. PROOF OF PROPOSITION 7.4

### A.1. Proof of Proposition 7.4 (i). 
Let us first establish an appropriate atomic decomposition for distributions in $\dot{H}^{1,p}$ with $\frac{n}{n+2} < p \le \frac{n}{n+1}$. For $\frac{n}{n+1} < p \le 1$, it was done by [AE23, Proposition 8.31].

**Definition A.1** ($\dot{H}^{1,p}$-atom). Let $\frac{n}{n+2} < p \le \frac{n}{n+1}$. A function $a \in L^2$ is called an $\dot{H}^{1,p}$-atom if

(i) there is a ball $B \subset \mathbb{R}^n$ so that $\text{supp}(a) \subset B$, which is called associated to $a$;

(ii) $a$ is of mean zero, i.e., $\int_{\mathbb{R}^n} a = 0$;

(iii) $\|\nabla a\|_2 \le |B|^{[2,p]}$.

Compared with the case for $\frac{n}{n+1} < p \le 1$ (see [AE23, Definition 8.30]), the only extra condition we impose is (ii).
**Lemma A.2** (Atomic decomposition for $\dot{H}^{1,p}$). Let $\frac{n}{n+2} < p \le \frac{n}{n+1}$. For any $f \in \dot{H}^{1,p} \cap W^{1,2}$, there exist $(\lambda_i) \in \ell^p$ and $\dot{H}^{1,p}$-atoms $a_i$ so that

$$f = \sum_{i \ge 1} \lambda_i a_i,$$

where the convergence holds in $\dot{W}^{1,2}$ (that is, with respect to the semi-norm $\|\nabla \cdot\|_2$). Moreover, the estimate holds as

$$\|f\|_{\dot{H}^{1,p}} \lesssim \|(\lambda_i)\|_{\ell^p}.$$

*Proof.* The proof is analogous to that of [AE23, Proposition 8.31]. We only give a sketch here. Let $D$ be the Dirac operator, which is defined on $L^2(\mathbb{R}^n; \mathbb{C}^{1+n})$ by

$$D := \begin{bmatrix} 0 & \operatorname{div} \\ -\nabla & 0 \end{bmatrix}.$$

The main strategy of their proof is to use the atomic decomposition of $D$-adapted Hardy space $H_D^p$ ($0 < p < \infty$) to the vector $g := D[f, 0]^T = -[0, \nabla f]^T$. The only difference is that we decompose $g$ with $(H_D^p, 1, 2)$-molecules instead of $(H_D^p, 1, 1)$-molecules, which is allowed by [AE23, Theorem 8.17]. The same localization arguments apply to obtain an $L^2$-convergent decomposition $\nabla f = \sum_{i \ge 1} \lambda_i \nabla a_i$, with $\lambda_i, a_i$ as required. $\square$

For any $p \in (0, \infty]$, denote by $p_*$ the lower Sobolev conjugate of $p$, i.e., $1/p_* := 1/p + 1/n$. For any $p \in (0, n)$, denote by $p^*$ the upper Sobolev conjugate of $p$, i.e., $1/p^* := 1/p - 1/n$.

*Proof of Proposition 7.4 (i).* We only need to show $H_L^{s,p}$ agrees with $\dot{H}^{s,p} \cap L^2$ with equivalent (quasi-)norms

$$\|f\|_{H_L^{s,p}} \lesssim \|f\|_{\dot{H}^{s,p}},$$

when $-1 \le s \le 1$ and $p_-(s, L) < p < p_+(s, L)$. First observe that by interpolation of $H_L^{s,p}$ (see [AA18, Theorem 4.28]), it suffices to consider three cases, $s = 0$ and $s = \pm 1$.

For $s = 0$, this is [AE23, Theorem 9.7], as $p_+(0, L) = p_+(L)$.

For $s = 1$, [AE23, Theorem 9.7] shows that it holds when $(\max\{p_-(L), 1\})_* < p < q_+(L) = p_+(1, L)$, so it remains to consider the case when $\frac{n}{n+1} \le p_-(L) < 1$ and $p_-(1, L) = (p_-(L))_* < p \le 1_*$. The method of proof is adapted from the argument for [AE23, §9.2, Part 5], and we only need to show that for any $\psi \in \Psi_\infty^\infty$, there is a constant $C > 0$ so that for any $\dot{H}^{1,p}$-atom $a$,

$$(A.1) \qquad \|\psi(tL)a\|_{T_1^p} \le C.$$

Let us verify it. Let $\psi$ be in $\Psi_\infty^\infty$, $a$ be an $\dot{H}^{1,p}$-atom, and $B$ be a ball of $\mathbb{R}^n$ associated to $a$. Note that

$$\|\psi(tL)a\|_{T_1^p} \lesssim \|\mathcal{A}(t^{-1}\psi(tL)a)\|_{L^p(4B)} + \|\mathcal{A}(t^{-1}\psi(tL)a)\|_{L^p((4B)^c)} =: I_1 + I_2,$$

where $\mathcal{A}$ is the conical square function defined by

$$\mathcal{A}(f)(x) := \left( \int_0^\infty \int_{B(x, t^{1/2})} |f(t, y)|^2 dt dy \right)^{1/2}.$$
**Lemma A.2** (Atomic decomposition for $\dot{H}^{1,p}$). Let $\frac{n}{n+2} < p \le \frac{n}{n+1}$. For any $f \in \dot{H}^{1,p} \cap W^{1,2}$, there exist $(\lambda_i) \in \ell^p$ and $\dot{H}^{1,p}$-atoms $a_i$ so that

$$f = \sum_{i \ge 1} \lambda_i a_i,$$

where the convergence holds in $\dot{W}^{1,2}$ (that is, with respect to the semi-norm $\|\nabla \cdot\|_2$). Moreover, the estimate holds as

$$\|f\|_{\dot{H}^{1,p}} \lesssim \|(\lambda_i)\|_{\ell^p}.$$

*Proof.* The proof is analogous to that of [AE23, Proposition 8.31]. We only give a sketch here. Let $D$ be the Dirac operator, which is defined on $L^2(\mathbb{R}^n; \mathbb{C}^{1+n})$ by

$$D := \begin{bmatrix} 0 & \operatorname{div} \\ -\nabla & 0 \end{bmatrix}.$$

The main strategy of their proof is to use the atomic decomposition of $D$-adapted Hardy space $H_D^p$ ($0 < p < \infty$) to the vector $g := D[f, 0]^T = -[0, \nabla f]^T$. The only difference is that we decompose $g$ with $(H_D^p, 1, 2)$-molecules instead of $(H_D^p, 1, 1)$-molecules, which is allowed by [AE23, Theorem 8.17]. The same localization arguments apply to obtain an $L^2$-convergent decomposition $\nabla f = \sum_{i \ge 1} \lambda_i \nabla a_i$, with $\lambda_i, a_i$ as required. $\square$

For any $p \in (0, \infty]$, denote by $p_*$ the lower Sobolev conjugate of $p$, i.e., $1/p_* := 1/p + 1/n$. For any $p \in (0, n)$, denote by $p^*$ the upper Sobolev conjugate of $p$, i.e., $1/p^* := 1/p - 1/n$.

*Proof of Proposition 7.4 (i).* We only need to show $H_L^{s,p}$ agrees with $\dot{H}^{s,p} \cap L^2$ with equivalent (quasi-)norms

$$\|f\|_{H_L^{s,p}} \lesssim \|f\|_{\dot{H}^{s,p}},$$

when $-1 \le s \le 1$ and $p_-(s, L) < p < p_+(s, L)$. First observe that by interpolation of $H_L^{s,p}$ (see [AA18, Theorem 4.28]), it suffices to consider three cases, $s = 0$ and $s = \pm 1$.

For $s = 0$, this is [AE23, Theorem 9.7], as $p_+(0, L) = p_+(L)$.

For $s = 1$, [AE23, Theorem 9.7] shows that it holds when $(\max\{p_-(L), 1\})_* < p < q_+(L) = p_+(1, L)$, so it remains to consider the case when $\frac{n}{n+1} \le p_-(L) < 1$ and $p_-(1, L) = (p_-(L))_* < p \le 1_*$. The method of proof is adapted from the argument for [AE23, §9.2, Part 5], and we only need to show that for any $\psi \in \Psi_\infty^\infty$, there is a constant $C > 0$ so that for any $\dot{H}^{1,p}$-atom $a$,

$$(A.1) \qquad \|\psi(tL)a\|_{T_1^p} \le C.$$

Let us verify it. Let $\psi$ be in $\Psi_\infty^\infty$, $a$ be an $\dot{H}^{1,p}$-atom, and $B$ be a ball of $\mathbb{R}^n$ associated to $a$. Note that

$$\|\psi(tL)a\|_{T_1^p} \lesssim \|\mathcal{A}(t^{-1}\psi(tL)a)\|_{L^p(4B)} + \|\mathcal{A}(t^{-1}\psi(tL)a)\|_{L^p((4B)^c)} =: I_1 + I_2,$$

where $\mathcal{A}$ is the conical square function defined by

$$\mathcal{A}(f)(x) := \left( \int_0^\infty \int_{B(x, t^{1/2})} |f(t, y)|^2 dt dy \right)^{1/2}.$$
The boundedness of $I_1$ is exactly as for [AE23, (9.30)]. Let us concentrate on $I_2$. To this end, we fix $(t, x) \in \mathbb{R}_+^{1+n}$ with $x \in (4B)^c$. As $a$ is of mean zero, Poincaré's inequality yields

$$ (A.2) \qquad \|a\|_2 \lesssim r(B) \|\nabla a\|_2 \lesssim |B|^{[2,p^*]}. $$

This estimate, together with the support condition and mean value 0, shows that $a$ belongs to $H^q$ for $\frac{n}{n+1} < q \le 1$ with the estimate

$$ (A.3) \qquad \|a\|_{H^q} \lesssim |B|^{[q,p^*]}. $$

Then we fix $p_-(L) < q < p^*$. The $H^q - L^2$ boundedness of $(\psi(tL))_{t>0}$ (see [AE23, Lemma 4.4]) yields

$$ (A.4) \qquad \|\psi(tL)a\|_2 \lesssim t^{\frac{n}{2}[2,q]} \|a\|_{H^q}. $$

Meanwhile, using $L^2 - L^2$ off-diagonal estimates,<fnref n="5" /> we get that for any $M > 0$,

$$ (A.5) \qquad \|\psi(tL)a\|_{L^2(B(x,t^{1/2}))} \lesssim \left(1 + \frac{\operatorname{dist}(B(x, t^{1/2}), B)^2}{t}\right)^{-M} \|a\|_2. $$

As $q < p^* < 2$, i.e., $n/q + 1 > n/p > n/2$, we pick $\theta \in (0, 1)$ so that

$$ (A.6) \qquad \rho := \frac{n}{q} + 1 - n\theta[q, 2] > \frac{n}{p}. $$

Note that $\theta$ can be chosen arbitrarily close to 0. We interpolate (A.4) and (A.5) with respect to $\theta$ to get

$$ \begin{aligned} & \|\psi(tL)a\|_{L^2(B(x,t^{1/2}))} \\ & \lesssim t^{\frac{n}{2}[2,q](1-\theta)} \left(1 + \frac{\operatorname{dist}(B(x, t^{1/2}), B)^2}{t}\right)^{-M\theta} \|a\|_{H^q}^{1-\theta} \|a\|_2^{\theta}. \end{aligned} $$

Taking $M > \frac{\rho}{2\theta}$, we integrate it over $t$ (with right weight and norm) to get

$$ \mathcal{A}(t^{-1}\psi(tL)a)(x) \lesssim \operatorname{dist}(x, B)^{-\rho} \|a\|_{H^q}^{1-\theta} \|a\|_2^{\theta}. $$

Gathering (A.2), (A.3), and definition of $\rho > n/p$ (cf. (A.6)), we obtain

$$ \begin{aligned} I_2 &= \|\mathcal{A}(t^{-1}\psi(tL)a)\|_{L^p((4B)^c)} \lesssim r(B)^{\frac{n}{p}-\rho} \|a\|_{H^q}^{1-\theta} \|a\|_2^{\theta} \\ &\lesssim |B|^{\frac{1}{p}-\frac{1}{n}-\frac{1}{q}+\theta[q,2]} |B|^{(1-\theta)[q,p^*]} |B|^{\theta[2,p^*]} \lesssim 1. \end{aligned} $$

This proves (A.1) and hence concludes the case $s = 1$.

For $s = -1$, we use duality. As $1 \le p_-(-1, L) < p < p_+(1, L) \le \infty$, we have $\max\{p_-(1, L^*), 1\} < p' < p_+(1, L^*) = q_+(L^*)$. Then for any $f \in \mathbb{H}_L^{-1,p}$, we have

$$ \|f\|_{\dot{H}^{-1,p}} = \sup_{g \in \dot{H}^{1,p'} \cap L^2(\mathbb{R}^n)} \frac{|\langle f, g \rangle|}{\|g\|_{\dot{H}^{1,p'}}} \approx \sup_{g \in \mathbb{H}_{L^*}^{1,p'}} \frac{|\langle f, g \rangle|}{\|g\|_{\mathbb{H}_{L^*}^{1,p'}}} \approx \|f\|_{\mathbb{H}_L^{-1,p}}. $$

The equality follows by density of $\dot{H}^{1,p'} \cap L^2$ in $\dot{H}^{1,p'}$ and the first equivalence by $\mathbb{H}_{L^*}^{1,p'} = \dot{H}^{1,p'} \cap L^2$ with $\|g\|_{\mathbb{H}_{L^*}^{1,p'}} \approx \|g\|_{\dot{H}^{1,p'}}$ (from Case $s = 1$ for $L^*$). The last equivalence holds by [AE23, Proposition 8.9]. This completes the proof. $\square$

<footnote n="5">⁵Here we use polynomial-order decay as in [AE23, Lemma 4.16].</footnote>
The boundedness of $I_1$ is exactly as for [AE23, (9.30)]. Let us concentrate on $I_2$. To this end, we fix $(t, x) \in \mathbb{R}_+^{1+n}$ with $x \in (4B)^c$. As $a$ is of mean zero, Poincaré's inequality yields

$$ (A.2) \qquad \|a\|_2 \lesssim r(B) \|\nabla a\|_2 \lesssim |B|^{[2,p^*]}. $$

This estimate, together with the support condition and mean value 0, shows that $a$ belongs to $H^q$ for $\frac{n}{n+1} < q \le 1$ with the estimate

$$ (A.3) \qquad \|a\|_{H^q} \lesssim |B|^{[q,p^*]}. $$

Then we fix $p_-(L) < q < p^*$. The $H^q - L^2$ boundedness of $(\psi(tL))_{t>0}$ (see [AE23, Lemma 4.4]) yields

$$ (A.4) \qquad \|\psi(tL)a\|_2 \lesssim t^{\frac{n}{2}[2,q]} \|a\|_{H^q}. $$

Meanwhile, using $L^2 - L^2$ off-diagonal estimates,<fnref n="5" /> we get that for any $M > 0$,

$$ (A.5) \qquad \|\psi(tL)a\|_{L^2(B(x,t^{1/2}))} \lesssim \left(1 + \frac{\operatorname{dist}(B(x, t^{1/2}), B)^2}{t}\right)^{-M} \|a\|_2. $$

As $q < p^* < 2$, i.e., $n/q + 1 > n/p > n/2$, we pick $\theta \in (0, 1)$ so that

$$ (A.6) \qquad \rho := \frac{n}{q} + 1 - n\theta[q, 2] > \frac{n}{p}. $$

Note that $\theta$ can be chosen arbitrarily close to 0. We interpolate (A.4) and (A.5) with respect to $\theta$ to get

$$ \begin{aligned} & \|\psi(tL)a\|_{L^2(B(x,t^{1/2}))} \\ & \lesssim t^{\frac{n}{2}[2,q](1-\theta)} \left(1 + \frac{\operatorname{dist}(B(x, t^{1/2}), B)^2}{t}\right)^{-M\theta} \|a\|_{H^q}^{1-\theta} \|a\|_2^{\theta}. \end{aligned} $$

Taking $M > \frac{\rho}{2\theta}$, we integrate it over $t$ (with right weight and norm) to get

$$ \mathcal{A}(t^{-1}\psi(tL)a)(x) \lesssim \operatorname{dist}(x, B)^{-\rho} \|a\|_{H^q}^{1-\theta} \|a\|_2^{\theta}. $$

Gathering (A.2), (A.3), and definition of $\rho > n/p$ (cf. (A.6)), we obtain

$$ \begin{aligned} I_2 &= \|\mathcal{A}(t^{-1}\psi(tL)a)\|_{L^p((4B)^c)} \lesssim r(B)^{\frac{n}{p}-\rho} \|a\|_{H^q}^{1-\theta} \|a\|_2^{\theta} \\ &\lesssim |B|^{\frac{1}{p}-\frac{1}{n}-\frac{1}{q}+\theta[q,2]} |B|^{(1-\theta)[q,p^*]} |B|^{\theta[2,p^*]} \lesssim 1. \end{aligned} $$

This proves (A.1) and hence concludes the case $s = 1$.

For $s = -1$, we use duality. As $1 \le p_-(-1, L) < p < p_+(1, L) \le \infty$, we have $\max\{p_-(1, L^*), 1\} < p' < p_+(1, L^*) = q_+(L^*)$. Then for any $f \in \mathbb{H}_L^{-1,p}$, we have

$$ \|f\|_{\dot{H}^{-1,p}} = \sup_{g \in \dot{H}^{1,p'} \cap L^2(\mathbb{R}^n)} \frac{|\langle f, g \rangle|}{\|g\|_{\dot{H}^{1,p'}}} \approx \sup_{g \in \mathbb{H}_{L^*}^{1,p'}} \frac{|\langle f, g \rangle|}{\|g\|_{\mathbb{H}_{L^*}^{1,p'}}} \approx \|f\|_{\mathbb{H}_L^{-1,p}}. $$

The equality follows by density of $\dot{H}^{1,p'} \cap L^2$ in $\dot{H}^{1,p'}$ and the first equivalence by $\mathbb{H}_{L^*}^{1,p'} = \dot{H}^{1,p'} \cap L^2$ with $\|g\|_{\mathbb{H}_{L^*}^{1,p'}} \approx \|g\|_{\dot{H}^{1,p'}}$ (from Case $s = 1$ for $L^*$). The last equivalence holds by [AE23, Proposition 8.9]. This completes the proof. $\square$

<footnote n="5">⁵Here we use polynomial-order decay as in [AE23, Lemma 4.16].</footnote>
A.2. **Proof of Proposition 7.4 (ii).** Recall that we assume $p_-(L) < 1$, which implies $p_+(L^*) = \infty$. When $1 < p < 2$ and $-1 \le s \le 0$, by duality, it is equivalent to prove $\dot{H}^{-s,p'} \cap L^2 \subset \mathbb{H}_{L^*}^{-s,p'}$ with $\|f\|_{\mathbb{H}_{L^*}^{-s,p'}} \lesssim \|f\|_{\dot{H}^{-s,p'}}$. For $s = 0$ and $s = -1$, this is explicitly proved in Part 2 and Part 9 of the proof of [AE23, Theorem 9.7]. Interpolation concludes the argument.

It remains to show that the extension of the identity map to the closure of $\mathbb{H}_L^{s,p}$ (for its norm) is injective in the restricted range described in the statement. More precisely, let $(f_j)$ be a Cauchy sequence in $\mathbb{H}_L^{s,p}$ that tends to 0 in $\dot{H}^{s,p}$. Our goal is to show $\|f_j\|_{\mathbb{H}_L^{s,p}}$ tends to 0.

To this end, we use duality. Pick $\phi \in C_c^\infty(\mathbb{R}_+^{1+n})$ and define $\Phi := \int_0^\infty e^{-tL^*} \phi(t) \frac{dt}{t}$. Fubini's theorem ensures

$$ \int_0^\infty \int_{\mathbb{R}^n} (e^{-tL} f_j)(y) \bar{\phi}(t, y) dy \frac{dt}{t} = \int_{\mathbb{R}^n} f_j(y) \overline{\Phi(y)} dy. $$

We claim $\Phi \in \dot{H}^{-s,p'}$. If so, then using the fact that $(f_j)$ tends to 0 in $\dot{H}^{s,p}$, we get $(\mathcal{E}_L(f_j))$ tends to 0 in $\mathcal{D}'(\mathbb{R}_+^{1+n})$ by arbitrariness of $\phi$. Moreover, as $(f_j)$ is a Cauchy sequence in $\mathbb{H}_L^{s,p}$, so is $(\mathcal{E}_L(f_j))$ in $T_{(s+1)/2}^p$, because the exponential function $e^{-z}$ is an admissible function for $\mathbb{H}_L^{s,p}$ when $s < 0$ and $p \le 2$. Hence, it tends to 0 in $T_{(s+1)/2}^p$, which implies $\lim_{j \to \infty} \|f_j\|_{\mathbb{H}_L^{s,p}} = 0$.

We finish by verifying the claim. Pick $0 < a < b < \infty$ so that $\text{supp}(\phi) \subset (a, b) \times \mathbb{R}^n$. Note that $\phi \in C_c^\infty(\mathbb{R}_+^{1+n})$ implies $\Phi \in L^2$, so we apply Theorem 3.2 to $\Delta\Phi$ to get

$$ (A.7) \qquad \|\Phi\|_{\dot{H}^{-s,p'}} \lesssim \|\tau \Delta e^{\tau \Delta} \Phi\|_{T_{(-s+1)/2}^{p'}}. $$

Pick $0 < a < b < \infty$ so that $\text{supp}(\phi) \subset (a, b) \times \mathbb{R}^n$. Using Hölder's inequality and Minkowski's inequality, we have (the implicit constants may depend on $a, b$)

$$ \begin{align*} \|\tau \Delta e^{\tau \Delta} \Phi\|_{T_{(-s+1)/2}^{p'}} & \lesssim \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x, \tau^{1/2})} \int_a^b |\tau^{\frac{s-1}{2}} \tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t)|^2 dt dy d\tau \right)^{p'/2} dx \right)^{1/p'} \\ & \lesssim \left( \int_a^b \int_0^\infty \left( \int_{\mathbb{R}^n} \left( \int_{B(x, \tau^{1/2})} |\tau^{\frac{s-1}{2}} \tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t)|^2 dy \right)^{p'/2} dx \right)^{2/p'} d\tau dt \right)^{1/2} \\ & \lesssim \left( \int_a^b \int_0^\infty \| \tau^{\frac{s-1}{2}} \tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t) \|_{p'}^2 d\tau dt \right)^{1/2} \end{align*} $$

When $\tau$ is large, uniform $L^{p'}$-boundedness of $(\tau \Delta e^{\tau \Delta})$ and $(e^{-tL^*})$ implies

$$ \|\tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t)\|_{p'} \lesssim \|\phi(t)\|_{p'}. $$

As $s < 0$, $s - 1 < -1$, so the convergence of the integral when $\tau \ge 1$ is ensured.

When $\tau$ is small, we need different methods to gain a positive power of $\tau$. For $2 \le q < q_+(L^*)'$, we use $L^q$-boundedness of $(e^{\tau \Delta} \text{div})$ and $(t^{1/2} \nabla e^{-tL^*})$ for the
A.2. **Proof of Proposition 7.4 (ii).** Recall that we assume $p_-(L) < 1$, which implies $p_+(L^*) = \infty$. When $1 < p < 2$ and $-1 \le s \le 0$, by duality, it is equivalent to prove $\dot{H}^{-s,p'} \cap L^2 \subset \mathbb{H}_{L^*}^{-s,p'}$ with $\|f\|_{\mathbb{H}^{-s,p'}_{L^*}} \lesssim \|f\|_{\dot{H}^{-s,p'}}$. For $s = 0$ and $s = -1$, this is explicitly proved in Part 2 and Part 9 of the proof of [AE23, Theorem 9.7]. Interpolation concludes the argument.

It remains to show that the extension of the identity map to the closure of $\mathbb{H}_L^{s,p}$ (for its norm) is injective in the restricted range described in the statement. More precisely, let $(f_j)$ be a Cauchy sequence in $\mathbb{H}_L^{s,p}$ that tends to 0 in $\dot{H}^{s,p}$. Our goal is to show $\|f_j\|_{\mathbb{H}_L^{s,p}}$ tends to 0.

To this end, we use duality. Pick $\phi \in C_c^\infty(\mathbb{R}_+^{1+n})$ and define $\Phi := \int_0^\infty e^{-tL^*} \phi(t) \frac{dt}{t}$. Fubini's theorem ensures

$$ \int_0^\infty \int_{\mathbb{R}^n} (e^{-tL} f_j)(y) \bar{\phi}(t, y) dy \frac{dt}{t} = \int_{\mathbb{R}^n} f_j(y) \overline{\Phi(y)} dy. $$

We claim $\Phi \in \dot{H}^{-s,p'}$. If so, then using the fact that $(f_j)$ tends to 0 in $\dot{H}^{s,p}$, we get $(\mathcal{E}_L(f_j))$ tends to 0 in $\mathcal{D}'(\mathbb{R}_+^{1+n})$ by arbitrariness of $\phi$. Moreover, as $(f_j)$ is a Cauchy sequence in $\mathbb{H}_L^{s,p}$, so is $(\mathcal{E}_L(f_j))$ in $T_{(s+1)/2}^p$, because the exponential function $e^{-z}$ is an admissible function for $\mathbb{H}_L^{s,p}$ when $s < 0$ and $p \le 2$. Hence, it tends to 0 in $T_{(s+1)/2}^p$, which implies $\lim_{j \to \infty} \|f_j\|_{\mathbb{H}_L^{s,p}} = 0$.

We finish by verifying the claim. Pick $0 < a < b < \infty$ so that $\text{supp}(\phi) \subset (a, b) \times \mathbb{R}^n$. Note that $\phi \in C_c^\infty(\mathbb{R}_+^{1+n})$ implies $\Phi \in L^2$, so we apply Theorem 3.2 to $\Delta\Phi$ to get

$$ (A.7) \qquad \|\Phi\|_{\dot{H}^{-s,p'}} \lesssim \|\tau \Delta e^{\tau \Delta} \Phi\|_{T_{(-s+1)/2}^{p'}}. $$

Pick $0 < a < b < \infty$ so that $\text{supp}(\phi) \subset (a, b) \times \mathbb{R}^n$. Using Hölder's inequality and Minkowski's inequality, we have (the implicit constants may depend on $a, b$)

$$ \begin{align*} \|\tau \Delta e^{\tau \Delta} \Phi\|_{T_{(-s+1)/2}^{p'}} & \lesssim \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x, \tau^{1/2})} \int_a^b |\tau^{\frac{s-1}{2}} \tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t)|^2 dt dy d\tau \right)^{p'/2} dx \right)^{1/p'} \\ & \lesssim \left( \int_a^b \int_0^\infty \left( \int_{\mathbb{R}^n} \left( \int_{B(x, \tau^{1/2})} |\tau^{\frac{s-1}{2}} \tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t)|^2 dy \right)^{p'/2} dx \right)^{2/p'} d\tau dt \right)^{1/2} \\ & \lesssim \left( \int_a^b \int_0^\infty \| \tau^{\frac{s-1}{2}} \tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t) \|_{p'}^2 d\tau dt \right)^{1/2} \end{align*} $$

When $\tau$ is large, uniform $L^{p'}$-boundedness of $(\tau \Delta e^{\tau \Delta})$ and $(e^{-tL^*})$ implies

$$ \|\tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t)\|_{p'} \lesssim \|\phi(t)\|_{p'}. $$

As $s < 0$, $s - 1 < -1$, so the convergence of the integral when $\tau \ge 1$ is ensured.

When $\tau$ is small, we need different methods to gain a positive power of $\tau$. For $2 \le q < q_+(L^*)'$, we use $L^q$-boundedness of $(e^{\tau \Delta} \text{div})$ and $(t^{1/2} \nabla e^{-tL^*})$ for the
decomposition to get

$$ (A.8) \qquad \begin{aligned} \|\tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t)\|_q &= \tau^{1/2} t^{-1/2} \left\| \left( \tau^{1/2} e^{\tau \Delta} \operatorname{div} \right) \left( t^{1/2} \nabla e^{-tL^*} \phi(t) \right) \right\|_q \\ &\lesssim \tau^{1/2} t^{-1/2} \|\phi(t)\|_q. \end{aligned} $$

Meanwhile, let $\alpha \in (0, 1)$ be a parameter to be determined with the constraint

$$ 0 < \alpha < n \left( \frac{1}{p_-(L)} - 1 \right). $$

Denote by $\dot{\Lambda}^\alpha$ the homogeneous $\alpha$-Hölder space. One can also use Gaussian decay of the kernel of $(\tau \Delta e^{\tau \Delta})$ and uniform $\dot{\Lambda}^\alpha$-boundedness of $(e^{-tL^*})$ (by duality from $H^p$-boundedness of $(e^{-tL})$) to get

$$ (A.9) \qquad \|\tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t)\|_{\infty} \lesssim \tau^{\alpha/2} \|e^{-tL^*} \phi(t)\|_{\dot{\Lambda}^{\alpha}} \lesssim \tau^{\alpha/2} \|\phi(t)\|_{\dot{\Lambda}^{\alpha}}. $$

Interpolating (A.8) and (A.9) yields

$$ (A.10) \qquad \|\tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t)\|_{p'} \lesssim_{\phi} \tau^{\frac{q}{2}(1-\frac{q}{p'})+\frac{q}{2p'}}. $$

Observe that when $s > -n(\frac{1}{p_-(L)} - 1) - \frac{q_+(L^*)}{p'}(1 - n(\frac{1}{p_-(L)} - 1))$, i.e., (7.5) is satisfied, there exist $\alpha \in (0, 1)$ and $q \in [2, q_+(L^*))$ so that

$$ \left\{ \begin{array}{l} 0 < \alpha < n(\frac{1}{p_-(L)} - 1) \\ s - 1 + \alpha(1 - \frac{q}{p'}) + \frac{q}{p'} > -1 \end{array} \right. . $$

So the convergence of the integral for small $\tau$ follows. We thus obtain $\Phi \in \dot{H}^{-s, p'}$ from (A.7). This completes the proof.

**Copyright.** A CC-BY 4.0 https://creativecommons.org/licenses/by/4.0/ public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

## REFERENCES

[AA11] P. Auscher and A. Axelsson. Remarks on maximal regularity. In *Parabolic problems*, volume 80 of *Progr. Nonlinear Differential Equations Appl.*, pages 45–55. Birkhäuser/Springer Basel AG, Basel, 2011.

[AA18] A. Amenta and P. Auscher. *Elliptic boundary value problems with fractional regularity data*, volume 37 of *CRM Monograph Series*. American Mathematical Society, Providence, RI, 2018. The first order approach.

[AE23] P. Auscher and M. Egert. *Boundary value problems and Hardy spaces for elliptic systems with block structure*, volume 346 of *Progress in Mathematics*. Birkhäuser/Springer, Cham, 2023.

[AF17] P. Auscher and D. Frey. On the well-posedness of parabolic equations of Navier-Stokes type with $BMO^{-1}$ data. *J. Inst. Math. Jussieu*, 16(5):947–985, 2017.

[AH23a] P. Auscher and H. Hou. On representation of solutions to the heat equation, 2023. arXiv:2310.19330.

[AH23b] P. Auscher and H. Hou. On well-posedness and maximal regularity for parabolic Cauchy problems on weighted tent spaces, 2023. arXiv:2311.04844.
decomposition to get

$$
\begin{equation} \tag{A.8}
\begin{aligned}
\|\tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t)\|_{q} &= \tau^{1/2} t^{-1/2} \left\| \left( \tau^{1/2} e^{\tau \Delta} \operatorname{div} \right) \left( t^{1/2} \nabla e^{-tL^*} \phi(t) \right) \right\|_{q} \\
&\lesssim \tau^{1/2} t^{-1/2} \|\phi(t)\|_{q}.
\end{aligned}
\end{equation}
$$

Meanwhile, let $\alpha \in (0, 1)$ be a parameter to be determined with the constraint

$$0 < \alpha < n \left( \frac{1}{p_-(L)} - 1 \right).$$

Denote by $\hat{\Lambda}^\alpha$ the homogeneous $\alpha$-Hölder space. One can also use Gaussian decay of the kernel of $(\tau \Delta e^{\tau \Delta})$ and uniform $\hat{\Lambda}^\alpha$-boundedness of $(e^{-tL^*})$ (by duality from $H^p$-boundedness of $(e^{-tL})$) to get

$$(A.9) \quad \|\tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t)\|_{\infty} \lesssim \tau^{\alpha/2} \|e^{-tL^*} \phi(t)\|_{\hat{\Lambda}^{\alpha}} \lesssim \tau^{\alpha/2} \|\phi(t)\|_{\hat{\Lambda}^{\alpha}}.$$

Interpolating (A.8) and (A.9) yields

$$(A.10) \quad \|\tau \Delta e^{\tau \Delta} e^{-tL^*} \phi(t)\|_{p'} \lesssim_{\phi} \tau^{\frac{q}{2}(1-\frac{q}{p'})+\frac{q}{2p'}}.$$

Observe that when $s > -n(\frac{1}{p_-(L)} - 1) - \frac{q_+(L^*)}{p'}(1 - n(\frac{1}{p_-(L)} - 1))$, i.e., (7.5) is satisfied, there exist $\alpha \in (0, 1)$ and $q \in [2, q_+(L^*))$ so that

$$0 < \alpha < n\left(\frac{1}{p_-(L)} - 1\right) \\ s - 1 + \alpha\left(1 - \frac{q}{p'}\right) + \frac{q}{2p'} > -1.$$

So the convergence of the integral for small $\tau$ follows. We thus obtain $\Phi \in \dot{H}^{-s, p'}$ from (A.7). This completes the proof.

**Copyright.** A CC-BY 4.0 https://creativecommons.org/licenses/by/4.0/
public copyright license has been applied by the authors to the present document
and will be applied to all subsequent versions up to the Author Accepted Manuscript
arising from this submission.

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[AA18] A. Amenta and P. Auscher. *Elliptic boundary value problems with fractional regularity data*, volume 37 of *CRM Monograph Series*. American Mathematical Society, Providence, RI, 2018. The first order approach.

[AE23] P. Auscher and M. Egert. *Boundary value problems and Hardy spaces for elliptic systems with block structure*, volume 346 of *Progress in Mathematics*. Birkhäuser/Springer, Cham, 2023.

[AF17] P. Auscher and D. Frey. On the well-posedness of parabolic equations of Navier-Stokes type with $BMO^{-1}$ data. *J. Inst. Math. Jussieu*, 16(5):947–985, 2017.

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[AM19] P. Auscher and M. Mourgoglou. Representation and uniqueness for boundary value elliptic problems via first order systems. *Rev. Mat. Iberoam.*, 35(1):241–315, 2019.

[Ame18] A. Amenta. Interpolation and embeddings of weighted tent spaces. *J. Fourier Anal. Appl.*, 24(1):108–140, 2018.

[AMP19] P. Auscher, S. Monniaux, and P. Portal. On existence and uniqueness for non-autonomous parabolic Cauchy problems with rough coefficients. *Ann. Sc. Norm. Super. Pisa Cl. Sci.* **(5)**, 19(2):387–471, 2019.

[AP23] P. Auscher and P. Portal. Stochastic and deterministic parabolic equations with bounded measurable coefficients in space and time: well-posedness and maximal regularity, 2023. arXiv:2310.06460.

[Aus07] P. Auscher. On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\mathbb{R}^n$ and related estimates. *Mem. Amer. Math. Soc.*, 186(871):xviii+75, 2007.

[Aus11] P. Auscher. Change of angle in tent spaces. *C. R. Math. Acad. Sci. Paris*, 349(5-6):297–301, 2011.

[AvNP14] P. Auscher, J. van Neerven, and P. Portal. Conical stochastic maximal $L^p$-regularity for $1 \le p < \infty$. *Math. Ann.*, 359(3-4):863–889, 2014.

[BCD11] H. Bahouri, J.-Y. Chemin, and R. Danchin. *Fourier analysis and non-linear partial differential equations*, volume 343 of *Grundlehren der mathematischen Wissenschaften* [*Fundamental Principles of Mathematical Sciences*]. Springer, Heidelberg, 2011.

[BM16] A. Barton and S. Mayboroda. Layer potentials and boundary-value problems for second order elliptic operators with data in Besov spaces. *Mem. Amer. Math. Soc.*, 243(1149):v+110, 2016.

[Bou13] G. Bourdaud. Realizations of homogeneous Besov and Lizorkin-Triebel spaces. *Math. Nachr.*, 286(5-6):476–491, 2013.

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[DV23] R. Danchin and I. Vasilyev. Density-dependent incompressible Navier-Stokes equations in critical tent spaces, 2023. arXiv:2305.09027.

[FS72] C. Fefferman and E. M. Stein. $H^p$ spaces of several variables. *Acta Math.*, 129(3-4):137-193, 1972.

[Gra14] L. Grafakos. *Modern Fourier analysis*, volume 250 of *Graduate Texts in Mathematics*. Springer, New York, third edition, 2014.

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[Lio57] J.-L. Lions. Sur les problèmes mixtes pour certains systèmes paraboliques dans des ouverts non cylindriques. *Ann. Inst. Fourier (Grenoble)*, 7:143–182, 1957.

[Lun95] A. Lunardi. *Analytic semigroups and optimal regularity in parabolic problems*, volume 16 of *Progress in Nonlinear Differential Equations and their Applications*. Birkhäuser Verlag, Basel, 1995.

[Mou15] M. Moussai. Realizations of homogeneous Besov and Triebel-Lizorkin spaces and an application to pointwise multipliers. *Anal. Appl. (Sin-gap)*, 13(2):149–183, 2015.

[Pee76] J. Peetre. *New thoughts on Besov spaces*, volume No. 1 of Duke University Mathematics Series. Duke University, Mathematics Department, Durham, NC, 1976.

[PV19] P. Portal and M. Veraar. Stochastic maximal regularity for rough time-dependent problems. *Stoch. Partial Differ. Equ. Anal. Comput.*, 7(4):541–597, 2019.

[Saw18] Y. Sawano. *Theory of Besov spaces*, volume 56 of *Developments in Mathematics*. Springer, Singapore, 2018.

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[Tri83] H. Triebel. *Theory of function spaces*, volume 78 of *Monographs in Mathematics*. Birkhäuser Verlag, Basel, 1983.

[Tri20] H. Triebel. *Theory of function spaces IV*, volume 107 of *Monographs in Mathematics*. Birkhäuser/Springer, Cham, 2020.

[Zat20] W. Zatoń. Tent space well-posedness for parabolic Cauchy problems with rough coefficients. *J. Differential Equations*, 269(12):11086–11164, 2020.

UNIVERSITÉ PARIS-SACLAY, CNRS, LABORATOIRE DE MATHÉMATIQUES D'ORSAY, 91405 ORSAY, FRANCE

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

UNIVERSITÉ PARIS-SACLAY, CNRS, LABORATOIRE DE MATHÉMATIQUES D'ORSAY, 91405 ORSAY, FRANCE

Email address: hedong.hou@universite-paris-saclay.fr
[HvNVW17] T. Hytönen, J. van Neerven, M. Veraar, and L. Weis. *Analysis in Banach spaces. Vol. II*, volume 67 of *Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics*. Springer, Cham, 2017. Probabilistic methods and operator theory.

[KM98] N. Kalton and M. Mitrea. Stability results on interpolation scales of quasi-Banach spaces and applications. *Trans. Amer. Math. Soc.*, 350(10):3903–3922, 1998.

[KP93] C. E. Kenig and J. Pipher. The Neumann problem for elliptic equations with nonsmooth coefficients. *Invent. Math.*, 113(3):447–509, 1993.

[KT01] H. Koch and D. Tataru. Well-posedness for the Navier-Stokes equations. *Adv. Math.*, 157(1):22–35, 2001.

[Lio57] J.-L. Lions. Sur les problèmes mixtes pour certains systèmes paraboliques dans des ouverts non cylindriques. *Ann. Inst. Fourier (Grenoble)*, 7:143–182, 1957.

[Lun95] A. Lunardi. *Analytic semigroups and optimal regularity in parabolic problems*, volume 16 of *Progress in Nonlinear Differential Equations and their Applications*. Birkhäuser Verlag, Basel, 1995.

[Mou15] M. Moussai. Realizations of homogeneous Besov and Triebel-Lizorkin spaces and an application to pointwise multipliers. *Anal. Appl. (Sin-gap)*, 13(2):149–183, 2015.

[Pee76] J. Peetre. *New thoughts on Besov spaces*, volume No. 1 of Duke University Mathematics Series. Duke University, Mathematics Department, Durham, NC, 1976.

[PV19] P. Portal and M. Veraar. Stochastic maximal regularity for rough time-dependent problems. *Stoch. Partial Differ. Equ. Anal. Comput.*, 7(4):541–597, 2019.

[Saw18] Y. Sawano. *Theory of Besov spaces*, volume 56 of *Developments in Mathematics*. Springer, Singapore, 2018.

[Str80] R. S. Strichartz. Bounded mean oscillation and Sobolev spaces. *Indi-ana Univ. Math. J.*, 29(4):539–558, 1980.

[Tri83] H. Triebel. *Theory of function spaces*, volume 78 of *Monographs in Mathematics*. Birkhäuser Verlag, Basel, 1983.

[Tri20] H. Triebel. *Theory of function spaces IV*, volume 107 of *Monographs in Mathematics*. Birkhäuser/Springer, Cham, 2020.

[Zat20] W. Zatoń. Tent space well-posedness for parabolic Cauchy problems with rough coefficients. *J. Differential Equations*, 269(12):11086–11164, 2020.

UNIVERSITÉ PARIS-SACLAY, CNRS, LABORATOIRE DE MATHÉMATIQUES D'ORSAY, 91405 ORSAY, FRANCE

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

UNIVERSITÉ PARIS-SACLAY, CNRS, LABORATOIRE DE MATHÉMATIQUES D'ORSAY, 91405 ORSAY, FRANCE

Email address: hedong.hou@universite-paris-saclay.fr