# ON WELL-POSEDNESS FOR PARABOLIC CAUCHY PROBLEMS OF LIONS TYPE WITH ROUGH INITIAL DATA

Pascal Auscher, Hedong Hou

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Pascal Auscher, Hedong Hou. ON WELL-POSEDNESS FOR PARABOLIC CAUCHY PROBLEMS OF LIONS TYPE WITH ROUGH INITIAL DATA. 2024. hal-04620829

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# On well-posedness for parabolic Cauchy problems of Lions type with rough initial data

Pascal Auscher, Hedong Hou

<figure><img src="image_1.png" /></figure>

## To cite this version:

Pascal Auscher, Hedong Hou. On well-posedness for parabolic Cauchy problems of Lions type with rough initial data. Mathematische Annalen, In press. hal-04620829

HAL Id: hal-04620829

https://hal.science/hal-04620829v1

Submitted on 22 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
# ON WELL-POSEDNESS FOR PARABOLIC CAUCHY PROBLEMS OF LIONS TYPE WITH ROUGH INITIAL DATA

PASCAL AUSCHER AND HEDONG HOU

ABSTRACT. We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of $p$ for which tempered distributions in homogeneous Hardy–Sobolev spaces $\dot{H}^{s,p}$ with regularity index $s \in (-1, 1)$ are initial data. Source terms of Lions’ type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces $\dot{B}_{p,p}^s$.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Function spaces</td><td>8</td></tr><tr><td>3. Heat equation</td><td>10</td></tr><tr><td>4. Basic facts about weak solutions</td><td>17</td></tr><tr><td>5. Properties of the Lions' operator</td><td>20</td></tr><tr><td>6. Traces and continuity</td><td>28</td></tr><tr><td>7. Homogeneous Cauchy problem</td><td>32</td></tr><tr><td>8. Uniqueness and representation</td><td>36</td></tr><tr><td>9. Results for homogeneous Besov spaces</td><td>40</td></tr><tr><td>10. An endpoint case $\beta = -1$</td><td>42</td></tr><tr><td>Appendix A. Proof of Proposition 7.4</td><td>45</td></tr><tr><td>References</td><td>49</td></tr></table>

## 1. INTRODUCTION

The goal of this work is to present a complete picture for existence and uniqueness of weak solutions (see Definition 4.1) to the Cauchy problem

$$ (1.1) \qquad \begin{cases} \partial_t u - \operatorname{div}_x A(x) \nabla_x u = f + \operatorname{div}_x F, & (t,x) \in (0, \infty) \times \mathbb{R}^n \\ u(0) = u_0 \end{cases}, $$

using tent spaces introduced by [CMS85] and their weighted generalizations (see Section 2.2 for definition). We assume the coefficient matrix $A \in L^\infty(\mathbb{R}^n; \operatorname{Mat}_n(\mathbb{C}))$

<footnote>Date: June 22, 2024.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. Primary 35K45; Secondary 42B37, 35K05, 42B30, 46E35.</footnote>

<footnote>Key words and phrases. Parabolic Cauchy problems, tent spaces, homogeneous Hardy–Sobolev spaces, heat equation, representation of solutions.</footnote>
# ON WELL-POSEDNESS FOR PARABOLIC CAUCHY PROBLEMS OF LIONS TYPE WITH ROUGH INITIAL DATA

PASCAL AUSCHER AND HEDONG HOU

ABSTRACT. We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of $p$ for which tempered distributions in homogeneous Hardy–Sobolev spaces $\dot{H}^{s,p}$ with regularity index $s \in (-1, 1)$ are initial data. Source terms of Lions’ type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces $\dot{B}_{p,p}^s$.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Function spaces</td><td>8</td></tr><tr><td>3. Heat equation</td><td>10</td></tr><tr><td>4. Basic facts about weak solutions</td><td>17</td></tr><tr><td>5. Properties of the Lions' operator</td><td>20</td></tr><tr><td>6. Traces and continuity</td><td>28</td></tr><tr><td>7. Homogeneous Cauchy problem</td><td>32</td></tr><tr><td>8. Uniqueness and representation</td><td>36</td></tr><tr><td>9. Results for homogeneous Besov spaces</td><td>40</td></tr><tr><td>10. An endpoint case $\beta = -1$</td><td>42</td></tr><tr><td>Appendix A. Proof of Proposition 7.4</td><td>45</td></tr><tr><td>References</td><td>49</td></tr></table>

## 1. INTRODUCTION

The goal of this work is to present a complete picture for existence and uniqueness of weak solutions (see Definition 4.1) to the Cauchy problem

$$ (1.1) \qquad \begin{cases} \partial_t u - \operatorname{div}_x A(x) \nabla_x u = f + \operatorname{div}_x F, & (t, x) \in (0, \infty) \times \mathbb{R}^n \\ u(0) = u_0 \end{cases}, $$

using tent spaces introduced by [CMS85] and their weighted generalizations (see Section 2.2 for definition). We assume the coefficient matrix $A \in L^\infty(\mathbb{R}^n; \operatorname{Mat}_n(\mathbb{C}))$

<footnote>Date: June 22, 2024.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. Primary 35K45; Secondary 42B37, 35K05, 42B30, 46E35.</footnote>

<footnote>Key words and phrases. Parabolic Cauchy problems, tent spaces, homogeneous Hardy–Sobolev spaces, heat equation, representation of solutions.</footnote>
is uniformly elliptic, that is, there are $\Lambda_0, \Lambda_1 > 0$ so that for a.e. $x \in \mathbb{R}^n$ and any $\xi, \eta \in \mathbb{C}^n$,

$$ (1.2) \qquad \Re(\langle A(x)\xi, \xi \rangle) \ge \Lambda_0 |\xi|^2, \quad |\langle A(x)\xi, \eta \rangle| \le \Lambda_1 |\xi||\eta|. $$

By the initial condition $u(0) = u_0$, we require that $u(t)$ tends to $u_0$ as $t \to 0$ in distributional sense, i.e., in $\mathcal{D}'(\mathbb{R}^n)$.

We refer the reader to the introduction of [AMP19] for motivation to include tools from harmonic analysis such as tent spaces when the initial data are in $L^p(\mathbb{R}^n)$ and the coefficients are not regular. There, well-posedness is established for weak solutions $u$ to the equation without source terms (even in the non-autonomous case) in a class defined by the non-tangential maximal Kenig-Pipher norm, as well as estimates of $\nabla u$ in tent spaces $T_0^p$ (with limited but explicit range of $p$). Uniqueness in the latter class is obtained in [Zat20]. It is worth pointing out that $L^p$ is not a trace space, with trace defined in relation to (real) interpolation theory, in contrast to (homogeneous) Besov spaces. So already, the above results do not belong to the realm of maximal regularity using the notion of mild solutions.

Source terms in tent spaces have been considered in nonlinear PDE's, see [KT01, AF17, DV23], and in deterministic estimates used for stochastic PDE's, see [AvNP14, PV19]. More recently, [AP23] shows existence of weak solutions with $\nabla u \in T_0^p$ when $F \in T_0^p$, $f = 0$ and $u_0 = 0$. For source terms $f$ (and $F = 0$), our prior work [AH23b] obtains well-posedness and maximal regularity of $u$ itself in a weighted tent space, the weight being an expression of regularity (see below). In this case, however, the initial data must be zero.

A subsequent forthcoming work of the second author will treat the non-autonomous case.

### 1.1. Main results.
In this paper, we treat initial data in spaces which are not necessarily trace spaces and carry regularity information by control of the gradient of weak solutions in a weighted tent space, where the weight is a power of the time variable. Indeed, we shall show that our approach gives access to initial data in homogeneous Hardy-Sobolev spaces defined by Littlewood-Paley decomposition (see Section 2.1). The regularity range is between $-1$ and $1$. Our method also applies for initial data in homogeneous Besov spaces.

The prototypical example illustrating this is the heat equation.

#### 1.1.1. Heat equation.
Our first result gives a flavor for the homogeneous Cauchy problem (By this, we mean having an initial data but null source terms), and we also provide representation of heat solutions by studying their traces.

**Theorem 1.1** (Heat equation and weighted tent spaces). Let $s \in \mathbb{R}$, $0 < p \le \infty$, and $g \in \mathcal{S}'(\mathbb{R}^n)$.

(i) (Weighted tent-space estimates) Suppose $s < 1$. If $g \in \dot{H}^{s,p}$, then the function $(t, x) \mapsto \nabla e^{t\Delta}g(x)$ belongs to $T_{s/2}^p$ with

$$ \|\nabla e^{t\Delta}g\|_{T_{s/2}^p} \sim \|g\|_{\dot{H}^{s,p}}. $$

(ii) (Representation of heat solutions) Let $u$ be a distributional solution to the heat equation on $\mathbb{R}_+^{1+n}$ with $\nabla u \in T_{s/2}^p$. Suppose $s > -1$ and $\frac{n}{n+s+1} \le p \le \infty$. Then there exists a unique $u_0 \in \mathcal{S}'$ so that $u(t) = e^{t\Delta}u_0$ for all $t > 0$. Moreover,

(1) If $s \ge 1$ and $\frac{n}{n+s-1} \le p \le \infty$, then $u$ is a constant.
is uniformly elliptic, that is, there are $\Lambda_0, \Lambda_1 > 0$ so that for a.e. $x \in \mathbb{R}^n$ and any $\xi, \eta \in \mathbb{C}^n$,

$$ (1.2) \qquad \Re(\langle A(x)\xi, \xi \rangle) \geq \Lambda_0 |\xi|^2, \quad |\langle A(x)\xi, \eta \rangle| \leq \Lambda_1 |\xi||\eta|. $$

By the initial condition $u(0) = u_0$, we require that $u(t)$ tends to $u_0$ as $t \to 0$ in distributional sense, i.e., in $\mathcal{D}'(\mathbb{R}^n)$.

We refer the reader to the introduction of [AMP19] for motivation to include tools from harmonic analysis such as tent spaces when the initial data are in $L^p(\mathbb{R}^n)$ and the coefficients are not regular. There, well-posedness is established for weak solutions $u$ to the equation without source terms (even in the non-autonomous case) in a class defined by the non-tangential maximal Kenig-Pipher norm, as well as estimates of $\nabla u$ in tent spaces $T_0^p$ (with limited but explicit range of $p$). Uniqueness in the latter class is obtained in [Zat20]. It is worth pointing out that $L^p$ is not a trace space, with trace defined in relation to (real) interpolation theory, in contrast to (homogeneous) Besov spaces. So already, the above results do not belong to the realm of maximal regularity using the notion of mild solutions.

Source terms in tent spaces have been considered in nonlinear PDE's, see [KT01, AF17, DV23], and in deterministic estimates used for stochastic PDE's, see [AvNP14, PV19]. More recently, [AP23] shows existence of weak solutions with $\nabla u \in T_0^p$ when $F \in T_0^p$, $f = 0$ and $u_0 = 0$. For source terms $f$ (and $F = 0$), our prior work [AH23b] obtains well-posedness and maximal regularity of $u$ itself in a weighted tent space, the weight being an expression of regularity (see below). In this case, however, the initial data must be zero.

A subsequent forthcoming work of the second author will treat the non-autonomous case.

### 1.1. Main results.
In this paper, we treat initial data in spaces which are not necessarily trace spaces and carry regularity information by control of the gradient of weak solutions in a weighted tent space, where the weight is a power of the time variable. Indeed, we shall show that our approach gives access to initial data in homogeneous Hardy-Sobolev spaces defined by Littlewood-Paley decomposition (see Section 2.1). The regularity range is between $-1$ and $1$. Our method also applies for initial data in homogeneous Besov spaces.

The prototypical example illustrating this is the heat equation.

#### 1.1.1. Heat equation.
Our first result gives a flavor for the homogeneous Cauchy problem (By this, we mean having an initial data but null source terms), and we also provide representation of heat solutions by studying their traces.

**Theorem 1.1** (Heat equation and weighted tent spaces). Let $s \in \mathbb{R}$, $0 < p \leq \infty$, and $g \in \mathcal{S}'(\mathbb{R}^n)$.

(i) (Weighted tent-space estimates) Suppose $s < 1$. If $g \in \dot{H}^{s,p}$, then the function $(t, x) \mapsto \nabla e^{t\Delta}g(x)$ belongs to $T_{s/2}^p$ with

$$ \|\nabla e^{t\Delta}g\|_{T_{s/2}^p} \sim \|g\|_{\dot{H}^{s,p}}. $$

(ii) (Representation of heat solutions) Let $u$ be a distributional solution to the heat equation on $\mathbb{R}_+^{1+n}$ with $\nabla u \in T_{s/2}^p$. Suppose $s > -1$ and $\frac{n}{n+s+1} \leq p \leq \infty$. Then there exists a unique $u_0 \in \mathcal{S}'$ so that $u(t) = e^{t\Delta}u_0$ for all $t > 0$. Moreover,

(1) If $s \geq 1$ and $\frac{n}{n+s-1} \leq p \leq \infty$, then $u$ is a constant.
(2) If $-1 < s < 1$ and $\frac{n}{n+s+1} \le p \le \infty$, then there exist $g \in \dot{H}^{s,p}$ and $c \in \mathbb{C}$ such that $u_0 = g+c$, so $u(t) = e^{t\Delta}g + c$ for all $t > 0$.

The first point (i) shows a precise correspondence between Sobolev regularity of the initial data and the choice of the power weight. The second point (ii) shows the regularity range $s < 1$ is essentially sharp.

When $s = 0$, the first point is reminiscent to the work of Fefferman–Stein [FS72] on defining Hardy spaces by so-called conical square functions when translated to parabolic setting. But their point was to take extensions not related to any kind of equations to obtain an intrinsic definition of Hardy spaces. Here, we have to restrict ourselves to extensions related to the heat equation. It is also reminiscent to Littlewood–Paley theory using rather vertical square functions. The reader can refer to [Tri20, §4.1] for more on this topic.

Combining these two points yields in particular well-posedness of the homogeneous Cauchy problem in the following strong form.

**Corollary 1.2 (Isomorphism).** Let $-1 < s < 1$ and $\frac{n}{n+s+1} \le p \le \infty$. The map $g \mapsto (e^{t\Delta}g)_{t>0}$ is a bijection from $\dot{H}^{s,p} + \mathbb{C}$ onto the space of distributional solutions $u$ to the heat equation with $\nabla u \in T_{s/2}^p$, and

$$ \|g\|_{\dot{H}^{s,p}} \approx \|\nabla u\|_{T_{s/2}^p}. $$

1.1.2. *Critical exponents for the parabolic problem.* To treat the more general parabolic problems (1.1), we introduce the operator $L$

$$ L := -\operatorname{div}(A\nabla) $$

on $L^2(\mathbb{R}^n)$ with domain

$$ D(L) := \{f \in W^{1,2}(\mathbb{R}^n) : \operatorname{div}(A\nabla f) \in L^2(\mathbb{R}^n)\}. $$

By the theory of maximal-accretive operators, $-L$ generates a bounded analytic semigroup $(e^{-tL})_{t \ge 0}$ on $L^2(\mathbb{R}^n)$. Our construction of weak solutions is based on an appropriate extension of the solution map $\mathcal{E}_L$, which is initially defined from $L^2(\mathbb{R}^n)$ to $L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2}(\mathbb{R}^n))$ by the semigroup as

$$ (1.3) \qquad \mathcal{E}_L(u_0)(t, x) := (e^{-tL}u_0)(x). $$

The $L^p$-theory of the semigroup is ruled by four critical numbers, which are introduced in [Aus07, Proposition 3.15] for $1 < p < \infty$, and later extended to $p > \frac{n}{n+1}$ in [AE23, §6] to include Hardy spaces $H^p(\mathbb{R}^n)$. These numbers are

* $p_\pm(L) \in [\frac{n}{n+1}, \infty]$ such that $(p_-(L), p_+(L))$ is the largest open set (an interval) of exponents $p$ for which the semigroup $(e^{-tL})_{t \ge 0}$ is uniformly bounded on $L^p$ when $p > 1$ and on $H^p$ when $p \le 1$;

* $q_\pm(L) \in [\frac{n}{n+1}, \infty]$ such that $(q_-(L), q_+(L))$ is the largest open set (an interval) of exponents $p$ for which the family $(t^{1/2}\nabla e^{-tL})_{t>0}$ is uniformly bounded on $L^p$ when $p > 1$ and on $H^p$ when $p \le 1$.

It is known that $p_-(L) = q_-(L) < \frac{2n}{n+2}$, $q_+(L) > 2$, and $p_+(L) \ge \frac{nq_+(L)}{n-q_+(L)}$ (by convention $p_+(L) = \infty$ if $q_+(L) \ge n$). The strict inequalities are best possible. We also have the duality relation $p_+(L^*) = \max\{p_-(L), 1\}'$, where $p'$ denotes the Hölder conjugate of $p \in [1, \infty]$. For the negative Laplacian, $p_-(-\Delta) = q_-(\Delta) = \frac{n}{n+1}$ and $p_+(\Delta) = q_+(\Delta) = \infty$.
(2) If $-1 < s < 1$ and $\frac{n}{n+s+1} \le p \le \infty$, then there exist $g \in \dot{H}^{s,p}$ and $c \in \mathbb{C}$ such that $u_0 = g + c$, so $u(t) = e^{t\Delta}g + c$ for all $t > 0$.

The first point (i) shows a precise correspondence between Sobolev regularity of the initial data and the choice of the power weight. The second point (ii) shows the regularity range $s < 1$ is essentially sharp.

When $s = 0$, the first point is reminiscent to the work of Fefferman–Stein [FS72] on defining Hardy spaces by so-called conical square functions when translated to parabolic setting. But their point was to take extensions not related to any kind of equations to obtain an intrinsic definition of Hardy spaces. Here, we have to restrict ourselves to extensions related to the heat equation. It is also reminiscent to Littlewood–Paley theory using rather vertical square functions. The reader can refer to [Tri20, §4.1] for more on this topic.

Combining these two points yields in particular well-posedness of the homogeneous Cauchy problem in the following strong form.

**Corollary 1.2 (Isomorphism).** Let $-1 < s < 1$ and $\frac{n}{n+s+1} \le p \le \infty$. The map $g \mapsto (e^{t\Delta}g)_{t>0}$ is a bijection from $\dot{H}^{s,p} + \mathbb{C}$ onto the space of distributional solutions $u$ to the heat equation with $\nabla u \in T_{s/2}^p$, and

$$ \|g\|_{\dot{H}^{s,p}} \approx \|\nabla u\|_{T_{s/2}^p}. $$

1.1.2. *Critical exponents for the parabolic problem.* To treat the more general parabolic problems (1.1), we introduce the operator $L$

$$ L := -\operatorname{div}(A\nabla) $$

on $L^2(\mathbb{R}^n)$ with domain

$$ D(L) := \{f \in W^{1,2}(\mathbb{R}^n) : \operatorname{div}(A\nabla f) \in L^2(\mathbb{R}^n)\}. $$

By the theory of maximal-accretive operators, $-L$ generates a bounded analytic semigroup $(e^{-tL})_{t \ge 0}$ on $L^2(\mathbb{R}^n)$. Our construction of weak solutions is based on an appropriate extension of the solution map $\mathcal{E}_L$, which is initially defined from $L^2(\mathbb{R}^n)$ to $L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2}(\mathbb{R}^n))$ by the semigroup as

$$ (1.3) \qquad \mathcal{E}_L(u_0)(t, x) := (e^{-tL}u_0)(x). $$

The $L^p$-theory of the semigroup is ruled by four critical numbers, which are introduced in [Aus07, Proposition 3.15] for $1 < p < \infty$, and later extended to $p > \frac{n}{n+1}$ in [AE23, §6] to include Hardy spaces $H^p(\mathbb{R}^n)$. These numbers are

* $p_\pm(L) \in [\frac{n}{n+1}, \infty]$ such that $(p_-(L), p_+(L))$ is the largest open set (an interval) of exponents $p$ for which the semigroup $(e^{-tL})_{t \ge 0}$ is uniformly bounded on $L^p$ when $p > 1$ and on $H^p$ when $p \le 1$;

* $q_\pm(L) \in [\frac{n}{n+1}, \infty]$ such that $(q_-(L), q_+(L))$ is the largest open set (an interval) of exponents $p$ for which the family $(t^{1/2}\nabla e^{-tL})_{t>0}$ is uniformly bounded on $L^p$ when $p > 1$ and on $H^p$ when $p \le 1$.

It is known that $p_-(L) = q_-(L) < \frac{2n}{n+2}$, $q_+(L) > 2$, and $p_+(L) \ge \frac{nq_+(L)}{n-q_+(L)}$ (by convention $p_+(L) = \infty$ if $q_+(L) \ge n$). The strict inequalities are best possible. We also have the duality relation $p_+(L^*) = \max\{p_-(L), 1\}'$, where $p'$ denotes the Hölder conjugate of $p \in [1, \infty]$. For the negative Laplacian, $p_-(\Delta) = q_-(\Delta) = \frac{n}{n+1}$ and $p_+(\Delta) = q_+(\Delta) = \infty$.
We extend the critical numbers to our $\dot{H}^{s,p}$-setting. For $-1 \le s \le 1$, define $p_{\pm}(s, L)$ as

$$ (1.4) \qquad \frac{1}{p_{-}(s, L)} := \begin{cases} \frac{1}{p_{-}(L)} + \frac{s}{n} & \text{if } 0 \le s \le 1 \\ \frac{1+s}{p_{-}(L)} - \frac{s}{q_{+}(L^{*})'} & \text{if } -1 \le s \le 0 \end{cases}, $$

and

$$ (1.5) \qquad p_{+}(s, L) := \max\{p_{-}(-s, L^{*}), 1\}^{\prime}. $$

Notice that $p_{\pm}(0, L) = p_{\pm}(L)$, $p_{-}(-1, L) = q_{+}(L^{*})' \in [1, 2)$, and that $p_{+}(1, L) = q_{+}(L) \in (2, \infty]$. In particular, $p_{-}(s, -\Delta) = \frac{n}{n+s+1}$ and $p_{+}(s, -\Delta) = \infty$.

We also introduce several other numbers which will parametrize our results. For convenience, we use a parameter $\beta$ whose relation to the regularity exponent $s$ is given by

$$ s = 2\beta + 1 $$

with $\beta > -1$ and no upper restriction. Define

$$ (1.6) \qquad \beta(L) := -\frac{1}{2} - \frac{n}{2} \left( \frac{1}{p_{-}(L)} - 1 \right) \ge -1. $$

Note that $\beta(L) \ge -1/2$ if and only if $p_{-}(L) \ge 1$. For $\beta > -1$, we introduce the numbers $p_{L}(\beta) \in (0, 2)$ given by

$$ (1.7) \qquad p_{L}(\beta) := \frac{np_{-}(L)}{n + (2\beta + 1)p_{-}(L)}, $$

and also $\tilde{p}_{L}(\beta) \in (0, 2)$ so that:

(i) When $p_{-}(L) \ge 1$, it is given by

$$ (1.8) \qquad \tilde{p}_{L}(\beta) := \begin{cases} p_{L}(\beta) & \text{if } \beta \ge -1/2 \\ p_{-}(2\beta + 1, L) & \text{if } -1 < \beta < -1/2 \end{cases}. $$

(ii) When $p_{-}(L) < 1$, it is given by

$$ (1.9) \qquad \tilde{p}_{L}(\beta) := \begin{cases} p_{L}(\beta) & \text{if } \beta \ge \beta(L) \\ \frac{(\beta(L)+1)q_{+}(L^{*})}{(\beta(L)+1)q_{+}(L^{*})+\beta-\beta(L)} & \text{if } -1 < \beta < \beta(L) \end{cases}. $$

Remark that $\tilde{p}_{L}(\beta(L)) = 1$ and $\tilde{p}_{L}(-1) = q_{+}(L^{*})' = p_{-}(-1, L)$. For the negative Laplacian, as $p_{-}(-\Delta) = \frac{n}{n+1}$, we have

$$ \tilde{p}_{-\Delta}(\beta) = p_{-\Delta}(\beta) = \frac{n}{n + 2\beta + 2}, \quad \forall \beta > -1. $$

Observe that for any $L$,

$$ \tilde{p}_{L}(\beta) \ge p_{L}(\beta) \ge \frac{n}{n + 2\beta + 2}. $$

To illustrate these exponents, we give graphic representations in Figure 1, distinguishing the two cases $p_{-}(L) \ge 1$ and $p_{-}(L) < 1$. In these figures, we write $p$ for $1/p$ to ease the presentation. When $p < 2$, we use red color for the graph of $p_{-}(2\beta + 1, L)$, blue for that of $\tilde{p}_{L}(\beta)$, and orange for that of $p_{L}(\beta)$. The orange shaded trapezoids are the regions of well-posedness for $\dot{H}^{2\beta+1,p}$-initial data, while the gray shaded is for constant initial data. Parallel lines to the orange one are lines of embedding for Hardy-Sobolev spaces and weighted tent spaces going downward.
We extend the critical numbers to our $\dot{H}^{s,p}$-setting. For $-1 \le s \le 1$, define $p_{\pm}(s, L)$ as

$$ (1.4) \qquad \frac{1}{p_{-}(s, L)} := \begin{cases} \frac{1}{p_{-}(L)} + \frac{s}{n} & \text{if } 0 \le s \le 1 \\ \frac{1+s}{p_{-}(L)} - \frac{s}{q_{+}(L^{*})'} & \text{if } -1 \le s \le 0 \end{cases}, $$

and

$$ (1.5) \qquad p_{+}(s, L) := \max\{p_{-}(-s, L^{*}), 1\}^{\prime}. $$

Notice that $p_{\pm}(0, L) = p_{\pm}(L)$, $p_{-}(-1, L) = q_{+}(L^{*})' \in [1, 2)$, and that $p_{+}(1, L) = q_{+}(L) \in (2, \infty]$. In particular, $p_{-}(s, -\Delta) = \frac{n}{n+s+1}$ and $p_{+}(s, -\Delta) = \infty$.

We also introduce several other numbers which will parametrize our results. For convenience, we use a parameter $\beta$ whose relation to the regularity exponent $s$ is given by

$$ s = 2\beta + 1 $$

with $\beta > -1$ and no upper restriction. Define

$$ (1.6) \qquad \beta(L) := -\frac{1}{2} - \frac{n}{2} \left( \frac{1}{p_{-}(L)} - 1 \right) \ge -1. $$

Note that $\beta(L) \ge -1/2$ if and only if $p_{-}(L) \ge 1$. For $\beta > -1$, we introduce the numbers $p_{L}(\beta) \in (0, 2)$ given by

$$ (1.7) \qquad p_{L}(\beta) := \frac{np_{-}(L)}{n + (2\beta + 1)p_{-}(L)}, $$

and also $\tilde{p}_{L}(\beta) \in (0, 2)$ so that:

(i) When $p_{-}(L) \ge 1$, it is given by

$$ (1.8) \qquad \tilde{p}_{L}(\beta) := \begin{cases} p_{L}(\beta) & \text{if } \beta \ge -1/2 \\ p_{-}(2\beta + 1, L) & \text{if } -1 < \beta < -1/2 \end{cases}. $$

(ii) When $p_{-}(L) < 1$, it is given by

$$ (1.9) \qquad \tilde{p}_{L}(\beta) := \begin{cases} p_{L}(\beta) & \text{if } \beta \ge \beta(L) \\ \frac{(\beta(L)+1)q_{+}(L^{*})}{(\beta(L)+1)q_{+}(L^{*})+\beta-\beta(L)} & \text{if } -1 < \beta < \beta(L) \end{cases}. $$

Remark that $\tilde{p}_{L}(\beta(L)) = 1$ and $\tilde{p}_{L}(-1) = q_{+}(L^{*})' = p_{-}(-1, L)$. For the negative Laplacian, as $p_{-}(-\Delta) = \frac{n}{n+1}$, we have

$$ \tilde{p}_{-\Delta}(\beta) = p_{-\Delta}(\beta) = \frac{n}{n + 2\beta + 2}, \quad \forall \beta > -1. $$

Observe that for any $L$,

$$ \tilde{p}_{L}(\beta) \ge p_{L}(\beta) \ge \frac{n}{n + 2\beta + 2}. $$

To illustrate these exponents, we give graphic representations in Figure 1, distinguishing the two cases $p_{-}(L) \ge 1$ and $p_{-}(L) < 1$. In these figures, we write $p$ for $1/p$ to ease the presentation. When $p < 2$, we use red color for the graph of $p_{-}(2\beta + 1, L)$, blue for that of $\tilde{p}_{L}(\beta)$, and orange for that of $p_{L}(\beta)$. The orange shaded trapezoids are the regions of well-posedness for $\dot{H}^{2\beta+1,p}$-initial data, while the gray shaded is for constant initial data. Parallel lines to the orange one are lines of embedding for Hardy-Sobolev spaces and weighted tent spaces going downward.
<figure><img src="image_2.png" /><figcaption>FIGURE 1. Regions of well-posedness</figcaption></figure>

Interestingly, the (smaller) set delimited by the red lines (excluded) and the black lines (included) has a special identification and is called the *identification range* for the operator $L$, and $\mathcal{E}_L$ defined in (1.3) can be extended to a semigroup on $\dot{H}^{s,p}$. The triangular region between the red and blue segments in the case $p_-(L) < 1$ is a region of embedding of an adapted space into $\dot{H}^{s,p}$ without knowing identification. See Section 7 and in particular Proposition 7.4.<fnref n="1" />

1.1.3. *Parabolic equations of type (1.1).* We begin with the homogeneous Cauchy problem ($F = 0, f = 0$).

**Theorem 1.3** (Well-posedness of homogeneous Cauchy problem). Let $-1 < \beta < 0$ and $\tilde{p}_L(\beta) < p \le \infty$. For any $v_0 \in \dot{H}^{2\beta+1,p}$, there is a unique global weak solution $v$ to the homogeneous Cauchy problem

$$ (\text{HC}) \qquad \begin{cases} \partial_t v - \operatorname{div}(A\nabla v) = 0, & (t, x) \in (0, \infty) \times \mathbb{R}^n \\ v(0) = v_0 & \end{cases}, $$

so that $\nabla v \in T_{\beta+1/2}^p$, and one has

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

Moreover, $v$ belongs to $C([0, \infty); \mathcal{S}')$. For $p_-(2\beta + 1, L) < p < p_+(2\beta + 1, L)$, it further holds that $v \in C_0([0, \infty); \dot{H}^{2\beta+1,p}) \cap C^\infty((0, \infty); \dot{H}^{2\beta+1,p})$. <fnref n="2" />

For $2\beta + 1 = 0$, [AMP19] treats this problem in the non-autonomous case. In the autonomous case here, we obtain a larger range.

Also, the last point gives the range in which $\dot{H}^{2\beta+1,p}$-regularity is preserved. In this range with $p \ge 1$ and $2\beta < n/p$,<fnref n="3" /> it also basically says that $v$ is a strong solution of the abstract Cauchy problem $\partial_t v + L_{\beta,p}v = 0, v(0) = v_0$ in the Banach space $\dot{H}^{2\beta+1,p}$ (where $-L_{\beta,p}$ is the generator of the extended semigroup on $\dot{H}^{2\beta+1,p}$), see

<footnote n="1">¹In the case where $p > 2$, the red colored broken lines for the values of $p_+(2\beta + 1, L)$ defined by (1.5) are obtained by symmetry about the point $(1/p, \beta) = (1/2, -1/2)$ from the ones for $\max\{p_-(2\beta - 1, L^*), 1\}$. Thus it depends on the value of $p_-(L^*)$. In the first (resp. second) picture, this corresponds to having $p_-(L^*) \ge 1$ (resp. $p_-(L^*) < 1$). As the values of $p_-(L)$ and $p_-(L^*)$ are independent, we should have in fact 4 such figures.</footnote>

<footnote n="2">²Here, $C_0([0, \infty); E)$ is the space of continuous functions with limit 0 as $t \to \infty$ in the prescribed topology of $E$.</footnote>

<footnote n="3">³When $2\beta - n/p \ge 0$, our space $\dot{H}^{2\beta+1,p}$ is not a Banach space. It is semi-normed.</footnote>
<figure><img src="image_3.png" /><figcaption>FIGURE 1. Regions of well-posedness</figcaption></figure>

Interestingly, the (smaller) set delimited by the red lines (excluded) and the black lines (included) has a special identification and is called the *identification range* for the operator $L$, and $\mathcal{E}_L$ defined in (1.3) can be extended to a semigroup on $\dot{H}^{s,p}$. The triangular region between the red and blue segments in the case $p_-(L) < 1$ is a region of embedding of an adapted space into $\dot{H}^{s,p}$ without knowing identification. See Section 7 and in particular Proposition 7.4.<fnref n="1" />

1.1.3. *Parabolic equations of type (1.1).* We begin with the homogeneous Cauchy problem ($F = 0, f = 0$).

**Theorem 1.3** (Well-posedness of homogeneous Cauchy problem). Let $-1 < \beta < 0$ and $\tilde{p}_L(\beta) < p \le \infty$. For any $v_0 \in \dot{H}^{2\beta+1,p}$, there is a unique global weak solution $v$ to the homogeneous Cauchy problem

$$ (\text{HC}) \qquad \begin{cases} \partial_t v - \operatorname{div}(A\nabla v) = 0, & (t, x) \in (0, \infty) \times \mathbb{R}^n \\ v(0) = v_0 & \end{cases}, $$

so that $\nabla v \in T_{\beta+1/2}^p$, and one has

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

Moreover, $v$ belongs to $C([0, \infty); \mathcal{S}')$. For $p_-(2\beta + 1, L) < p < p_+(2\beta + 1, L)$, it further holds that $v \in C_0([0, \infty); \dot{H}^{2\beta+1,p}) \cap C^\infty((0, \infty); \dot{H}^{2\beta+1,p})$. <fnref n="2" />

For $2\beta + 1 = 0$, [AMP19] treats this problem in the non-autonomous case. In the autonomous case here, we obtain a larger range.

Also, the last point gives the range in which $\dot{H}^{2\beta+1,p}$-regularity is preserved. In this range with $p \ge 1$ and $2\beta < n/p$,<fnref n="3" /> it also basically says that $v$ is a strong solution of the abstract Cauchy problem $\partial_t v + L_{\beta,p}v = 0, v(0) = v_0$ in the Banach space $\dot{H}^{2\beta+1,p}$ (where $-L_{\beta,p}$ is the generator of the extended semigroup on $\dot{H}^{2\beta+1,p}$), see

<footnote n="1">¹In the case where $p > 2$, the red colored broken lines for the values of $p_+(2\beta + 1, L)$ defined by (1.5) are obtained by symmetry about the point $(1/p, \beta) = (1/2, -1/2)$ from the ones for $\max\{p_-(-2\beta - 1, L^*), 1\}$. Thus it depends on the value of $p_-(L^*)$. In the first (resp. second) picture, this corresponds to having $p_-(L^*) \ge 1$ (resp. $p_-(L^*) < 1$). As the values of $p_-(L)$ and $p_-(L^*)$ are independent, we should have in fact 4 such figures.</footnote>

<footnote n="2">²Here, $C_0([0, \infty); E)$ is the space of continuous functions with limit 0 as $t \to \infty$ in the prescribed topology of $E$.</footnote>

<footnote n="3">³When $2\beta - n/p \ge 0$, our space $\dot{H}^{2\beta+1,p}$ is not a Banach space. It is semi-normed.</footnote>
[Lun95, Definition 4.1.1]. Note that $p < 1$ is possible here, even though there is no abstract theory for strong solutions in quasi-Banach spaces as far as we understand. The second result is a converse statement.

**Theorem 1.4 (Representation).** Let $\beta > -1$ and $\tilde{p}_L(\beta) < p \le \infty$. Let $u$ be a weak solution to the equation

$$ \partial_t u - \operatorname{div}(A\nabla u) = 0, \quad (t, x) \in (0, \infty) \times \mathbb{R}^n, $$

with $\nabla u \in T_{\beta+1/2}^p$. Then $u$ has a trace $u_0 \in \mathcal{S}'$ in the sense that $u(t)$ converges to $u_0$ in $\mathcal{S}'$ as $t \to 0$. Moreover,

(i) If $\beta \ge 0$ and $\frac{n}{n+2\beta} \le p \le \infty$, then $u$ is a constant.

(ii) If $-1 < \beta < 0$, then there exist $g \in \dot{H}^{2\beta+1,p}$ and $c \in \mathbb{C}$ so that $u_0 = g + c$ and $u = \mathcal{E}_L(g) + c$, where $\mathcal{E}_L$ is an appropriate extension of the semigroup solution map given by (1.3).

As we shall see in Proposition 6.2, existence of the trace $u_0$ only requires $\tilde{p}_{-\Delta}(\beta) = \frac{n}{n+2\beta+2} < p \le \infty$. The condition $p > \tilde{p}_L(\beta)$ is used to prove the rest. Also note that in (ii), the extension $\mathcal{E}_L$ may not be a semigroup if $p$ does not lie in the interval $(p_-(2\beta + 1, L), p_+(2\beta + 1, L))$.

Theorems 1.3 and 1.4 also go beyond well-posedness of the homogeneous Cauchy problem. For convenience, define $\mathcal{E}_L(1) = 1$ (constant functions).

**Corollary 1.5 (Isomorphism for parabolic equations).** Let $-1 < \beta < 0$ and $\tilde{p}_L(\beta) < p \le \infty$. The map $g \mapsto \mathcal{E}_L(g) = u$ is a bijection from $\dot{H}^{2\beta+1,p} + \mathbb{C}$ onto the space of global weak solutions to $\partial_t u - \operatorname{div}(A\nabla u) = 0$ with $\nabla u \in T_{\beta+1/2}^p$, and

$$ \|g\|_{\dot{H}^{2\beta+1,p}} \stackrel{\sim}{\approx} \|\nabla u\|_{T_{\beta+1/2}^p}. $$

Next, we consider the inhomogeneous Cauchy problem (which means null initial data) in the spirit of Lions' theory with source terms of divergence form (that is, $f = 0$), for which the weak solutions will be constructed by Duhamel's formula.

**Theorem 1.6 (Well-posedness of Lions' equation).** Let $\beta > -1$ and $\tilde{p}_L(\beta) < p \le \infty$. Let $F$ be in $T_{\beta+1/2}^p$. There exists a unique global weak solution $u$ to the Cauchy problem (which we call Lions' equation)

$$ \left\{ \begin{array}{ll} \partial_t u - \operatorname{div}(A\nabla u) = \operatorname{div} F, & (t, x) \in (0, \infty) \times \mathbb{R}^n, \\ u(0) = 0 & \end{array} \right. , $$

with $\nabla u \in T_{\beta+1/2}^p$. Moreover, $u$ belongs to $T_{\beta+1}^p \cap C([0, \infty); \mathcal{S}')$.

To finish this introduction, let us state the announced complete picture for (1.1). For $\gamma > -1/2$, define

$$ (1.10) \qquad p_L^\flat(\gamma) := \frac{n \max\{p_-(L), 1\}}{n + (2\gamma + 1) \max\{p_-(L), 1\}}. $$

**Theorem 1.7 (Well-posedness of Cauchy problems of type (1.1)).** Let $\beta > -1$ and $\tilde{p}_L(\beta) < p \le \infty$. Let $\gamma > -1/2$ and $p_L^\flat(\gamma) < q \le \infty$. Suppose

$$ (1.11) \qquad \gamma \ge \beta \quad \text{and} \quad 2\beta - \frac{n}{p} = 2\gamma - \frac{n}{q}. $$
[Lun95, Definition 4.1.1]. Note that $p < 1$ is possible here, even though there is no abstract theory for strong solutions in quasi-Banach spaces as far as we understand. The second result is a converse statement.

**Theorem 1.4 (Representation).** Let $\beta > -1$ and $\tilde{p}_L(\beta) < p \le \infty$. Let $u$ be a weak solution to the equation

$$ \partial_t u - \operatorname{div}(A\nabla u) = 0, \quad (t, x) \in (0, \infty) \times \mathbb{R}^n, $$

with $\nabla u \in T_{\beta+1/2}^p$. Then $u$ has a trace $u_0 \in \mathcal{S}'$ in the sense that $u(t)$ converges to $u_0$ in $\mathcal{S}'$ as $t \to 0$. Moreover,

(i) If $\beta \ge 0$ and $\frac{n}{n+2\beta} \le p \le \infty$, then $u$ is a constant.

(ii) If $-1 < \beta < 0$, then there exist $g \in \dot{H}^{2\beta+1,p}$ and $c \in \mathbb{C}$ so that $u_0 = g + c$ and $u = \mathcal{E}_L(g) + c$, where $\mathcal{E}_L$ is an appropriate extension of the semigroup solution map given by (1.3).

As we shall see in Proposition 6.2, existence of the trace $u_0$ only requires $\tilde{p}_{-\Delta}(\beta) = \frac{n}{n+2\beta+2} < p \le \infty$. The condition $p > \tilde{p}_L(\beta)$ is used to prove the rest. Also note that in (ii), the extension $\mathcal{E}_L$ may not be a semigroup if $p$ does not lie in the interval $(p_-(2\beta+1, L), p_+(2\beta+1, L))$.

Theorems 1.3 and 1.4 also go beyond well-posedness of the homogeneous Cauchy problem. For convenience, define $\mathcal{E}_L(1) = 1$ (constant functions).

**Corollary 1.5 (Isomorphism for parabolic equations).** Let $-1 < \beta < 0$ and $\tilde{p}_L(\beta) < p \le \infty$. The map $g \mapsto \mathcal{E}_L(g) = u$ is a bijection from $\dot{H}^{2\beta+1,p} + \mathbb{C}$ onto the space of global weak solutions to $\partial_t u - \operatorname{div}(A\nabla u) = 0$ with $\nabla u \in T_{\beta+1/2}^p$, and

$$ \|g\|_{\dot{H}^{2\beta+1,p}} \sim \|\nabla u\|_{T_{\beta+1/2}^p}. $$

Next, we consider the inhomogeneous Cauchy problem (which means null initial data) in the spirit of Lions' theory with source terms of divergence form (that is, $f = 0$), for which the weak solutions will be constructed by Duhamel's formula.

**Theorem 1.6 (Well-posedness of Lions' equation).** Let $\beta > -1$ and $\tilde{p}_L(\beta) < p \le \infty$. Let $F$ be in $T_{\beta+1/2}^p$. There exists a unique global weak solution $u$ to the Cauchy problem (which we call Lions' equation)

$$ \left\{ \begin{array}{ll} \partial_t u - \operatorname{div}(A\nabla u) = \operatorname{div} F, & (t, x) \in (0, \infty) \times \mathbb{R}^n, \\ u(0) = 0 & \end{array} \right. , $$

with $\nabla u \in T_{\beta+1/2}^p$. Moreover, $u$ belongs to $T_{\beta+1}^p \cap C([0, \infty); \mathcal{S}')$.

To finish this introduction, let us state the announced complete picture for (1.1). For $\gamma > -1/2$, define

$$ (1.10) \qquad p_L^\flat(\gamma) := \frac{n \max\{p_-(L), 1\}}{n + (2\gamma + 1) \max\{p_-(L), 1\}}. $$

**Theorem 1.7 (Well-posedness of Cauchy problems of type (1.1)).** Let $\beta > -1$ and $\tilde{p}_L(\beta) < p \le \infty$. Let $\gamma > -1/2$ and $p_L^\flat(\gamma) < q \le \infty$. Suppose

$$ (1.11) \qquad \gamma \ge \beta \quad \text{and} \quad 2\beta - \frac{n}{p} = 2\gamma - \frac{n}{q}. $$
(i) If $\beta < 0$, then for any $u_0 \in \dot{H}^{2\beta+1,p}$, $F \in T_{\beta+1/2}^p$, and $f \in T_\gamma^q$, there exists a unique global weak solution $u$ to the Cauchy problem

$$ \left\{ \begin{array}{l} \partial_t u - \operatorname{div}(A \nabla u) = f + \operatorname{div} F, \quad (t, x) \in (0, \infty) \times \mathbb{R}^n, \\ u(0) = u_0 \end{array} \right. , $$

with $\nabla u \in T_{\beta+1/2}^p$, and one has the estimate

$$ (1.12) \qquad \|\nabla u\|_{T_{\beta+1/2}^p} \lesssim \|u_0\|_{\dot{H}^{2\beta+1,p}} + \|F\|_{T_{\beta+1/2}^p} + \|f\|_{T_\gamma^q}. $$

Moreover, $u$ belongs to $C([0, \infty); \mathcal{S}')$ and $u - \mathcal{E}_L(u_0) \in T_{\beta+1}^p$.

(ii) If $\beta \ge 0$, then the same statement holds when $u_0$ is constant (for which $\|u_0\|_{\dot{H}^{2\beta+1,p}} = 0$).

The proof is a simple combination of Theorems 1.3, 1.6, and [AH23b, Theorem 1.2], where we showed that for source term $f \in T_\gamma^q$ and $\gamma > -1/2$, the corresponding solution $u_1$ satisfies $\nabla u_1 \in T_{\gamma+1/2}^q$. The constraint $\gamma > -1/2$ is essentially sharp. So the conditions on $(\beta, p)$ and $(\gamma, q)$ in (1.11) allow one to use embedding between weighted tent spaces (see Section 2.2) to conclude that $\nabla u_1$ belongs to $T_{\beta+1/2}^p$. The continuity in $\mathcal{S}'$ is not proved there for $u_1$ but it can be shown analogously to the proof of Theorem 5.1 (d).

Moreover, when $\beta \ge 0$, Theorem 1.4 (ii) implies that constant initial data are the only ones compatible with the solution class $\nabla u \in T_{\beta+1/2}^p$.

## 1.2. Organization.

The paper is organized as follows. Section 2 summarizes basic properties of the function spaces to be used, and in particular provides the construction of our homogeneous Hardy–Sobolev spaces $\dot{H}^{s,p}$ in detail.

Section 3 is concerned with the heat equation. We provide the tent-space estimates of heat extensions and their gradients, and also study the representation of heat (distributional) solutions. The combination of these two parts implies Theorem 1.1.

Sections 5, 6 (resp. 7) are devoted to proving existence of the weak solutions asserted in Theorem 1.6 (resp. Theorem 1.3), see Theorem 5.1 (resp. Theorem 7.1). Uniqueness is established in Theorem 8.1. Section 8 also presents the proof of Theorem 1.4.

Section 9 contains extension to initial data in homogeneous Besov spaces, and Section 10 discusses the homogeneous Cauchy problem at the endpoint for regularity index $\beta = -1$.

## 1.3. Notation.

$$ [q, r] := \frac{1}{q} - \frac{1}{r}, $$

if there is no confusion with closed intervals. We say $X \lesssim Y$ (or $X \lesssim_A Y$, resp.) if $X \le CY$ with an irrelevant constant $C$ (or depending on $A$, resp.), and say $X \asymp Y$ if $X \lesssim Y$ and $Y \lesssim X$.

Write $\mathbb{R}_+^{1+n} := \mathbb{R}_+ \times \mathbb{R}^n = (0, \infty) \times \mathbb{R}^n$. For any (Euclidean) ball $B \subset \mathbb{R}^n$, write $r(B)$ for the radius of $B$. For any function $f$ defined on $\mathbb{R}_+^{1+n}$, denote by $f(t)$ the function $x \mapsto f(t, x)$ for any $t > 0$.
(i) If $\beta < 0$, then for any $u_0 \in \dot{H}^{2\beta+1,p}$, $F \in T_{\beta+1/2}^p$, and $f \in T_\gamma^q$, there exists a unique global weak solution $u$ to the Cauchy problem

$$ \left\{ \begin{array}{l} \partial_t u - \operatorname{div}(A \nabla u) = f + \operatorname{div} F, \quad (t, x) \in (0, \infty) \times \mathbb{R}^n, \\ u(0) = u_0 \end{array} \right. , $$

with $\nabla u \in T_{\beta+1/2}^p$, and one has the estimate

$$ (1.12) \qquad \|\nabla u\|_{T_{\beta+1/2}^p} \lesssim \|u_0\|_{\dot{H}^{2\beta+1,p}} + \|F\|_{T_{\beta+1/2}^p} + \|f\|_{T_\gamma^q}. $$

Moreover, $u$ belongs to $C([0, \infty); \mathcal{S}')$ and $u - \mathcal{E}_L(u_0) \in T_{\beta+1}^p$.

(ii) If $\beta \ge 0$, then the same statement holds when $u_0$ is constant (for which $\|u_0\|_{\dot{H}^{2\beta+1,p}} = 0$).

The proof is a simple combination of Theorems 1.3, 1.6, and [AH23b, Theorem 1.2], where we showed that for source term $f \in T_\gamma^q$ and $\gamma > -1/2$, the corresponding solution $u_1$ satisfies $\nabla u_1 \in T_{\gamma+1/2}^q$. The constraint $\gamma > -1/2$ is essentially sharp. So the conditions on $(\beta, p)$ and $(\gamma, q)$ in (1.11) allow one to use embedding between weighted tent spaces (see Section 2.2) to conclude that $\nabla u_1$ belongs to $T_{\beta+1/2}^p$. The continuity in $\mathcal{S}'$ is not proved there for $u_1$ but it can be shown analogously to the proof of Theorem 5.1 (d).

Moreover, when $\beta \ge 0$, Theorem 1.4 (ii) implies that constant initial data are the only ones compatible with the solution class $\nabla u \in T_{\beta+1/2}^p$.

## 1.2. Organization.

The paper is organized as follows. Section 2 summarizes basic properties of the function spaces to be used, and in particular provides the construction of our homogeneous Hardy–Sobolev spaces $\dot{H}^{s,p}$ in detail.

Section 3 is concerned with the heat equation. We provide the tent-space estimates of heat extensions and their gradients, and also study the representation of heat (distributional) solutions. The combination of these two parts implies Theorem 1.1.

Sections 5, 6 (resp. 7) are devoted to proving existence of the weak solutions asserted in Theorem 1.6 (resp. Theorem 1.3), see Theorem 5.1 (resp. Theorem 7.1). Uniqueness is established in Theorem 8.1. Section 8 also presents the proof of Theorem 1.4.

Section 9 contains extension to initial data in homogeneous Besov spaces, and Section 10 discusses the homogeneous Cauchy problem at the endpoint for regularity index $\beta = -1$.

## 1.3. Notation.

$$ [q, r] := \frac{1}{q} - \frac{1}{r}, $$

if there is no confusion with closed intervals. We say $X \lesssim Y$ (or $X \lesssim_A Y$, resp.) if $X \le CY$ with an irrelevant constant $C$ (or depending on $A$, resp.), and say $X \asymp Y$ if $X \lesssim Y$ and $Y \lesssim X$.

Write $\mathbb{R}_+^{1+n} := \mathbb{R}_+ \times \mathbb{R}^n = (0, \infty) \times \mathbb{R}^n$. For any (Euclidean) ball $B \subset \mathbb{R}^n$, write $r(B)$ for the radius of $B$. For any function $f$ defined on $\mathbb{R}_+^{1+n}$, denote by $f(t)$ the function $x \mapsto f(t, x)$ for any $t > 0$.
Let $(X, \mu)$ be a measure space. For any measurable subset $E \subset X$ with finite positive measure and $f \in L^1(E, \mu)$, we write

$$ \int_E f d\mu := \frac{1}{\mu(E)} \int_E f d\mu. $$

Write $\| \cdot \|_p$ as an abbreviation for the norm $\| \cdot \|_{L^p(X, \mu)}$. For any $\beta \in \mathbb{R}$ and $E \subset \mathbb{R}^{1+n}$, denote by $L_{\beta}^p(E)$ the space $L^p(E, t^{-p\beta}dtdy)$.

We use the sans-serif font c in the scripts of function spaces in short of “with compact support” in the prescribed set, and loc if the prescribed property holds on all compact subsets of the prescribed set. Often, we omit the domain of the function space if it is clear from the context.

## 2. FUNCTION SPACES

We begin with introducing the spaces for the initial data and next, the spaces for the solutions. Denote by $\mathcal{S}$ the space of Schwartz functions on $\mathbb{R}^n$ and by $\mathcal{S}'$ the space of tempered distributions. For any $f \in \mathcal{S}'$, write $\hat{f}$ or $\mathcal{F}(f)$ for the Fourier transform of $f$.

### 2.1. Homogeneous Hardy-Sobolev spaces.
Our homogeneous Hardy-Sobolev spaces can be roughly regarded as “realizations” of Triebel-Lizorkin spaces in $\mathcal{S}'$. We follow the formulation in [Saw18, §2.4.3], which was originated from an observation of J. Peetre [Pee76, p.56], see also [Bou13, Mou15].

Denote by $C$ the annulus $\{\xi \in \mathbb{R}^n : 2^{-1} \le |\xi| \le 2^2\}$ and by $2^jC$ the annulus $\{\xi \in \mathbb{R}^n : 2^{j-1} \le |\xi| \le 2^{j+2}\}$ for any $j \in \mathbb{Z}$. Let $\chi$ be in $C_c^\infty(\mathbb{R}^n)$ so that $\text{supp}(\chi) \subset C$ and for any $\xi \ne 0$,

$$ \sum_{j \in \mathbb{Z}} \chi(2^{-j}\xi) = 1. $$

Let $\Delta_j$ be the $j$-th Littlewood-Paley operator associated with $\chi$, given by

$$ (2.1) \qquad \Delta_j f := \mathcal{F}^{-1}(\chi(2^{-j}\cdot)\mathcal{F}(f)), \quad \forall f \in \mathcal{S}'. $$

Suppose $s \in \mathbb{R}$ and $0 < p \le \infty$. Let $\mathcal{S}_{s,p}$ be the collection of sequences of measurable functions $(f_j)_{j \in \mathbb{Z}}$ on $\mathbb{R}^n$ so that

$$ (2.2) \qquad \|(f_j)\|_{\mathcal{S}_{s,p}} := \left\| \left( \sum_j |2^{js} f_j|^2 \right)^{1/2} \right\|_p < \infty. $$

It is clear that $f_j \in L^p$ for any $j \in \mathbb{Z}$, whenever $(f_j) \in \mathcal{S}_{s,p}$.

Let $\mathcal{P}$ be the space of polynomials $C[x_1, \dots, x_n]$, $\mathcal{P}_0 := \{0\}$, and $\mathcal{P}_m$ be the subspace of $\mathcal{P}$ consisting of polynomials of degree less than $m$ for $m \ge 1$. For $p \ne \infty$, the Triebel-Lizorkin space $\dot{F}_{p,2}^s$ consists of $f \in \mathcal{S}'/\mathcal{P}$ for which

$$ \|f\|_{\dot{F}_{p,2}^s} := \|(\Delta_j f)\|_{\mathcal{S}_{s,p}} < \infty. $$

For $p = \infty$, the space $\dot{F}_{\infty,2}^s$ consists of $f \in \mathcal{S}'/\mathcal{P}$ so that there exists $(f_j) \in \mathcal{S}_{s,\infty}$ satisfying that $f = \sum_j \Delta_j f_j$ in $\mathcal{S}'/\mathcal{P}$. The norm is given by

$$ \|f\|_{\dot{F}_{\infty,2}^s} := \inf \|(f_j)\|_{\mathcal{S}_{s,\infty}}, $$

where the infimum is taken among all $(f_j) \in \mathcal{S}_{s,\infty}$ so that $\sum_j \Delta_j f_j = f$ in $\mathcal{S}'/\mathcal{P}$.
Let $(X, \mu)$ be a measure space. For any measurable subset $E \subset X$ with finite positive measure and $f \in L^1(E, \mu)$, we write

$$ \int_E f d\mu := \frac{1}{\mu(E)} \int_E f d\mu. $$

Write $\| \cdot \|_p$ as an abbreviation for the norm $\| \cdot \|_{L^p(X, \mu)}$. For any $\beta \in \mathbb{R}$ and $E \subset \mathbb{R}^{1+n}$, denote by $L_p^\beta(E)$ the space $L^p(E, t^{-p\beta}dtdy)$.

We use the sans-serif font c in the scripts of function spaces in short of “with compact support” in the prescribed set, and loc if the prescribed property holds on all compact subsets of the prescribed set. Often, we omit the domain of the function space if it is clear from the context.

## 2. FUNCTION SPACES

We begin with introducing the spaces for the initial data and next, the spaces for the solutions. Denote by $\mathcal{S}$ the space of Schwartz functions on $\mathbb{R}^n$ and by $\mathcal{S}'$ the space of tempered distributions. For any $f \in \mathcal{S}'$, write $\hat{f}$ or $\mathcal{F}(f)$ for the Fourier transform of $f$.

### 2.1. Homogeneous Hardy-Sobolev spaces.
Our homogeneous Hardy-Sobolev spaces can be roughly regarded as “realizations” of Triebel-Lizorkin spaces in $\mathcal{S}'$. We follow the formulation in [Saw18, §2.4.3], which was originated from an observation of J. Peetre [Pee76, p.56], see also [Bou13, Mou15].

Denote by $C$ the annulus $\{\xi \in \mathbb{R}^n : 2^{-1} \le |\xi| \le 2^2\}$ and by $2^jC$ the annulus $\{\xi \in \mathbb{R}^n : 2^{j-1} \le |\xi| \le 2^{j+2}\}$ for any $j \in \mathbb{Z}$. Let $\chi$ be in $C_c^\infty(\mathbb{R}^n)$ so that $\text{supp}(\chi) \subset C$ and for any $\xi \ne 0$,

$$ \sum_{j \in \mathbb{Z}} \chi(2^{-j}\xi) = 1. $$

Let $\Delta_j$ be the $j$-th Littlewood-Paley operator associated with $\chi$, given by

$$ (2.1) \qquad \Delta_j f := \mathcal{F}^{-1}(\chi(2^{-j} \cdot) \mathcal{F}(f)), \quad \forall f \in \mathcal{S}'. $$

Suppose $s \in \mathbb{R}$ and $0 < p \le \infty$. Let $\mathcal{S}_{s,p}$ be the collection of sequences of measurable functions $(f_j)_{j \in \mathbb{Z}}$ on $\mathbb{R}^n$ so that

$$ (2.2) \qquad \|(f_j)\|_{\mathcal{S}_{s,p}} := \left\| \left( \sum_j |2^{js} f_j|^2 \right)^{1/2} \right\|_p < \infty. $$

It is clear that $f_j \in L^p$ for any $j \in \mathbb{Z}$, whenever $(f_j) \in \mathcal{S}_{s,p}$.

Let $\mathcal{P}$ be the space of polynomials $C[x_1, \dots, x_n]$, $\mathcal{P}_0 := \{0\}$, and $\mathcal{P}_m$ be the subspace of $\mathcal{P}$ consisting of polynomials of degree less than $m$ for $m \ge 1$. For $p \ne \infty$, the Triebel-Lizorkin space $\dot{F}_{p,2}^s$ consists of $f \in \mathcal{S}'/\mathcal{P}$ for which

$$ \|f\|_{\dot{F}_{p,2}^s} := \|(\Delta_j f)\|_{\mathcal{S}_{s,p}} < \infty. $$

For $p = \infty$, the space $\dot{F}_{\infty,2}^s$ consists of $f \in \mathcal{S}'/\mathcal{P}$ so that there exists $(f_j) \in \mathcal{S}_{s,\infty}$ satisfying that $f = \sum_j \Delta_j f_j$ in $\mathcal{S}'/\mathcal{P}$. The norm is given by

$$ \|f\|_{\dot{F}_{\infty,2}^s} := \inf \|(f_j)\|_{\mathcal{S}_{s,\infty}}, $$

where the infimum is taken among all $(f_j) \in \mathcal{S}_{s,\infty}$ so that $\sum_j \Delta_j f_j = f$ in $\mathcal{S}'/\mathcal{P}$.
Define $\nu(s,p) := \max\{0, [s - \frac{n}{p}] + 1\}$. For any $f \in \dot{F}_{p,2}^s$, the Littlewood-Paley series $\sum_j \Delta_j f$ converges in $\mathcal{S}'/\mathcal{P}_{\nu(s,p)}$. Moreover, it induces an isometric embedding $\iota: \dot{F}_{p,2}^s \to \mathcal{S}'/\mathcal{P}_{\nu(s,p)}$ given by

$$ \iota(f) := \sum_j \Delta_j f. $$

**Definition 2.1 (Homogeneous Hardy-Sobolev spaces).** Let $s \in \mathbb{R}$ and $0 < p \le \infty$. The homogeneous Hardy-Sobolev space $\dot{H}^{s,p}$ consists of $f \in \mathcal{S}'$ whose class in $\mathcal{S}'/\mathcal{P}_{\nu(s,p)}$ belongs to $\iota(\dot{F}_{p,2}^s)$. The (quasi-)semi-norm of $\dot{H}^{s,p}$ is given by

$$ \|f\|_{\dot{H}^{s,p}} := \|[f]\|_{\dot{F}_{p,2}^s}, $$

where $[f]$ denotes the class of $f$ in $\mathcal{S}'/\mathcal{P}$.

Note that for $s = 0$, up to equivalent (quasi-)norms, $\dot{H}^{0,p}$ agrees with the Hardy space $H^p$ if $p \le 1$, the Lebesgue space $L^p$ if $1 < p < \infty$, and BMO if $p = \infty$. It follows by classical results of Littlewood-Paley theory, see e.g., [Gra14, Theorem 1.3.8] and [Tri83, §5.2.4]. Moreover, for any $s \in \mathbb{R}$, $\dot{H}^{s,\infty}/\mathcal{P}_{\nu(s,\infty)}$ is isomorphic to $\text{BMO}^s$ introduced by Strichartz [Str80].

The reader can refer to [AE23, §2.5] and the references there for basic properties of the spaces $\dot{H}^{s,p}$, for instance, duality, complex interpolation, lifting property, and Sobolev embedding. In particular, let us mention a dense subspace of $\dot{H}^{s,p}$. Denote by $\mathcal{S}_\infty$ the subspace of $\mathcal{S}$ consisting of $\phi \in \mathcal{S}$ so that for any multi-index $\alpha$, $\partial^\alpha \hat{\phi}(0) = 0$. It is a dense subspace of $\dot{H}^{s,p}$ if $p < \infty$, or weak*-dense if $p = \infty$.

**2.2. Tent spaces and slice spaces.** We adapt the original definition of (un-weighted) tent spaces in [CMS85] to our parabolic setting. For any $\beta \in \mathbb{R}$ and $p \in (0, \infty)$, the (parabolic) tent space $T_\beta^p$ consists of (possibly $\mathbb{C}^n$-valued) measurable functions $F$ on $\mathbb{R}_+^{1+n}$ for which

$$ \|F\|_{T_\beta^p} := \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x,t^{1/2})} |t^{-\beta} F(t,y)|^2 dt dy \right)^{p/2} dx \right)^{1/p} < \infty. $$

For $0 < p \le 1$, the space $T_{\beta,([p,1])}^\infty$ consists of measurable functions $F$ on $\mathbb{R}_+^{1+n}$ for which

$$ \|F\|_{T_{\beta,([p,1])}^{\infty}} := \sup_{B} \frac{1}{|B|^{[p,1]}} \left( \int_{0}^{r(B)^2} \int_{B} |t^{-\beta} F(t,y)|^2 dt dy \right)^{1/2} < \infty, $$

where the supremum is taken over all balls $B$ of $\mathbb{R}^n$. We also write $T_\beta^\infty$ for $T_{\beta, (0)}^\infty$. Recall that $[p, 1]$ is defined in Section 1.3 by $\frac{1}{p} - 1$.

The reader can refer to [AH23b, §3.1] for a short summary of basic properties of tent spaces, such as duality and a dense class. Here we also recall two properties:

(i) (Interpolation) Let $\beta_0, \beta_1 \in \mathbb{R}$, $0 < p_0, p_1 \le \infty$ and $\theta \in (0, 1)$. If $1/p = (1-\theta)/p_0 + \theta/p_1$ and $\beta = (1-\theta)\beta_0 + \theta\beta_1$, then the complex interpolation space $[T_{\beta_0}^{p_0}, T_{\beta_1}^{p_1}]_\theta$ (in the sense of Kalton-Mitrea method, see [KM98]) identifies with $T_\beta^p$. See [Ame18, Theorem 2.12].

(ii) (Embedding) Let $\beta_0 > \beta_1$ and $0 < p_0 < p_1 \le \infty$. Suppose

$$ 2\beta_0 - \frac{n}{p_0} = 2\beta_1 - \frac{n}{p_1}. $$
Define $\nu(s,p) := \max\{0, [s - \frac{n}{p}] + 1\}$. For any $f \in \dot{F}_{p,2}^s$, the Littlewood-Paley series $\sum_j \Delta_j f$ converges in $\mathcal{S}'/\mathcal{P}_{\nu(s,p)}$. Moreover, it induces an isometric embedding $\iota: \dot{F}_{p,2}^s \to \mathcal{S}'/\mathcal{P}_{\nu(s,p)}$ given by

$$ \iota(f) := \sum_j \Delta_j f. $$

**Definition 2.1 (Homogeneous Hardy-Sobolev spaces).** Let $s \in \mathbb{R}$ and $0 < p \le \infty$. The homogeneous Hardy-Sobolev space $\dot{H}^{s,p}$ consists of $f \in \mathcal{S}'$ whose class in $\mathcal{S}'/\mathcal{P}_{\nu(s,p)}$ belongs to $\iota(\dot{F}_{p,2}^s)$. The (quasi-)semi-norm of $\dot{H}^{s,p}$ is given by

$$ \|f\|_{\dot{H}^{s,p}} := \|[f]\|_{\dot{F}_{p,2}^s}, $$

where $[f]$ denotes the class of $f$ in $\mathcal{S}'/\mathcal{P}$.

Note that for $s = 0$, up to equivalent (quasi-)norms, $\dot{H}^{0,p}$ agrees with the Hardy space $H^p$ if $p \le 1$, the Lebesgue space $L^p$ if $1 < p < \infty$, and BMO if $p = \infty$. It follows by classical results of Littlewood-Paley theory, see e.g., [Gra14, Theorem 1.3.8] and [Tri83, §5.2.4]. Moreover, for any $s \in \mathbb{R}$, $\dot{H}^{s,\infty}/\mathcal{P}_{\nu(s,\infty)}$ is isomorphic to $\text{BMO}^s$ introduced by Strichartz [Str80].

The reader can refer to [AE23, §2.5] and the references there for basic properties of the spaces $\dot{H}^{s,p}$, for instance, duality, complex interpolation, lifting property, and Sobolev embedding. In particular, let us mention a dense subspace of $\dot{H}^{s,p}$. Denote by $\mathcal{S}_\infty$ the subspace of $\mathcal{S}$ consisting of $\phi \in \mathcal{S}$ so that for any multi-index $\alpha$, $\partial^\alpha \hat{\phi}(0) = 0$. It is a dense subspace of $\dot{H}^{s,p}$ if $p < \infty$, or weak*-dense if $p = \infty$.

**2.2. Tent spaces and slice spaces.** We adapt the original definition of (un-weighted) tent spaces in [CMS85] to our parabolic setting. For any $\beta \in \mathbb{R}$ and $p \in (0, \infty)$, the (parabolic) tent space $T_\beta^p$ consists of (possibly $\mathbb{C}^n$-valued) measurable functions $F$ on $\mathbb{R}_+^{1+n}$ for which

$$ \|F\|_{T_\beta^p} := \left( \int_{\mathbb{R}^n} \left( \int_0^\infty \int_{B(x,t^{1/2})} |t^{-\beta} F(t,y)|^2 dt dy \right)^{p/2} dx \right)^{1/p} < \infty. $$

For $0 < p \le 1$, the space $T_{\beta,([p,1])}^\infty$ consists of measurable functions $F$ on $\mathbb{R}_+^{1+n}$ for which

$$ \|F\|_{T_{\beta,([p,1])}^{\infty}} := \sup_B \frac{1}{|B|^{[p,1]}} \left( \int_0^{r(B)^2} \int_B |t^{-\beta} F(t,y)|^2 dt dy \right)^{1/2} < \infty, $$

where the supremum is taken over all balls $B$ of $\mathbb{R}^n$. We also write $T_\beta^\infty$ for $T_{\beta, (0)}^\infty$. Recall that $[p, 1]$ is defined in Section 1.3 by $\frac{1}{p} - 1$.

The reader can refer to [AH23b, §3.1] for a short summary of basic properties of tent spaces, such as duality and a dense class. Here we also recall two properties:

(i) (Interpolation) Let $\beta_0, \beta_1 \in \mathbb{R}$, $0 < p_0, p_1 \le \infty$ and $\theta \in (0, 1)$. If $1/p = (1 - \theta)/p_0 + \theta/p_1$ and $\beta = (1 - \theta)\beta_0 + \theta\beta_1$, then the complex interpolation space $[T_{\beta_0}^{p_0}, T_{\beta_1}^{p_1}]_\theta$ (in the sense of Kalton-Mitrea method, see [KM98]) identifies with $T_\beta^p$. See [Ame18, Theorem 2.12].

(ii) (Embedding) Let $\beta_0 > \beta_1$ and $0 < p_0 < p_1 \le \infty$. Suppose

$$ 2\beta_0 - \frac{n}{p_0} = 2\beta_1 - \frac{n}{p_1}. $$
Then $T_{\beta_0}^{p_0}$ embeds into $T_{\beta_1}^{p_1}$. See [Ame18, Theorem 2.19].

We shall also use another family of spaces studied in [AM19, §3], called slice spaces. Let $1 \le p \le \infty$ and $\delta > 0$. The (parabolic) slice space $E_{\delta}^p$ consists of measurable (possibly $C^n$-valued) functions $f$ on $\mathbb{R}^n$ for which

$$ \|f\|_{E_{\delta}^p} := \left\| \left( \fint_{B(\cdot, \delta^{1/2})} |f(y)|^2 dy \right)^{1/2} \right\|_p < \infty. $$

We also define the space $E_{\delta}^{1,p}$ as the collection of $L_{\text{loc}}^2$-functions $g$ on $\mathbb{R}^n$ so that $\nabla g$ (in the sense of distributions) belongs to $E_{\delta}^p$, equipped with the semi-norm

$$ \|g\|_{E_{\delta}^{1,p}} := \|\nabla g\|_{E_{\delta}^p}. $$

Also define $E_{\delta}^{-1,p} := \{g \in \mathcal{D}'(\mathbb{R}^n) : g = \text{div}(G) \text{ for some } G \in E_{\delta}^p\}$ with the norm

$$ \|g\|_{E_{\delta}^{-1,p}} := \inf\{\|G\|_{E_{\delta}^p} : G \in E_{\delta}^p, g = \text{div} G\}. $$

The $L^2$-inner product on $\mathbb{R}^n$ realizes $E_{\delta}^{p'}$ (resp. $E_{\delta}^{-1,p'}$) as the dual of $E_{\delta}^p$ (resp. $E_{\delta}^{1,p}$) when $1 \le p < \infty$.

### 3. HEAT EQUATION

In this section, we consider the Cauchy problem of the heat equation

$$ \begin{cases} \partial_t u - \Delta u = 0 & \text{in } \mathcal{D}'(\mathbb{R}_+^{1+n}) \\ u(0) = f & \end{cases}. $$

Recall that any distributional solution to the heat equation on $\mathbb{R}_+^{1+n}$ is in fact smooth by hypoellipticity, see for instance [Hör03, §4.4].

**3.1. Heat extension and homogeneous Hardy–Sobolev spaces.** For any $f \in \mathcal{S}'$, the function $(t, x) \mapsto (e^{t\Delta}f)(x)$ belongs to $C^{\infty}(\mathbb{R}_+^{1+n}) \cap C([0, \infty); \mathcal{S}')$. We call it the heat extension of $f$, denoted by $\mathcal{E}_{-\Delta}(f)$.

**Proposition 3.1** (Heat extension on $\dot{H}^{s,p}$). Let $s \in \mathbb{R}$ and $0 < p \le \infty$. Then $\mathcal{E}_{-\Delta}$ is a bounded and continuous map from $\dot{H}^{s,p}$ to $C_0([0, \infty); \dot{H}^{s,p}) \cap C^{\infty}((0, \infty); \dot{H}^{s,p})$ with the estimate

$$ (3.1) \qquad \sup_{t \ge 0} \| \mathcal{E}_{-\Delta}(f)(t) \|_{\dot{H}^{s,p}} \lesssim \| f \|_{\dot{H}^{s,p}}. $$

Here, for $p = \infty$, the space $\dot{H}^{s,\infty}$ is equipped with the weak*-topology against elements of $\dot{H}^{-s,1}$.

**Proof.** First consider the case $0 < p < \infty$. Let $(\Delta_j)$ be the Littlewood–Paley operator defined in (2.1). It is well-known that for any $s \in \mathbb{R}$, there are implicit constants such that for any $f \in \mathcal{S}_{\infty}$,

$$ \sup_{t \ge 0} \left\| \left( \sum_{j \in \mathbb{Z}} |2^{js} \Delta_j (e^{t\Delta} f)|^2 \right)^{1/2} \right\|_p \lesssim \left\| \left( \sum_{j \in \mathbb{Z}} |2^{js} \Delta_j f|^2 \right)^{1/2} \right\|_p. $$

By density, this inequality extends to all $\dot{H}^{s,p}$, so we have

$$ \sup_{t \ge 0} \| \mathcal{E}_{-\Delta}(f)(t) \|_{\dot{H}^{s,p}} \lesssim \| f \|_{\dot{H}^{s,p}}. $$
Then $T_{\beta_0}^{p_0}$ embeds into $T_{\beta_1}^{p_1}$. See [Ame18, Theorem 2.19].

We shall also use another family of spaces studied in [AM19, §3], called slice spaces. Let $1 \le p \le \infty$ and $\delta > 0$. The (parabolic) slice space $E_{\delta}^p$ consists of measurable (possibly $C^n$-valued) functions $f$ on $\mathbb{R}^n$ for which

$$ \|f\|_{E_{\delta}^p} := \left\| \left( \fint_{B(\cdot, \delta^{1/2})} |f(y)|^2 dy \right)^{1/2} \right\|_p < \infty. $$

We also define the space $E_{\delta}^{1,p}$ as the collection of $L_{\text{loc}}^2$-functions $g$ on $\mathbb{R}^n$ so that $\nabla g$ (in the sense of distributions) belongs to $E_{\delta}^p$, equipped with the semi-norm

$$ \|g\|_{E_{\delta}^{1,p}} := \|\nabla g\|_{E_{\delta}^{p}}. $$

Also define $E_{\delta}^{-1,p} := \{g \in \mathcal{D}'(\mathbb{R}^n) : g = \text{div}(G) \text{ for some } G \in E_{\delta}^p\}$ with the norm

$$ \|g\|_{E_{\delta}^{-1,p}} := \inf\{\|G\|_{E_{\delta}^{p}} : G \in E_{\delta}^{p}, g = \text{div} G\}. $$

The $L^2$-inner product on $\mathbb{R}^n$ realizes $E_{\delta}^{p'}$ (resp. $E_{\delta}^{-1,p'}$) as the dual of $E_{\delta}^p$ (resp. $E_{\delta}^{1,p}$) when $1 \le p < \infty$.

### 3. HEAT EQUATION

In this section, we consider the Cauchy problem of the heat equation

$$ \begin{cases} \partial_t u - \Delta u = 0 & \text{in } \mathcal{D}'(\mathbb{R}_+^{1+n}) \\ u(0) = f & \end{cases}. $$

Recall that any distributional solution to the heat equation on $\mathbb{R}_+^{1+n}$ is in fact smooth by hypoellipticity, see for instance [Hör03, §4.4].

**3.1. Heat extension and homogeneous Hardy–Sobolev spaces.** For any $f \in \mathcal{S}'$, the function $(t, x) \mapsto (e^{t\Delta}f)(x)$ belongs to $C^{\infty}(\mathbb{R}_+^{1+n}) \cap C([0, \infty); \mathcal{S}')$. We call it the heat extension of $f$, denoted by $\mathcal{E}_{-\Delta}(f)$.

**Proposition 3.1** (Heat extension on $\dot{H}^{s,p}$). Let $s \in \mathbb{R}$ and $0 < p \le \infty$. Then $\mathcal{E}_{-\Delta}$ is a bounded and continuous map from $\dot{H}^{s,p}$ to $C_0([0, \infty); \dot{H}^{s,p}) \cap C^{\infty}((0, \infty); \dot{H}^{s,p})$ with the estimate

$$ (3.1) \qquad \sup_{t \ge 0} \| \mathcal{E}_{-\Delta}(f)(t) \|_{\dot{H}^{s,p}} \lesssim \|f\|_{\dot{H}^{s,p}}. $$

Here, for $p = \infty$, the space $\dot{H}^{s,\infty}$ is equipped with the weak*-topology against elements of $\dot{H}^{-s,1}$.

**Proof.** First consider the case $0 < p < \infty$. Let $(\Delta_j)$ be the Littlewood–Paley operator defined in (2.1). It is well-known that for any $s \in \mathbb{R}$, there are implicit constants such that for any $f \in \mathcal{S}_{\infty}$,

$$ \sup_{t \ge 0} \left\| \left( \sum_{j \in \mathbb{Z}} |2^{js} \Delta_j (e^{t\Delta} f)|^2 \right)^{1/2} \right\|_p \lesssim \left\| \left( \sum_{j \in \mathbb{Z}} |2^{js} \Delta_j f|^2 \right)^{1/2} \right\|_p. $$

By density, this inequality extends to all $\dot{H}^{s,p}$, so we have

$$ \sup_{t \ge 0} \| \mathcal{E}_{-\Delta}(f)(t) \|_{\dot{H}^{s,p}} \lesssim \|f\|_{\dot{H}^{s,p}}. $$
Moreover, one can find that $\mathcal{E}_{-\Delta}$ is a bounded map from $\mathcal{S}_{\infty}$ to $C_0([0, \infty); \mathcal{S}_{\infty}) \cap C^{\infty}((0, \infty); \mathcal{S}_{\infty})$. Hence, by density, it is bounded from $\dot{H}^{s,p}$ to $C_0([0, \infty); \dot{H}^{s,p}) \cap C^{\infty}((0, \infty); \dot{H}^{s,p})$.

For $p = \infty$, we proceed by weak*-duality to obtain boundedness and regularity. This completes the proof. □

We now provide an intermediate result describing when the heat extension of a tempered distribution belongs to a weighted tent space in terms of its Hardy–Sobolev regularity. To the best of our knowledge, such a precise result and the next ones below do not appear in the literature.

**Theorem 3.2** (Characterization of $\dot{H}^{s,p}$ via heat extension). Let $s \in \mathbb{R}$ and $0 < p \le \infty$.

(i) Suppose $s < 0$. If $f \in \dot{H}^{s,p}$ (in particular $f \in \mathcal{S}'$), then $\mathcal{E}_{-\Delta}(f)$ belongs to $T_{(s+1)/2}^p$ with the estimate

$$ (3.2) \qquad \|\mathcal{E}_{-\Delta}(f)\|_{T_{(s+1)/2}^p} \lesssim \|f\|_{\dot{H}^{s,p}}. $$

(ii) Suppose $f \in \mathcal{S}'$ and $\mathcal{E}_{-\Delta}(f)$ belongs to $T_{(s+1)/2}^p$.

(1) If $s \ge 0$ and $\frac{n}{n+s} \le p \le \infty$, then $f = 0$.

(2) If $s < 0$, then $f$ belongs to $\dot{H}^{s,p}$ with the estimate

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

Consequently, when $s < 0$, a tempered distribution $f \in \mathcal{S}'$ lies in $\dot{H}^{s,p}$ if and only if $\mathcal{E}_{-\Delta}(f)$ lies in $T_{(s+1)/2}^p$, with the equivalence of norms as

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

**Remark 3.3.** The restriction on $s$ in (i) is as in Triebel–Lizorkin theory. For (ii), whenever $s \in \mathbb{R}$, $\mathcal{E}_{-\Delta}(f) \in T_{(s+1)/2}^p$ implies that there is a representative in the class of $f$ in $\mathcal{S}'/\mathcal{P}$ that belongs to $\dot{H}^{s,p}$. But for $s \ge 0$ and $p < \frac{n}{n+s}$, we do not know any better conclusion.

In the sequel, for a function $F(t, x)$, we also write $\|F(t)\|_{T_{\beta}^p}$ for $\|F\|_{T_{\beta}^p}$ to emphasize the time variable.

**Proof.** Let us start from some general facts on relations of Hardy spaces and tent spaces. Let $\psi$ be in $C^{\infty}(\mathbb{R}^n)$ satisfying that

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

(b) there exists some $\varepsilon > 0$ so that for any multi-index $\alpha$,

$$ (3.3) \qquad |\partial_x^\alpha \psi(x)| \lesssim (1 + |x|)^{-n-\varepsilon-|\alpha|}. $$

Set $\psi_t(x) := t^{-n/2}\psi(t^{-1/2}x)$. One can deduce from classical arguments that the map $Q_{\psi}$ defined by

$$ Q_{\psi}(f)(t, x) := (\psi_t * f)(x) $$

is bounded from $\dot{H}^{0,p}$ to $T_{1/2}^p$, for $0 < p \le \infty$, or equivalently, from $H^p$ to $T_{1/2}^p$ when $0 < p \le 1$, from $L^p$ to $T_{1/2}^p$ when $1 < p < \infty$, and from BMO to $T_{1/2}^{\infty}$.

Indeed, for $p = 2$, it is direct from Fourier transform as

$$ \|Q_{\psi} f\|_{T_{1/2}^2} \approx \|Q_{\psi} f\|_{L_{1/2}^2(\mathbb{R}_+^{1+n})} \approx \int_{\mathbb{R}^n} \left( \int_0^\infty |\hat{\psi}(t^{1/2}\xi)|^2 \frac{dt}{t} \right) |\hat{f}(\xi)|^2 d\xi \approx \|f\|_2^2. $$
Moreover, one can find that $\mathcal{E}_{-\Delta}$ is a bounded map from $\mathcal{S}_{\infty}$ to $C_0([0, \infty); \mathcal{S}_{\infty}) \cap C^{\infty}((0, \infty); \mathcal{S}_{\infty})$. Hence, by density, it is bounded from $\dot{H}^{s,p}$ to $C_0([0, \infty); \dot{H}^{s,p}) \cap C^{\infty}((0, \infty); \dot{H}^{s,p})$.

For $p = \infty$, we proceed by weak*-duality to obtain boundedness and regularity. This completes the proof. □

We now provide an intermediate result describing when the heat extension of a tempered distribution belongs to a weighted tent space in terms of its Hardy–Sobolev regularity. To the best of our knowledge, such a precise result and the next ones below do not appear in the literature.

**Theorem 3.2** (Characterization of $\dot{H}^{s,p}$ via heat extension). Let $s \in \mathbb{R}$ and $0 < p \le \infty$.

(i) Suppose $s < 0$. If $f \in \dot{H}^{s,p}$ (in particular $f \in \mathcal{S}'$), then $\mathcal{E}_{-\Delta}(f)$ belongs to $T_{(s+1)/2}^p$ with the estimate

$$ (3.2) \qquad \|\mathcal{E}_{-\Delta}(f)\|_{T_{(s+1)/2}^p} \lesssim \|f\|_{\dot{H}^{s,p}}. $$

(ii) Suppose $f \in \mathcal{S}'$ and $\mathcal{E}_{-\Delta}(f)$ belongs to $T_{(s+1)/2}^p$.

(1) If $s \ge 0$ and $\frac{n}{n+s} \le p \le \infty$, then $f = 0$.

(2) If $s < 0$, then $f$ belongs to $\dot{H}^{s,p}$ with the estimate

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

Consequently, when $s < 0$, a tempered distribution $f \in \mathcal{S}'$ lies in $\dot{H}^{s,p}$ if and only if $\mathcal{E}_{-\Delta}(f)$ lies in $T_{(s+1)/2}^p$, with the equivalence of norms as

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

**Remark 3.3.** The restriction on $s$ in (i) is as in Triebel–Lizorkin theory. For (ii), whenever $s \in \mathbb{R}$, $\mathcal{E}_{-\Delta}(f) \in T_{(s+1)/2}^p$ implies that there is a representative in the class of $f$ in $\mathcal{S}'/\mathcal{P}$ that belongs to $\dot{H}^{s,p}$. But for $s \ge 0$ and $p < \frac{n}{n+s}$, we do not know any better conclusion.

In the sequel, for a function $F(t, x)$, we also write $\|F(t)\|_{T_{\beta}^p}$ for $\|F\|_{T_{\beta}^p}$ to emphasize the time variable.

**Proof.** Let us start from some general facts on relations of Hardy spaces and tent spaces. Let $\psi$ be in $C^{\infty}(\mathbb{R}^n)$ satisfying that

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

(b) there exists some $\varepsilon > 0$ so that for any multi-index $\alpha$,

$$ (3.3) \qquad |\partial_x^\alpha \psi(x)| \lesssim (1 + |x|)^{-n-\varepsilon-|\alpha|}. $$

Set $\psi_t(x) := t^{-n/2}\psi(t^{-1/2}x)$. One can deduce from classical arguments that the map $Q_{\psi}$ defined by

$$ Q_{\psi}(f)(t, x) := (\psi_t * f)(x) $$

is bounded from $\dot{H}^{0,p}$ to $T_{1/2}^p$, for $0 < p \le \infty$, or equivalently, from $H^p$ to $T_{1/2}^p$ when $0 < p \le 1$, from $L^p$ to $T_{1/2}^p$ when $1 < p < \infty$, and from BMO to $T_{1/2}^{\infty}$.

Indeed, for $p = 2$, it is direct from Fourier transform as

$$ \|Q_{\psi} f\|_{T_{1/2}^2} \approx \|Q_{\psi} f\|_{L_{1/2}^2(\mathbb{R}_+^{1+n})} \approx \int_{\mathbb{R}^n} \left( \int_0^\infty |\hat{\psi}(t^{1/2}\xi)|^2 \frac{dt}{t} \right) |\hat{f}(\xi)|^2 d\xi \approx \|f\|_2^2. $$
For $p = \infty$, it holds by a well-known Carleson measure argument, see for instance, [Gra14, Theorem 3.3.8(c)]. For $0 < p \le 1$, it follows from atomic decomposition of tent spaces (see [CMS85, Proposition 5]), using decay and regularity of the kernel $t^{-1/2}\psi(t^{-1/2}(x-y))$ given by (3.3) and the moment conditions of $H^p$-atoms. Details are left to the reader. The rest follows by interpolation.

Moreover, let $\phi \in \mathcal{S}_{\infty}$. For $0 < p < \infty$, we infer from [CMS85, Theorem 6] (adapted to the parabolic scaling) that for any $F \in T_{1/2}^p$, the integral

$$S_{\phi}(F) := \int_{0}^{\infty} \phi_t * F(t) \frac{dt}{t}$$

converges in $\mathcal{S}'$, and hence induces a bounded operator from $T_{1/2}^p$ to $L^p$ if $1 < p < \infty$ and to $H^p$ if $0 < p \le 1$. For $p = \infty$, $S_{\phi}(F)$ is defined in BMO by testing against $h \in H^1$ using

$$\langle S_{\phi}(F), h \rangle := \iint_{\mathbb{R}_{+}^{1+n}} F(t, x) \overline{Q_{\tilde{\phi}}h(t, x)} \frac{dtdx}{t},$$

where $\tilde{\phi}(x) := \overline{\phi(-x)}$. Since $\tilde{\phi}$ also satisfies the conditions (a) and (b), by duality, one can easily get $S_{\phi}$ is bounded from $T_{1/2}^{\infty}$ to BMO.

Then we proceed to prove (i), assuming $s < 0$. For $0 < p < \infty$, we use density. Pick $f \in \mathcal{S}_{\infty}$ and define $g := (-\Delta)^{-s/2}f$, so we have for any $t > 0$,

$$\mathcal{E}_{-\Delta}(f)(t) = e^{t\Delta}f = (-\Delta)^{s/2}e^{t\Delta}g = t^{s/2}(-t\Delta)^{-s/2}e^{t\Delta}g = t^{s/2}\psi_t * g,$$

where $\psi$ denotes the kernel of $(-\Delta)^{-s/2}e^{\Delta}$. One can easily verify that $\psi$ satisfies the conditions (a) and (b) with $\varepsilon = -s > 0$. Then we get

$$\|\mathcal{E}_{-\Delta}(f)\|_{T_{(s+1)/2}^p} = \|Q_{\psi}(g)\|_{T_{1/2}^p} \lesssim \|g\|_{\dot{H}^{0,p}} = \|(-\Delta)^{-s/2}f\|_{\dot{H}^{0,p}} \asymp \|f\|_{\dot{H}^{s,p}}.$$

Since $\mathcal{S}_{\infty}$ is dense in $\dot{H}^{s,p}$, this inequality extends to all $f \in \dot{H}^{s,p}$.

For $p = \infty$, we use weak*-density. For any $f \in \dot{H}^{s,\infty}$, pick $(f_k)$ as a sequence in $\mathcal{S}_{\infty}$ that converges weakly* to $f$ in $\dot{H}^{s,\infty}$, hence in $\mathcal{S}'$, since $\dot{H}^{s,\infty}$ embeds into $\mathcal{S}'$ when $s < 0$. Using the above computation, we have

$$\|\mathcal{E}_{-\Delta}(f_k)\|_{T_{(s+1)/2}^{\infty}} \lesssim \|f_k\|_{\dot{H}^{s,\infty}}.$$

In particular, $(\mathcal{E}_{-\Delta}(f_k))$ is bounded in $T_{(s+1)/2}^{\infty}$. Let $F$ be a weak*-accumulation point of $(\mathcal{E}_{-\Delta}(f_k))$ in $T_{(s+1)/2}^{\infty}$. Let $G \in C_c^{\infty}(\mathbb{R}_+^{1+n})$. Then

$$\iint_{\mathbb{R}_{+}^{1+n}} (e^{t\Delta}f_k)(x) \overline{G}(t, x) dtdx = \int_{0}^{\infty} \langle f_k, e^{t\Delta}G(t) \rangle dt,$$

where the pairing is in the sense of tempered distributions and Schwartz functions. As $t \mapsto e^{t\Delta}G(t) \in C((0, \infty); \mathcal{S})$ and the integral is supported on a compact subset of $(0, \infty)$, we obtain at the limit

$$\iint_{\mathbb{R}_{+}^{1+n}} F(t, x) \overline{G}(t, x) dtdx = \int_{0}^{\infty} \langle f, e^{t\Delta}G(t) \rangle dt.$$

So $F = \mathcal{E}_{-\Delta}(f)$ and $\|\mathcal{E}_{-\Delta}(f)\|_{T_{(s+1)/2}^{\infty}} \lesssim \|f\|_{\dot{H}^{s,\infty}}$. This finishes the proof of (i).
For $p = \infty$, it holds by a well-known Carleson measure argument, see for instance, [Gra14, Theorem 3.3.8(c)]. For $0 < p \le 1$, it follows from atomic decomposition of tent spaces (see [CMS85, Proposition 5]), using decay and regularity of the kernel $t^{-1/2}\psi(t^{-1/2}(x-y))$ given by (3.3) and the moment conditions of $H^p$-atoms. Details are left to the reader. The rest follows by interpolation.

Moreover, let $\phi \in \mathcal{S}_{\infty}$. For $0 < p < \infty$, we infer from [CMS85, Theorem 6] (adapted to the parabolic scaling) that for any $F \in T_{1/2}^p$, the integral

$$S_{\phi}(F) := \int_{0}^{\infty} \phi_t * F(t) \frac{dt}{t}$$

converges in $\mathcal{S}'$, and hence induces a bounded operator from $T_{1/2}^p$ to $L^p$ if $1 < p < \infty$ and to $H^p$ if $0 < p \le 1$. For $p = \infty$, $S_{\phi}(F)$ is defined in BMO by testing against $h \in H^1$ using

$$\langle S_{\phi}(F), h \rangle := \iint_{\mathbb{R}_{+}^{1+n}} F(t, x) \overline{Q_{\tilde{\phi}}h(t, x)} \frac{dtdx}{t},$$

where $\tilde{\phi}(x) := \overline{\phi(-x)}$. Since $\tilde{\phi}$ also satisfies the conditions (a) and (b), by duality, one can easily get $S_{\phi}$ is bounded from $T_{1/2}^{\infty}$ to BMO.

Then we proceed to prove (i), assuming $s < 0$. For $0 < p < \infty$, we use density. Pick $f \in \mathcal{S}_{\infty}$ and define $g := (-\Delta)^{-s/2}f$, so we have for any $t > 0$,

$$\mathcal{E}_{-\Delta}(f)(t) = e^{t\Delta}f = (-\Delta)^{s/2}e^{t\Delta}g = t^{s/2}(-t\Delta)^{-s/2}e^{t\Delta}g = t^{s/2}\psi_t * g,$$

where $\psi$ denotes the kernel of $(-\Delta)^{-s/2}e^{\Delta}$. One can easily verify that $\psi$ satisfies the conditions (a) and (b) with $\varepsilon = -s > 0$. Then we get

$$\|\mathcal{E}_{-\Delta}(f)\|_{T_{(s+1)/2}^p} = \|Q_{\psi}(g)\|_{T_{1/2}^p} \lesssim \|g\|_{\dot{H}^{0,p}} = \|(-\Delta)^{-s/2}f\|_{\dot{H}^{0,p}} \asymp \|f\|_{\dot{H}^{s,p}}.$$

Since $\mathcal{S}_{\infty}$ is dense in $\dot{H}^{s,p}$, this inequality extends to all $f \in \dot{H}^{s,p}$.

For $p = \infty$, we use weak*-density. For any $f \in \dot{H}^{s,\infty}$, pick $(f_k)$ as a sequence in $\mathcal{S}_{\infty}$ that converges weakly* to $f$ in $\dot{H}^{s,\infty}$, hence in $\mathcal{S}'$, since $\dot{H}^{s,\infty}$ embeds into $\mathcal{S}'$ when $s < 0$. Using the above computation, we have

$$\|\mathcal{E}_{-\Delta}(f_k)\|_{T_{(s+1)/2}^{\infty}} \lesssim \|f_k\|_{\dot{H}^{s,\infty}}.$$

In particular, $(\mathcal{E}_{-\Delta}(f_k))$ is bounded in $T_{(s+1)/2}^{\infty}$. Let $F$ be a weak*-accumulation point of $(\mathcal{E}_{-\Delta}(f_k))$ in $T_{(s+1)/2}^{\infty}$. Let $G \in C_c^{\infty}(\mathbb{R}_+^{1+n})$. Then

$$\iint_{\mathbb{R}_{+}^{1+n}} (e^{t\Delta}f_k)(x) \overline{G}(t, x) dtdx = \int_{0}^{\infty} \langle f_k, e^{t\Delta}G(t) \rangle dt,$$

where the pairing is in the sense of tempered distributions and Schwartz functions. As $t \mapsto e^{t\Delta}G(t) \in C((0, \infty); \mathcal{S})$ and the integral is supported on a compact subset of $(0, \infty)$, we obtain at the limit

$$\iint_{\mathbb{R}_{+}^{1+n}} F(t, x) \overline{G}(t, x) dtdx = \int_{0}^{\infty} \langle f, e^{t\Delta}G(t) \rangle dt.$$

So $F = \mathcal{E}_{-\Delta}(f)$ and $\|\mathcal{E}_{-\Delta}(f)\|_{T_{(s+1)/2}^{\infty}} \lesssim \|f\|_{\dot{H}^{s,\infty}}$. This finishes the proof of (i).
Next, we prove (ii) and begin with (2). Let $s < 0$ and $f \in \mathcal{S}'$ with $F(t, x) := (t^{-s/2}e^{t\Delta}f)(x) \in T_{1/2}^p$. Pick $\phi \in \mathcal{S}_\infty$ with the non-degeneracy condition

$$ (3.4) \qquad \int_0^\infty \hat{\phi}(r\xi)|r\xi|^{-s}e^{-|r\xi|^2}\frac{dr}{r} = 1, \quad \forall \xi \neq 0. $$

We know from the above discussion that $g := S_\phi(F)$ lies in $\dot{H}^{0,p}$. Define $\psi := (-\Delta)^{s/2}\phi$. Computing in $\mathcal{S}'/\mathcal{P}$ and using (3.4), we have

$$ (-\Delta)^{s/2}g = S_\psi(e^{t\Delta}f) = f \quad \text{in } \mathcal{S}'/\mathcal{P}. $$

Thus, there is a representative $\tilde{f} \in \mathcal{S}'$ in the class of $f$ in $\mathcal{S}'/\mathcal{P}$ which also lies in $\dot{H}^{s,p}$. By (i), we know that $e^{t\Delta}\tilde{f}$ lies in $T_{(s+1)/2}^p$, thus, so does $e^{t\Delta}(\tilde{f} - f)$. As $\tilde{f} - f$ is a polynomial, $e^{t\Delta}(\tilde{f} - f)(x) = P(t, x)$ is also a polynomial, and it is easy to see that $P(t, x) = 0$ from $\|P\|_{T_{(s+1)/2}^p} < \infty$. Finally, as $e^{t\Delta}(\tilde{f} - f) \to \tilde{f} - f$ in $\mathcal{S}'$ when $t \to 0$, it follows that $\tilde{f} = f$. This proves (2).

To establish (1), we divide the discussion into two cases.

Case 1: $(s, p) \neq (0, \infty)$. For $0 < a < 1$, define

$$ I(a) := \int_a^{2a} \langle e^{t\Delta}f, \phi \rangle dt. $$

We claim that $I(a)$ tends to 0 as $a \to 0$. Meanwhile, $\langle e^{t\Delta}f, \phi \rangle$ tends to $\langle f, \phi \rangle$ as $t \to 0$, so we get $f = 0$. Let us verify the claim. For $N \ge 0$, denote by $\mathcal{P}_N$ the semi-norm on $\mathcal{S}$ defined by

$$ (3.5) \qquad \mathcal{P}_N(\phi) := \sup_{|\alpha|+|\gamma| \le N} \sup_{x \in \mathbb{R}^n} |x^\alpha \partial^\gamma \phi(x)|. $$

For $1 < p \le \infty$, pick $N > n/p'$ so that

$$ \begin{aligned} \|\mathbf{1}_{(a,2a)}\phi\|_{T_{-(s+1)/2}^{p'}} & \lesssim a^{\frac{s}{2}+1} \left( \int_{\mathbb{R}^n} \langle x \rangle^{-Np'} \left( \sup_{y \in B(x,(2a)^{1/2})} \langle y \rangle^N |\phi(y)| \right)^{p'} dx \right)^{1/p'} \\ & \lesssim a^{\frac{s}{2}+1} \mathcal{P}_N(\phi), \end{aligned} $$

where $\langle x \rangle$ is the Japanese bracket, i.e., $\langle x \rangle := (1 + |x|^2)^{1/2}$. We get

$$ \begin{aligned} I(a) & \lesssim a^{-1} \|\mathbf{1}_{(a,2a)} e^{t\Delta} f\|_{T_{(s+1)/2}^p} \|\mathbf{1}_{(a,2a)} \phi\|_{T_{-(s+1)/2}^{p'}} \\ & \lesssim a^{\frac{s}{2}} \mathcal{P}_N(\phi) \|\mathbf{1}_{(a,2a)} e^{t\Delta} f\|_{T_{(s+1)/2}^p}. \end{aligned} $$

When $s > 0$, it follows that $I(a)$ tends to 0 if $a \to 0$ as $\|\mathbf{1}_{(a,2a)}e^{t\Delta}f\|_{T_{(s+1)/2}^p} \le \|e^{t\Delta}f\|_{T_{(s+1)/2}^p}$. When $s = 0$ but $p \ne \infty$, it also holds as $\|\mathbf{1}_{(a,2a)}e^{t\Delta}f\|_{T_{(s+1)/2}^p}$ tends to 0 if $a \to 0$.

For $\frac{n}{n+s} \le p \le 1$, i.e., $s - n[p, 1] \ge 0$, note that

$$ \|\mathbf{1}_{(a,2a)}\phi\|_{T_{-(s+1)/2},([p,1])} \lesssim a \mathcal{P}_0(\phi), $$

so the claim follows as

$$ I(a) \lesssim a^{-1} \|\mathbf{1}_{(a,2a)} e^{t\Delta} f\|_{T_{(s+1)/2}^p} \|\mathbf{1}_{(a,2a)} \phi\|_{T_{-(s+1)/2},([p,1])} \lesssim \phi \|\mathbf{1}_{(a,2a)} e^{t\Delta} f\|_{T_{(s+1)/2}^p}, $$
Next, we prove (ii) and begin with (2). Let $s < 0$ and $f \in \mathcal{S}'$ with $F(t, x) := (t^{-s/2}e^{t\Delta}f)(x) \in T_{1/2}^p$. Pick $\phi \in \mathcal{S}_\infty$ with the non-degeneracy condition

$$ (3.4) \qquad \int_0^\infty \hat{\phi}(r\xi)|r\xi|^{-s}e^{-|r\xi|^2}\frac{dr}{r} = 1, \quad \forall \xi \neq 0. $$

We know from the above discussion that $g := S_\phi(F)$ lies in $\dot{H}^{0,p}$. Define $\psi := (-\Delta)^{s/2}\phi$. Computing in $\mathcal{S}'/\mathcal{P}$ and using (3.4), we have

$$ (-\Delta)^{s/2}g = S_\psi(e^{t\Delta}f) = f \quad \text{in } \mathcal{S}'/\mathcal{P}. $$

Thus, there is a representative $\tilde{f} \in \mathcal{S}'$ in the class of $f$ in $\mathcal{S}'/\mathcal{P}$ which also lies in $\dot{H}^{s,p}$. By (i), we know that $e^{t\Delta}\tilde{f}$ lies in $T_{(s+1)/2}^p$, thus, so does $e^{t\Delta}(\tilde{f} - f)$. As $\tilde{f} - f$ is a polynomial, $e^{t\Delta}(\tilde{f} - f)(x) = P(t, x)$ is also a polynomial, and it is easy to see that $P(t, x) = 0$ from $\|P\|_{T_{(s+1)/2}^p} < \infty$. Finally, as $e^{t\Delta}(\tilde{f} - f) \to \tilde{f} - f$ in $\mathcal{S}'$ when $t \to 0$, it follows that $\tilde{f} = f$. This proves (2).

To establish (1), we divide the discussion into two cases.

Case 1: $(s, p) \neq (0, \infty)$. For $0 < a < 1$, define

$$ I(a) := \int_a^{2a} \langle e^{t\Delta}f, \phi \rangle dt. $$

We claim that $I(a)$ tends to 0 as $a \to 0$. Meanwhile, $\langle e^{t\Delta}f, \phi \rangle$ tends to $\langle f, \phi \rangle$ as $t \to 0$, so we get $f = 0$. Let us verify the claim. For $N \ge 0$, denote by $\mathcal{P}_N$ the semi-norm on $\mathcal{S}$ defined by

$$ (3.5) \qquad \mathcal{P}_N(\phi) := \sup_{|\alpha|+|\gamma| \le N} \sup_{x \in \mathbb{R}^n} |x^\alpha \partial^\gamma \phi(x)|. $$

For $1 < p \le \infty$, pick $N > n/p'$ so that

$$ \begin{aligned} \|\mathbf{1}_{(a,2a)}\phi\|_{T_{-(s+1)/2}^{p'}} & \lesssim a^{\frac{s}{2}+1} \left( \int_{\mathbb{R}^n} \langle x \rangle^{-Np'} \left( \sup_{y \in B(x,(2a)^{1/2})} \langle y \rangle^N |\phi(y)| \right)^{p'} dx \right)^{1/p'} \\ & \lesssim a^{\frac{s}{2}+1} \mathcal{P}_N(\phi), \end{aligned} $$

where $\langle x \rangle$ is the Japanese bracket, i.e., $\langle x \rangle := (1 + |x|^2)^{1/2}$. We get

$$ \begin{aligned} I(a) & \lesssim a^{-1} \|\mathbf{1}_{(a,2a)} e^{t\Delta} f\|_{T_{(s+1)/2}^p} \|\mathbf{1}_{(a,2a)} \phi\|_{T_{-(s+1)/2}^{p'}} \\ & \lesssim a^{\frac{s}{2}} \mathcal{P}_N(\phi) \|\mathbf{1}_{(a,2a)} e^{t\Delta} f\|_{T_{(s+1)/2}^p}. \end{aligned} $$

When $s > 0$, it follows that $I(a)$ tends to 0 if $a \to 0$ as $\|\mathbf{1}_{(a,2a)}e^{t\Delta}f\|_{T_{(s+1)/2}^p} \le \|e^{t\Delta}f\|_{T_{(s+1)/2}^p}$. When $s = 0$ but $p \ne \infty$, it also holds as $\|\mathbf{1}_{(a,2a)}e^{t\Delta}f\|_{T_{(s+1)/2}^p}$ tends to 0 if $a \to 0$.

For $\frac{n}{n+s} \le p \le 1$, i.e., $s - n[p, 1] \ge 0$, note that

$$ \|\mathbf{1}_{(a,2a)}\phi\|_{T_{-(s+1)/2},([p,1])} \lesssim a \mathcal{P}_0(\phi), $$

so the claim follows as

$$ I(a) \lesssim a^{-1} \|\mathbf{1}_{(a,2a)} e^{t\Delta} f\|_{T_{(s+1)/2}^p} \|\mathbf{1}_{(a,2a)} \phi\|_{T_{-(s+1)/2},([p,1])} \lesssim_{\phi} \|\mathbf{1}_{(a,2a)} e^{t\Delta} f\|_{T_{(s+1)/2}^p}, $$
which also tends to 0 when $a \to 0$.

Case 2: $s = 0$ and $p = \infty$. For any $\phi \in C_c^\infty(\mathbb{R}^n)$, pick a ball $B \subset \mathbb{R}^n$ containing $\text{supp}(\phi)$. Note that

$$ \int_0^{r(B)^2} |\langle e^{t\Delta} f, \phi \rangle|^2 \frac{dt}{t} \lesssim \|e^{t\Delta} f\|_{T_{1/2}^\infty}^2 \mathcal{P}_0(\phi)^2 |B|^2 < \infty. $$

The fact that $\langle e^{t\Delta}f, \phi \rangle$ tends to $\langle f, \phi \rangle$ if $t \to 0$ forces $f = 0$ for the integral to converge, so $f = 0$ in $\mathcal{D}'(\mathbb{R}^n)$. This completes the proof. $\square$

Replacing the heat extension by its gradient gives us the following

**Corollary 3.4** (Characterization of $\dot{H}^{s,p}$ via gradient of heat extension). Let $s \in \mathbb{R}$ and $0 < p \le \infty$.

(i) Suppose $s < 1$ and $f \in \dot{H}^{s,p}$. Then $\nabla \mathcal{E}_{-\Delta}(f)$ lies in $T_{s/2}^p$ with

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

(ii) Suppose $f \in \mathcal{S}'$ with $\nabla \mathcal{E}_{-\Delta}(f) \in T_{s/2}^p$.

(1) If $s \ge 1$ and $\frac{n}{n+s-1} \le p \le \infty$, then $f$ is a constant.

(2) If $s < 1$, then there exists some constant $c \in \mathbb{C}$ so that $f - c \in \dot{H}^{s,p}$ with

$$ \|f - c\|_{\dot{H}^{s,p}} \approx \|\nabla \mathcal{E}_{-\Delta}(f)\|_{T_{s/2}^p}. $$

Consequently, when $s < 1$, a tempered distribution $f \in \mathcal{S}'$ belongs to $\dot{H}^{s,p} + \mathbb{C}$ if and only if $\nabla \mathcal{E}_{-\Delta}(f)$ lies in $T_{s/2}^p$, with the equivalence of norms as

$$ \inf_{c \in \mathbb{C}} \|f - c\|_{\dot{H}^{s,p}} \approx \|\nabla \mathcal{E}_{-\Delta}(f)\|_{T_{s/2}^p}. $$

*Proof.* A tempered distribution $f \in \mathcal{S}'$ lies in $\dot{H}^{s,p} + \mathbb{C}$ if and only if $\nabla f \in \dot{H}^{s-1,p}$, with the equivalence $\inf_{c \in \mathbb{C}} \|f - c\|_{\dot{H}^{s,p}} \approx \|\nabla f\|_{\dot{H}^{s-1,p}}$. So the corollary follows from applying Theorem 3.2 to $\nabla f$ instead of $f$ and using the above equivalence. $\square$

In fact, working modulo constants, one can equip $\dot{H}^{s,p} + \mathbb{C}$ with the “norm modulo constants” given by $\inf_{c \in \mathbb{C}} \|f - c\|_{\dot{H}^{s,p}}$. When $\nu(s,p) \ge 0$, constants are contained in $\dot{H}^{s,p}$, so $\dot{H}^{s,p} + \mathbb{C} = \dot{H}^{s,p}$, and for any $c \in \mathbb{C}$, $\|f - c\|_{\dot{H}^{s,p}} = \|f\|_{\dot{H}^{s,p}}$.

**3.2. Representation of heat solutions.** We prove here Theorem 1.1. In Corollary 3.4, we started from an element of $\dot{H}^{s,p}$ and described in which range of exponents this is equivalent to the gradient of its heat extension being in a weighted tent space. The question is now whether all heat solutions are of this form. We prove this is true in some (different) range of exponents, by establishing a trace result followed by a representation via heat semigroup.

**Proposition 3.5** (Representation of heat solutions). Let $s > -1$ and $\frac{n}{n+s+1} \le p \le \infty$. Let $u$ be a distributional solution to the heat equation on $\mathbb{R}_+^{1+n}$ with $\nabla u \in T_{s/2}^p$. Then there exists a unique $u_0 \in \mathcal{S}'$ so that $u(t) = e^{t\Delta}u_0$ for all $t > 0$.

*Remark 3.6.* Note that we work in $\mathcal{S}'$, and not just modulo polynomials. This induces restrictions on $s$ and $p$ in the proof.

Let us start the proof by a lemma on the growth of local $L^2$-norms of functions with $\nabla u \in T_{s/2}^p$.
which also tends to 0 when $a \to 0$.

Case 2: $s = 0$ and $p = \infty$. For any $\phi \in C_c^\infty(\mathbb{R}^n)$, pick a ball $B \subset \mathbb{R}^n$ containing $\text{supp}(\phi)$. Note that

$$ \int_0^{r(B)^2} |\langle e^{t\Delta} f, \phi \rangle|^2 \frac{dt}{t} \lesssim \|e^{t\Delta} f\|_{T_{1/2}^\infty}^2 \mathcal{P}_0(\phi)^2 |B|^2 < \infty. $$

The fact that $\langle e^{t\Delta} f, \phi \rangle$ tends to $\langle f, \phi \rangle$ if $t \to 0$ forces $f = 0$ for the integral to converge, so $f = 0$ in $\mathcal{D}'(\mathbb{R}^n)$. This completes the proof. $\square$

Replacing the heat extension by its gradient gives us the following

**Corollary 3.4** (Characterization of $\dot{H}^{s,p}$ via gradient of heat extension). Let $s \in \mathbb{R}$ and $0 < p \le \infty$.

(i) Suppose $s < 1$ and $f \in \dot{H}^{s,p}$. Then $\nabla \mathcal{E}_{-\Delta}(f)$ lies in $T_{s/2}^p$ with

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

(ii) Suppose $f \in \mathcal{S}'$ with $\nabla \mathcal{E}_{-\Delta}(f) \in T_{s/2}^p$.

(1) If $s \ge 1$ and $\frac{n}{n+s-1} \le p \le \infty$, then $f$ is a constant.

(2) If $s < 1$, then there exists some constant $c \in \mathbb{C}$ so that $f - c \in \dot{H}^{s,p}$ with

$$ \|f - c\|_{\dot{H}^{s,p}} \approx \|\nabla \mathcal{E}_{-\Delta}(f)\|_{T_{s/2}^p}. $$

Consequently, when $s < 1$, a tempered distribution $f \in \mathcal{S}'$ belongs to $\dot{H}^{s,p} + \mathbb{C}$ if and only if $\nabla \mathcal{E}_{-\Delta}(f)$ lies in $T_{s/2}^p$, with the equivalence of norms as

$$ \inf_{c \in \mathbb{C}} \|f - c\|_{\dot{H}^{s,p}} \approx \|\nabla \mathcal{E}_{-\Delta}(f)\|_{T_{s/2}^p}. $$

*Proof.* A tempered distribution $f \in \mathcal{S}'$ lies in $\dot{H}^{s,p} + \mathbb{C}$ if and only if $\nabla f \in \dot{H}^{s-1,p}$, with the equivalence $\inf_{c \in \mathbb{C}} \|f - c\|_{\dot{H}^{s,p}} \approx \|\nabla f\|_{\dot{H}^{s-1,p}}$. So the corollary follows from applying Theorem 3.2 to $\nabla f$ instead of $f$ and using the above equivalence. $\square$

In fact, working modulo constants, one can equip $\dot{H}^{s,p} + \mathbb{C}$ with the “norm modulo constants” given by $\inf_{c \in \mathbb{C}} \|f - c\|_{\dot{H}^{s,p}}$. When $\nu(s,p) \ge 0$, constants are contained in $\dot{H}^{s,p}$, so $\dot{H}^{s,p} + \mathbb{C} = \dot{H}^{s,p}$, and for any $c \in \mathbb{C}$, $\|f - c\|_{\dot{H}^{s,p}} = \|f\|_{\dot{H}^{s,p}}$.

**3.2. Representation of heat solutions.** We prove here Theorem 1.1. In Corollary 3.4, we started from an element of $\dot{H}^{s,p}$ and described in which range of exponents this is equivalent to the gradient of its heat extension being in a weighted tent space. The question is now whether all heat solutions are of this form. We prove this is true in some (different) range of exponents, by establishing a trace result followed by a representation via heat semigroup.

**Proposition 3.5** (Representation of heat solutions). Let $s > -1$ and $\frac{n}{n+s+1} \le p \le \infty$. Let $u$ be a distributional solution to the heat equation on $\mathbb{R}_+^{1+n}$ with $\nabla u \in T_{s/2}^p$. Then there exists a unique $u_0 \in \mathcal{S}'$ so that $u(t) = e^{t\Delta}u_0$ for all $t > 0$.

*Remark 3.6.* Note that we work in $\mathcal{S}'$, and not just modulo polynomials. This induces restrictions on $s$ and $p$ in the proof.

Let us start the proof by a lemma on the growth of local $L^2$-norms of functions with $\nabla u \in T_{s/2}^p$.
**Lemma 3.7.** Let $s \in \mathbb{R}$ and $0 < p \le \infty$. Let $u$ be in $L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$ with $\nabla u \in T_{s/2}^p$. Then for $0 < a < b < \infty$ and $R > 1$,

$$ (3.6) \quad \int_a^b \int_{B(0,R)} |u|^2 \lesssim_{a,b,p,s} R^{3n+2} \left( \|\nabla u\|_{T_{s/2}^p}^2 + \|u\|_{L^2((a,b) \times B(0,1))}^2 \right). $$

*Proof.* Poincaré's inequality yields

$$ \int_{B(0,R)} \left| u - \int_{B(0,R)} u \right|^2 \lesssim R^2 \int_{B(0,R)} |\nabla u|^2. $$

Here and in the sequel, unspecified measures in the integral are unweighted Lebesgue measure. One can also easily verify that

$$ \left| \int_{B(0,R)} u - \int_{B(0,1)} u \right| \lesssim R \int_{B(0,R)} |\nabla u|. $$

Then we have

$$ \begin{aligned} \int_a^b \int_{B(0,R)} |u|^2 &\lesssim \int_a^b \int_{B(0,R)} \left| u - \int_{B(0,R)} u \right|^2 \\ &+ \int_a^b \int_{B(0,R)} \left| \int_{B(0,R)} u - \int_{B(0,1)} u \right|^2 + R^n \int_a^b \int_{B(0,1)} |u|^2 \\ &\lesssim R^{2n+2} \int_a^b \int_{B(0,R)} |\nabla u|^2 + R^n \int_a^b \int_{B(0,1)} |u|^2. \end{aligned} $$

Then (3.6) follows from the claim that for any $R > 1$,

$$ (3.7) \quad \int_a^b \int_{B(0,R)} |\nabla u|^2 \lesssim_{a,b,p,s} R^n \|\nabla u\|_{T_{s/2}^p}^2. $$

Let us verify the claim. For $2 \le p \le \infty$, [AH23b, Lemma 4.9] implies

$$ \left\| \left( \int_a^b \int_{B(\cdot, b^{1/2})} |\nabla u(t, y)|^2 dt dy \right)^{1/2} \right\|_p \lesssim_{a,b,s} \|\nabla u\|_{T_{s/2}^p}^2. $$

Denote by $r$ the Hölder conjugate of $p/2$, and (3.7) follows from

$$ \begin{aligned} \int_a^b \int_{B(0,R)} |\nabla u|^2 &\lesssim \int_{B(0,R+b^{1/2})} dx \int_a^b \int_{B(x,b^{1/2})} |\nabla u(t,y)|^2 dt dy \\ &\lesssim_{a,b,p,s} R^{n/r} \|\nabla u\|_{T_{s/2}^p}^2. \end{aligned} $$

For $0 < p \le 2$, (3.7) follows by [AH23b, Lemma 4.8] as

$$ \int_a^b \int_{B(0,R)} |\nabla u|^2 \lesssim_{a,b,p,s} \|\nabla u\|_{T_{s/2}^p}^2. $$

This completes the proof.

□

*Proof of Proposition 3.5.* The conclusion holds from applying [AH23a, Theorem 1.1]. Let us verify the size condition and the uniform control condition there.
**Lemma 3.7.** Let $s \in \mathbb{R}$ and $0 < p \le \infty$. Let $u$ be in $L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$ with $\nabla u \in T_{s/2}^p$. Then for $0 < a < b < \infty$ and $R > 1$,

$$ (3.6) \quad \int_a^b \int_{B(0,R)} |u|^2 \lesssim_{a,b,p,s} R^{3n+2} \left( \|\nabla u\|_{T_{s/2}^p}^2 + \|u\|_{L^2((a,b) \times B(0,1))}^2 \right). $$

*Proof.* Poincaré's inequality yields

$$ \int_{B(0,R)} \left| u - \int_{B(0,R)} u \right|^2 \lesssim R^2 \int_{B(0,R)} |\nabla u|^2. $$

Here and in the sequel, unspecified measures in the integral are unweighted Lebesgue measure. One can also easily verify that

$$ \left| \int_{B(0,R)} u - \int_{B(0,1)} u \right| \lesssim R \int_{B(0,R)} |\nabla u|. $$

Then we have

$$ \begin{aligned} \int_a^b \int_{B(0,R)} |u|^2 &\lesssim \int_a^b \int_{B(0,R)} \left| u - \int_{B(0,R)} u \right|^2 \\ &+ \int_a^b \int_{B(0,R)} \left| \int_{B(0,R)} u - \int_{B(0,1)} u \right|^2 + R^n \int_a^b \int_{B(0,1)} |u|^2 \\ &\lesssim R^{2n+2} \int_a^b \int_{B(0,R)} |\nabla u|^2 + R^n \int_a^b \int_{B(0,1)} |u|^2. \end{aligned} $$

Then (3.6) follows from the claim that for any $R > 1$,

$$ (3.7) \quad \int_a^b \int_{B(0,R)} |\nabla u|^2 \lesssim_{a,b,p,s} R^n \|\nabla u\|_{T_{s/2}^p}^2. $$

Let us verify the claim. For $2 \le p \le \infty$, [AH23b, Lemma 4.9] implies

$$ \left\| \left( \int_a^b \int_{B(\cdot, b^{1/2})} |\nabla u(t, y)|^2 dt dy \right)^{1/2} \right\|_p \lesssim_{a,b,s} \|\nabla u\|_{T_{s/2}^p}^2. $$

Denote by $r$ the Hölder conjugate of $p/2$, and (3.7) follows from

$$ \begin{aligned} \int_a^b \int_{B(0,R)} |\nabla u|^2 &\lesssim \int_{B(0,R+b^{1/2})} dx \int_a^b \int_{B(x,b^{1/2})} |\nabla u(t,y)|^2 dt dy \\ &\lesssim_{a,b,p,s} R^{n/r} \|\nabla u\|_{T_{s/2}^p}^2. \end{aligned} $$

For $0 < p \le 2$, (3.7) follows by [AH23b, Lemma 4.8] as

$$ \int_a^b \int_{B(0,R)} |\nabla u|^2 \lesssim_{a,b,p,s} \|\nabla u\|_{T_{s/2}^p}^2. $$

This completes the proof.

□

*Proof of Proposition 3.5.* The conclusion holds from applying [AH23a, Theorem 1.1]. Let us verify the size condition and the uniform control condition there.
Recall that $u$ is smooth. Lemma 3.7 implies that for $0 < a < b < \infty$, there exists $\gamma \in (0, 1/4)$ so that for any $R > 0$,

$$ \left( \int_a^b \int_{B(0,R)} |u|^2 \right)^{1/2} \lesssim_{a,b,\gamma} \exp \left( \frac{\gamma R^2}{b-a} \right), $$

which is exactly the size condition. For the uniform control condition, we show that there exists $N > 0$ so that for any $\phi \in C_c^\infty(\mathbb{R}^n)$,

$$ (3.8) \quad \sup_{0<t<1/2} |\langle u(t), \phi \rangle| \lesssim \mathcal{P}_N(\phi) \left( \|\nabla u\|_{T_{s/2}^p} + \|u\|_{L^2((1,2)\times B(0,1))} \right), $$

where $\mathcal{P}_N(\phi)$ is defined in the proof of Theorem 3.2. To this end, fix $0 < t < 1/2 < 1 < t' < 2$ and $\phi \in C_c^\infty(\mathbb{R}^n)$. Note that

$$ \int_1^2 \int_{\mathbb{R}^n} |u||\phi| \le \int_1^2 \int_{|x|<1} |u||\phi| + \sum_{k=1}^\infty \int_1^2 \int_{2^{k-1} \le |x| < 2^k} |u||\phi|. $$

Denote by $I_k$ the $k$-th term for $k \ge 0$. For $k = 0$, we have

$$ I_0 \lesssim \|u\|_{L^2((1,2)\times B(0,1))} \mathcal{P}_0(\phi). $$

For $k \ge 1$, using (3.6), we get

$$ \begin{aligned} I_k &\le \left( \int_1^2 \int_{|x|<2^k} |u|^2 \right)^{1/2} \left( \int_1^2 \int_{2^{k-1} \le |x| < 2^k} |\phi|^2 \right)^{1/2} \\ &\lesssim 2^{k(2n+1)} \left( \|\nabla u\|_{T_{s/2}^p} + \|u\|_{L^2((1,2)\times B(0,1))} \right) \sup_{2^{k-1} \le |x| < 2^k} |\phi(x)| \\ &\lesssim 2^{-k} \mathcal{P}_{N_1}(\phi) \left( \|\nabla u\|_{T_{s/2}^p} + \|u\|_{L^2((1,2)\times B(0,1))} \right) \end{aligned} $$

for some $N_1 > 0$. We hence obtain

$$ \int_1^2 \int_{\mathbb{R}^n} |u||\phi| \lesssim \mathcal{P}_{N_1}(\phi) \left( \|\nabla u\|_{T_{s/2}^p} + \|u\|_{L^2((1,2)\times B(0,1))} \right). $$

Next, we claim that there exists $N_2 > 0$ so that

$$ (3.9) \quad \int_0^2 \int_{\mathbb{R}^n} |\nabla u||\nabla \phi| \lesssim \mathcal{P}_{N_2}(\phi) \|\nabla u\|_{T_{s/2}^p}. $$

Then (3.8) follows by using the mean-value inequality in time and the equation for $u$. Indeed, pick $N > \max\{N_1, N_2\}$, and we get

$$ \begin{aligned} |\langle u(t), \phi \rangle| &\le \int_1^2 |\langle u(t'), \phi \rangle| dt' + \int_1^2 |\langle u(t'), \phi \rangle - \langle u(t), \phi \rangle| dt' \\ &\le \int_1^2 |\langle u(t'), \phi \rangle| dt' + \int_0^2 \int_{\mathbb{R}^n} |\nabla u||\nabla \phi| \\ &\lesssim \mathcal{P}_N(\phi) \left( \|\nabla u\|_{T_{s/2}^p} + \|u\|_{L^2((1,2)\times B(0,1))} \right) \end{aligned} $$

as desired.
Recall that $u$ is smooth. Lemma 3.7 implies that for $0 < a < b < \infty$, there exists $\gamma \in (0, 1/4)$ so that for any $R > 0$,

$$ \left( \int_a^b \int_{B(0,R)} |u|^2 \right)^{1/2} \lesssim_{a,b,\gamma} \exp \left( \frac{\gamma R^2}{b-a} \right), $$

which is exactly the size condition. For the uniform control condition, we show that there exists $N > 0$ so that for any $\phi \in C_c^\infty(\mathbb{R}^n)$,

$$ (3.8) \quad \sup_{0<t<1/2} |\langle u(t), \phi \rangle| \lesssim \mathcal{P}_N(\phi) \left( \|\nabla u\|_{T_{s/2}^p} + \|u\|_{L^2((1,2)\times B(0,1))} \right), $$

where $\mathcal{P}_N(\phi)$ is defined in the proof of Theorem 3.2. To this end, fix $0 < t < 1/2 < 1 < t' < 2$ and $\phi \in C_c^\infty(\mathbb{R}^n)$. Note that

$$ \int_1^2 \int_{\mathbb{R}^n} |u||\phi| \le \int_1^2 \int_{|x|<1} |u||\phi| + \sum_{k=1}^\infty \int_1^2 \int_{2^{k-1} \le |x| < 2^k} |u||\phi|. $$

Denote by $I_k$ the $k$-th term for $k \ge 0$. For $k = 0$, we have

$$ I_0 \lesssim \|u\|_{L^2((1,2)\times B(0,1))} \mathcal{P}_0(\phi). $$

For $k \ge 1$, using (3.6), we get

$$ \begin{aligned} I_k &\le \left( \int_1^2 \int_{|x|<2^k} |u|^2 \right)^{1/2} \left( \int_1^2 \int_{2^{k-1} \le |x| < 2^k} |\phi|^2 \right)^{1/2} \\ &\lesssim 2^{k(2n+1)} \left( \|\nabla u\|_{T_{s/2}^p} + \|u\|_{L^2((1,2)\times B(0,1))} \right) \sup_{2^{k-1} \le |x| < 2^k} |\phi(x)| \\ &\lesssim 2^{-k} \mathcal{P}_{N_1}(\phi) \left( \|\nabla u\|_{T_{s/2}^p} + \|u\|_{L^2((1,2)\times B(0,1))} \right) \end{aligned} $$

for some $N_1 > 0$. We hence obtain

$$ \int_1^2 \int_{\mathbb{R}^n} |u||\phi| \lesssim \mathcal{P}_{N_1}(\phi) \left( \|\nabla u\|_{T_{s/2}^p} + \|u\|_{L^2((1,2)\times B(0,1))} \right). $$

Next, we claim that there exists $N_2 > 0$ so that

$$ (3.9) \quad \int_0^2 \int_{\mathbb{R}^n} |\nabla u||\nabla \phi| \lesssim \mathcal{P}_{N_2}(\phi) \|\nabla u\|_{T_{s/2}^p}. $$

Then (3.8) follows by using the mean-value inequality in time and the equation for $u$. Indeed, pick $N > \max\{N_1, N_2\}$, and we get

$$ \begin{aligned} |\langle u(t), \phi \rangle| &\le \int_1^2 |\langle u(t'), \phi \rangle| dt' + \int_1^2 |\langle u(t'), \phi \rangle - \langle u(t), \phi \rangle| dt' \\ &\le \int_1^2 |\langle u(t'), \phi \rangle| dt' + \int_0^2 \int_{\mathbb{R}^n} |\nabla u||\nabla \phi| \\ &\lesssim \mathcal{P}_N(\phi) \left( \|\nabla u\|_{T_{s/2}^p} + \|u\|_{L^2((1,2)\times B(0,1))} \right) \end{aligned} $$

as desired.
Let us verify (3.9). We use duality of tent spaces. Let $s > -1$. For $1 < p \le \infty$, pick $N_2 > n/p' + 1$ and we get

$$ \begin{aligned} \|\mathbf{1}_{(0,2)} \nabla \phi\|_{T_{-s/2}^{p'}} & \lesssim \left(\int_0^2 t^s dt\right)^{1/2} \left(\int_{\mathbb{R}^n} \left(\sup_{y \in B(x,2)} |\nabla \phi(y)|\right)^{p'} dx\right)^{1/p'} \\ & \lesssim \left(\int_{\mathbb{R}^n} \langle x \rangle^{-(N_2-1)p'} \left(\sup_{y \in B(x,2)} \langle y \rangle^{N_2-1} |\nabla \phi(y)|\right)^{p'} dx\right)^{1/p'} \lesssim \mathcal{P}_{N_2}(\phi). \end{aligned} $$

For $\frac{n}{n+s+1} \le p \le 1$, i.e., $s+1-n[p, 1] \ge 0$, we have

$$ \|\mathbf{1}_{(0,2)} \nabla \phi\|_{T_{-s/2,([p,1])}^{\infty}} \lesssim \|\nabla \phi\|_{\infty} \le \mathcal{P}_{N_2}(\phi). $$

This proves (3.9) and hence completes the proof. □

*Proof of Theorem 1.1.* Assertion (i) has been shown in Corollary 3.4. For (ii), the existence of $u_0 \in \mathcal{S}'$ comes from Proposition 3.5. Corollary 3.4 and Proposition 3.1 show the desired properties (1) and (2). □

## 4. BASIC FACTS ABOUT WEAK SOLUTIONS

The goal of the rest of the paper is to study the Cauchy problems

$$ (4.1) \qquad \begin{cases} \partial_t u - \operatorname{div}(A \nabla u) = \operatorname{div} F \\ u(0) = u_0 \end{cases}, $$

where A satisfies (1.2). In this section, we review the definition of weak solutions, prove a priori energy inequalities, and recall elements of the $L^2$-theory. Denote by $W^{1,2}$ the inhomogeneous Sobolev space with the norm $\|f\|_{W^{1,2}} := \|f\|_2 + \|\nabla f\|_2$.

**Definition 4.1** (Weak solutions). Let $0 \le a < b \le \infty$, $\Omega$ be an open subset of $\mathbb{R}^n$, and $Q := (a, b) \times \Omega$. Let $f$ and $F$ be in $\mathcal{D}'(Q)$. A function $u \in L_{\text{loc}}^2((a, b); W_{\text{loc}}^{1,2}(\Omega))$ is called a weak solution to the equation

$$ \partial_t u - \operatorname{div}(A \nabla u) = f + \operatorname{div} F $$

with source term $f + \operatorname{div} F$, if for any $\phi \in C_c^\infty(Q)$,

$$ (4.2) \qquad - \iint_Q u \partial_t \phi + \iint_Q (A \nabla u) \cdot \nabla \phi = (f, \phi) - (F, \nabla \phi). $$

The pairs on the right-hand side are understood as pairings of distributions and test functions on $Q$. We say $u$ is a global weak solution if (4.2) holds for $Q = \mathbb{R}_+^{1+n}$.

Let $a = 0$ and $u_0 \in \mathcal{D}'(\Omega)$. By the initial condition $u(0) = u_0$, we further impose that the weak solution $u(t)$ converges to $u_0$ in $\mathcal{D}'(\Omega)$ as $t \to 0$.

Similarly, there is a corresponding definition of weak solutions for the dual backward equation $-\partial_t u - \operatorname{div}(A^* \nabla u) = f + \operatorname{div} F$.

### 4.1. Energy inequalities.
Let us recall without proof a form of the classical Cauchy-riopoli's inequality.
Let us verify (3.9). We use duality of tent spaces. Let $s > -1$. For $1 < p \le \infty$, pick $N_2 > n/p' + 1$ and we get

$$ \begin{aligned} \|\mathbf{1}_{(0,2)} \nabla \phi\|_{T_{-s/2}^{p'}} & \lesssim \left(\int_0^2 t^s dt\right)^{1/2} \left(\int_{\mathbb{R}^n} \left(\sup_{y \in B(x,2)} |\nabla \phi(y)|\right)^{p'} dx\right)^{1/p'} \\ & \lesssim \left(\int_{\mathbb{R}^n} \langle x \rangle^{-(N_2-1)p'} \left(\sup_{y \in B(x,2)} \langle y \rangle^{N_2-1} |\nabla \phi(y)|\right)^{p'} dx\right)^{1/p'} \lesssim \mathcal{P}_{N_2}(\phi). \end{aligned} $$

For $\frac{n}{n+s+1} \le p \le 1$, i.e., $s+1-n[p, 1] \ge 0$, we have

$$ \|\mathbf{1}_{(0,2)} \nabla \phi\|_{T_{-s/2,([p,1])}^{\infty}} \lesssim \|\nabla \phi\|_{\infty} \le \mathcal{P}_{N_2}(\phi). $$

This proves (3.9) and hence completes the proof. □

*Proof of Theorem 1.1.* Assertion (i) has been shown in Corollary 3.4. For (ii), the existence of $u_0 \in \mathcal{S}'$ comes from Proposition 3.5. Corollary 3.4 and Proposition 3.1 show the desired properties (1) and (2). □

## 4. BASIC FACTS ABOUT WEAK SOLUTIONS

The goal of the rest of the paper is to study the Cauchy problems

$$ (4.1) \qquad \begin{cases} \partial_t u - \operatorname{div}(A \nabla u) = \operatorname{div} F, \\ u(0) = u_0 \end{cases}, $$

where A satisfies (1.2). In this section, we review the definition of weak solutions, prove a priori energy inequalities, and recall elements of the $L^2$-theory. Denote by $W^{1,2}$ the inhomogeneous Sobolev space with the norm $\|f\|_{W^{1,2}} := \|f\|_2 + \|\nabla f\|_2$.

**Definition 4.1** (Weak solutions). Let $0 \le a < b \le \infty$, $\Omega$ be an open subset of $\mathbb{R}^n$, and $Q := (a, b) \times \Omega$. Let $f$ and $F$ be in $\mathcal{D}'(Q)$. A function $u \in L_{\text{loc}}^2((a, b); W_{\text{loc}}^{1,2}(\Omega))$ is called a weak solution to the equation

$$ \partial_t u - \operatorname{div}(A \nabla u) = f + \operatorname{div} F $$

with source term $f + \operatorname{div} F$, if for any $\phi \in C_c^\infty(Q)$,

$$ (4.2) \qquad - \iint_Q u \partial_t \phi + \iint_Q (A \nabla u) \cdot \nabla \phi = (f, \phi) - (F, \nabla \phi). $$

The pairs on the right-hand side are understood as pairings of distributions and test functions on $Q$. We say $u$ is a global weak solution if (4.2) holds for $Q = \mathbb{R}_+^{1+n}$.

Let $a = 0$ and $u_0 \in \mathcal{D}'(\Omega)$. By the initial condition $u(0) = u_0$, we further impose that the weak solution $u(t)$ converges to $u_0$ in $\mathcal{D}'(\Omega)$ as $t \to 0$.

Similarly, there is a corresponding definition of weak solutions for the dual backward equation $-\partial_t u - \operatorname{div}(A^* \nabla u) = f + \operatorname{div} F$.

### 4.1. Energy inequalities.
Let us recall without proof a form of the classical Cauchy-riopoli's inequality.
**Lemma 4.2.** Let $0 < a < b < \infty$, $B \subset \mathbb{R}^n$ be a ball, and $f, F$ be in $L^2((a, b) \times 2B)$. Let $u \in L^2((a, b); W^{1,2}(2B))$ be a weak solution to the equation $\partial_t u - \text{div}(A\nabla u) = f + \text{div} F$ in $(a, b) \times 2B$. Then $u$ lies in $C([a, b]; L^2(B))$ with

$$ \|u(b)\|_{L^2(B)}^2 \lesssim \left( \frac{1}{r(B)^2} + \frac{1}{b-a} \right) \int_a^b \|u(s)\|_{L^2(2B)}^2 ds \\ + r(B)^2 \int_a^b \|f(s)\|_{L^2(2B)}^2 ds + \int_a^b \|F(s)\|_{L^2(2B)}^2 ds. $$

Moreover, for any $c \in (a, b)$, it holds that

$$ \int_c^b \|\nabla u(s)\|_{L^2(B)}^2 ds \lesssim \frac{1}{c-a} \left(1 + \frac{b-a}{r(B)^2}\right) \int_a^b \|u(s)\|_{L^2(2B)}^2 ds \\ + \frac{r(B)^2(b-a)}{c-a} \int_a^b \|f(s)\|_{L^2(2B)}^2 ds + \frac{b-a}{c-a} \int_a^b \|F(s)\|_{L^2(2B)}^2 ds. $$

The implicit constants are independent of $a, b, c$, and $B$.

There is also a corresponding version for weak solutions to the backward equation. In the sequel, we refer to “Caccioppoli’s inequality” in both cases.

For any $(t, x) \in \mathbb{R}_+^{1+n}$, the set $W(t, x) := (t, 2t) \times B(x, t^{1/2})$ is called a (parabolic) Whitney cube at $(t, x)$.

**Corollary 4.3.** Let $\beta \in \mathbb{R}$ and $0 < p \le \infty$. For any global weak solution $u$ on $\mathbb{R}_+^{1+n}$ to $\partial_t u - \text{div}(A\nabla u) = f + \text{div} F$, the a priori energy inequality holds as

$$ (4.3) \qquad \|\nabla u\|_{T_{\beta+1/2}^p} \lesssim \|u\|_{T_{\beta+1}^p} + \|F\|_{T_{\beta+1/2}^p} + \|f\|_{T_{\beta}^p}. $$

The same inequality occurs for global weak solutions to the backward equation.

*Proof.* When $p < \infty$, using Caccioppoli’s inequality on local Whitney cubes and the change of aperture property of tent-space norms, we get

$$ \begin{align*} \|\nabla u\|_{T_{\beta+1/2}^p} &\lesssim \left( \int_{\mathbb{R}^n} \left( \int_0^\infty t^{-2\beta-1} dt \int_{t/2}^t \int_{B(x,t^{1/2})} |\nabla u|^2 \right)^{p/2} dx \right)^{1/p} \\ &\lesssim \left( \int_{\mathbb{R}^n} \left( \int_0^\infty t^{-2\beta-2} dt \int_{t/4}^t \int_{B(x,2t^{1/2})} |u|^2 \right)^{p/2} dx \right)^{1/p} \\ &\quad + \left( \int_{\mathbb{R}^n} \left( \int_0^\infty t^{-2\beta-1} dt \int_{t/4}^t \int_{B(x,2t^{1/2})} |F|^2 \right)^{p/2} dx \right)^{1/p} \\ &\quad + \left( \int_{\mathbb{R}^n} \left( \int_0^\infty t^{-2\beta} dt \int_{t/4}^t \int_{B(x,2t^{1/2})} |f|^2 \right)^{p/2} dx \right)^{1/p} \\ &\lesssim \|u\|_{T_{\beta+1}^p} + \|F\|_{T_{\beta+1/2}^p} + \|f\|_{T_{\beta}^p}. \end{align*} $$

When $p = \infty$, we take a covering of the Carleson boxes $(0, r(B)^2) \times B$ by local Whitney cubes and apply Caccioppoli’s inequality on each, noting that the enlarged local Whitney cubes are contained in $(0, 16r(B)^2) \times 4B$ with bounded overlapping. Detailed verification is left to the reader. □
**Lemma 4.2.** Let $0 < a < b < \infty$, $B \subset \mathbb{R}^n$ be a ball, and $f, F$ be in $L^2((a, b) \times 2B)$. Let $u \in L^2((a, b); W^{1,2}(2B))$ be a weak solution to the equation $\partial_t u - \text{div}(A\nabla u) = f + \text{div} F$ in $(a, b) \times 2B$. Then $u$ lies in $C([a, b]; L^2(B))$ with

$$ \|u(b)\|_{L^2(B)}^2 \lesssim \left( \frac{1}{r(B)^2} + \frac{1}{b-a} \right) \int_a^b \|u(s)\|_{L^2(2B)}^2 ds \\ + r(B)^2 \int_a^b \|f(s)\|_{L^2(2B)}^2 ds + \int_a^b \|F(s)\|_{L^2(2B)}^2 ds. $$

Moreover, for any $c \in (a, b)$, it holds that

$$ \int_c^b \|\nabla u(s)\|_{L^2(B)}^2 ds \lesssim \frac{1}{c-a} \left(1 + \frac{b-a}{r(B)^2}\right) \int_a^b \|u(s)\|_{L^2(2B)}^2 ds \\ + \frac{r(B)^2(b-a)}{c-a} \int_a^b \|f(s)\|_{L^2(2B)}^2 ds + \frac{b-a}{c-a} \int_a^b \|F(s)\|_{L^2(2B)}^2 ds. $$

The implicit constants are independent of $a, b, c$, and $B$.

There is also a corresponding version for weak solutions to the backward equation. In the sequel, we refer to “Caccioppoli’s inequality” in both cases.

For any $(t, x) \in \mathbb{R}_+^{1+n}$, the set $W(t, x) := (t, 2t) \times B(x, t^{1/2})$ is called a (parabolic) Whitney cube at $(t, x)$.

**Corollary 4.3.** Let $\beta \in \mathbb{R}$ and $0 < p \le \infty$. For any global weak solution $u$ on $\mathbb{R}_+^{1+n}$ to $\partial_t u - \text{div}(A\nabla u) = f + \text{div} F$, the a priori energy inequality holds as

$$ (4.3) \qquad \|\nabla u\|_{T_{\beta+1/2}^p} \lesssim \|u\|_{T_{\beta+1}^p} + \|F\|_{T_{\beta+1/2}^p} + \|f\|_{T_{\beta}^p}. $$

The same inequality occurs for global weak solutions to the backward equation.

*Proof.* When $p < \infty$, using Caccioppoli’s inequality on local Whitney cubes and the change of aperture property of tent-space norms, we get

$$ \begin{aligned} \|\nabla u\|_{T_{\beta+1/2}^p} & \lesssim \left( \int_{\mathbb{R}^n} \left( \int_0^\infty t^{-2\beta-1} dt \int_{t/2}^t \int_{B(x,t^{1/2})} |\nabla u|^2 \right)^{p/2} dx \right)^{1/p} \\ & \lesssim \left( \int_{\mathbb{R}^n} \left( \int_0^\infty t^{-2\beta-2} dt \int_{t/4}^t \int_{B(x,2t^{1/2})} |u|^2 \right)^{p/2} dx \right)^{1/p} \\ & \quad + \left( \int_{\mathbb{R}^n} \left( \int_0^\infty t^{-2\beta-1} dt \int_{t/4}^t \int_{B(x,2t^{1/2})} |F|^2 \right)^{p/2} dx \right)^{1/p} \\ & \quad + \left( \int_{\mathbb{R}^n} \left( \int_0^\infty t^{-2\beta} dt \int_{t/4}^t \int_{B(x,2t^{1/2})} |f|^2 \right)^{p/2} dx \right)^{1/p} \\ & \lesssim \|u\|_{T_{\beta+1}^p} + \|F\|_{T_{\beta+1/2}^p} + \|f\|_{T_{\beta}^p}. \end{aligned} $$

When $p = \infty$, we take a covering of the Carleson boxes $(0, r(B)^2) \times B$ by local Whitney cubes and apply Caccioppoli’s inequality on each, noting that the enlarged local Whitney cubes are contained in $(0, 16r(B)^2) \times 4B$ with bounded overlapping. Detailed verification is left to the reader. □
4.2. $L^2$-theory. In this section, we summarize the $L^2$-theory as a starting point. The weak solutions are given by Duhamel's formula and semigroup theory, see Section 1.1.2 for the definition of operators $L$ and $\mathcal{E}_L$. We also define the Lions' operator $\mathcal{R}_{1/2}^L : L^2(\mathbb{R}_+^{1+n}) \to L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$ by the $L^2$-valued Bochner integrals (where $L^2$ denotes $L^2(\mathbb{R}^n)$)

$$ (4.4) \qquad \mathcal{R}_{1/2}^L(F)(t) := \int_0^t e^{-(t-s)L} \operatorname{div} F(s) ds, \quad \forall t > 0. $$

**Proposition 4.4** ($L^2$-theory). Let $u_0 \in L^2$ and $F \in L^2(\mathbb{R}_+^{1+n})$. Then there exists a unique global weak solution $u$ to the Cauchy problem (4.1) with $\nabla u \in L^2(\mathbb{R}_+^{1+n})$. Moreover,

(i) $u$ belongs to $C_0([0, \infty); L^2)$ with the estimate

$$ \sup_{t \ge 0} \|u(t)\|_2 + \|\nabla u\|_{L^2(\mathbb{R}_+^{1+n})} \lesssim \|u_0\|_2 + \|F\|_{L^2(\mathbb{R}_+^{1+n})}. $$

(ii) (Duhamel's formula) $u = \mathcal{E}_L(u_0) + \mathcal{R}_{1/2}^L(F)$.

Existence originates from the work of J.-L. Lions [Lio57, Théorème II.3.1], and uniqueness in this larger (likely the largest) class is established in [AMP19, Theorem 3.11]. The reader can refer to [AP23, Theorem 2.2] for a detailed survey of different proofs of this theorem.

This also allows us to obtain several equivalent expressions of these operators, using different interpretations of the equation. This will come into play when studying their extensions. Denote by $\mathbb{I}$ the identity matrix in $\text{Mat}_n(\mathbb{C})$. Define the operator $\mathcal{L}_1^L : L^2(\mathbb{R}_+^{1+n}) \to L_{\text{loc}}^2((0, \infty); L^2)$ by the $L^2$-valued Bochner integrals

$$ (4.5) \qquad \mathcal{L}_1^L(f)(t) := \int_0^t e^{-(t-s)L} f(s) ds, \quad \forall t > 0. $$

**Corollary 4.5** (Explicit formulae). Let $u_0 \in L^2$ and $F \in L^2(\mathbb{R}_+^{1+n})$.

(i) Define $\tilde{F} := (A - \mathbb{I})\nabla \mathcal{R}_{1/2}^L(F) + F$ on $\mathbb{R}_+^{1+n}$. Then

$$ (4.6) \qquad \mathcal{R}_{1/2}^L(F) = \mathcal{R}_{1/2}^{-\Delta}(\tilde{F}) = \operatorname{div} \mathcal{L}_1^{-\Delta}(\tilde{F}) \quad \text{in } \mathcal{P}'(\mathbb{R}_+^{1+n}). $$

(ii) It holds that

$$ (4.7) \qquad \mathcal{E}_L(u_0) = \mathcal{E}_{-\Delta}(u_0) + \mathcal{R}_{1/2}^L((A - \mathbb{I})\nabla \mathcal{E}_{-\Delta}(u_0)) $$

$$ (4.8) \qquad = \mathcal{E}_{-\Delta}(u_0) - \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla \mathcal{E}_L(u_0)). $$

*Proof.* First consider (i). Proposition 4.4 says that $u := \mathcal{R}_{1/2}^L(F)$ is a global weak solution to the Cauchy problem

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

with $\nabla u \in L^2(\mathbb{R}_+^{1+n})$. We hence infer that $\tilde{F}$ lies in $L^2(\mathbb{R}_+^{1+n})$, and then $\tilde{u} := \mathcal{R}_{1/2}^{-\Delta}(\tilde{F})$ is a global weak solution to the Cauchy problem

$$ \left\{ \begin{array}{l} \partial_t \tilde{u} - \Delta \tilde{u} = \operatorname{div} \tilde{F} = \operatorname{div}((A - \mathbb{I})\nabla u) + \operatorname{div} F \\ \tilde{u}(0) = 0 \end{array} \right. $$
4.2. $L^2$-theory. In this section, we summarize the $L^2$-theory as a starting point. The weak solutions are given by Duhamel's formula and semigroup theory, see Section 1.1.2 for the definition of operators $L$ and $\mathcal{E}_L$. We also define the Lions' operator $\mathcal{R}_{1/2}^L: L^2(\mathbb{R}_+^{1+n}) \to L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$ by the $L^2$-valued Bochner integrals (where $L^2$ denotes $L^2(\mathbb{R}^n)$)

$$ (4.4) \qquad \mathcal{R}_{1/2}^L(F)(t) := \int_0^t e^{-(t-s)L} \operatorname{div} F(s) ds, \quad \forall t > 0. $$

**Proposition 4.4** ($L^2$-theory). Let $u_0 \in L^2$ and $F \in L^2(\mathbb{R}_+^{1+n})$. Then there exists a unique global weak solution $u$ to the Cauchy problem (4.1) with $\nabla u \in L^2(\mathbb{R}_+^{1+n})$. Moreover,

(i) $u$ belongs to $C_0([0, \infty); L^2)$ with the estimate

$$ \sup_{t \ge 0} \|u(t)\|_2 + \|\nabla u\|_{L^2(\mathbb{R}_+^{1+n})} \lesssim \|u_0\|_2 + \|F\|_{L^2(\mathbb{R}_+^{1+n})}. $$

(ii) (Duhamel's formula) $u = \mathcal{E}_L(u_0) + \mathcal{R}_{1/2}^L(F)$.

Existence originates from the work of J.-L. Lions [Lio57, Théorème II.3.1], and uniqueness in this larger (likely the largest) class is established in [AMP19, Theorem 3.11]. The reader can refer to [AP23, Theorem 2.2] for a detailed survey of different proofs of this theorem.

This also allows us to obtain several equivalent expressions of these operators, using different interpretations of the equation. This will come into play when studying their extensions. Denote by $\mathbb{I}$ the identity matrix in $\text{Mat}_n(\mathbb{C})$. Define the operator $\mathcal{L}_1^L: L^2(\mathbb{R}_+^{1+n}) \to L_{\text{loc}}^2((0, \infty); L^2)$ by the $L^2$-valued Bochner integrals

$$ (4.5) \qquad \mathcal{L}_1^L(f)(t) := \int_0^t e^{-(t-s)L} f(s) ds, \quad \forall t > 0. $$

**Corollary 4.5** (Explicit formulae). Let $u_0 \in L^2$ and $F \in L^2(\mathbb{R}_+^{1+n})$.

(i) Define $\tilde{F} := (A - \mathbb{I})\nabla \mathcal{R}_{1/2}^L(F) + F$ on $\mathbb{R}_+^{1+n}$. Then

$$ (4.6) \qquad \mathcal{R}_{1/2}^L(F) = \mathcal{R}_{1/2}^{-\Delta}(\tilde{F}) = \operatorname{div} \mathcal{L}_1^{-\Delta}(\tilde{F}) \quad \text{in } \mathcal{P}'(\mathbb{R}_+^{1+n}). $$

(ii) It holds that

$$ (4.7) \qquad \mathcal{E}_L(u_0) = \mathcal{E}_{-\Delta}(u_0) + \mathcal{R}_{1/2}^L((A - \mathbb{I})\nabla \mathcal{E}_{-\Delta}(u_0)) $$

$$ (4.8) \qquad = \mathcal{E}_{-\Delta}(u_0) - \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla \mathcal{E}_L(u_0)). $$

*Proof.* First consider (i). Proposition 4.4 says that $u := \mathcal{R}_{1/2}^L(F)$ is a global weak solution to the Cauchy problem

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

with $\nabla u \in L^2(\mathbb{R}_+^{1+n})$. We hence infer that $\tilde{F}$ lies in $L^2(\mathbb{R}_+^{1+n})$, and then $\tilde{u} := \mathcal{R}_{1/2}^{-\Delta}(\tilde{F})$ is a global weak solution to the Cauchy problem

$$ \left\{ \begin{array}{l} \partial_t \tilde{u} - \Delta \tilde{u} = \operatorname{div} \tilde{F} = \operatorname{div}((A - \mathbb{I})\nabla u) + \operatorname{div} F \\ \tilde{u}(0) = 0 \end{array} \right. $$
with $\nabla \tilde{u} \in L^2(\mathbb{R}_+^{1+n})$. Therefore, $w := u - \tilde{u}$ is a global weak solution to the heat equation with null source term and null initial data. Since $\nabla w \in L^2(\mathbb{R}_+^{1+n})$, we deduce from uniqueness in Proposition 4.4 for the heat equation that $w = 0$, proving the first equality in (4.6). For the last, as $\tilde{F} \in L^2(\mathbb{R}_+^{1+n})$, for a.e. $t > 0$, we have

$$ \begin{aligned} \mathcal{R}_{1/2}^{-\Delta}(\tilde{F})(t) &= \int_{0}^{t} e^{(t-s)\Delta} \operatorname{div} \tilde{F}(s) ds \\ &= \operatorname{div} \int_{0}^{t} e^{(t-s)\Delta} \tilde{F}(s) ds = \operatorname{div} \mathcal{L}_{1}^{-\Delta}(\tilde{F})(t) \end{aligned} $$

in the sense of distributions in $\mathbb{R}^n$ and (i) follows.

Next, we consider (ii). Proposition 4.4 asserts that $v := \mathcal{E}_L(u_0)$ is the unique global solution to the homogeneous Cauchy problem

$$ (\text{HC}) \qquad \begin{cases} \partial_t v - \operatorname{div}(A \nabla v) = 0 \\ v(0) = u_0 \end{cases}, $$

with $\nabla v \in L^2(\mathbb{R}_+^{1+n})$. Meanwhile, as $\nabla \mathcal{E}_{-\Delta}(u_0) \in L^2(\mathbb{R}_+^{1+n})$, it also shows that $\tilde{v} := \mathcal{R}_{1/2}^L((A - \mathbb{I})\nabla \mathcal{E}_{-\Delta}(u_0))$ is a global solution to the Cauchy problem

$$ \begin{cases} \partial_t \tilde{v} - \operatorname{div}(A \nabla \tilde{v}) = \operatorname{div}((A - \mathbb{I})\nabla \mathcal{E}_{-\Delta}(u_0)) \\ \tilde{v}(0) = 0 \end{cases}. $$

Notice that $\mathcal{E}_{-\Delta}(u_0) + \tilde{v}$ is a global weak solution to the homogeneous Cauchy problem (HC) in the same class as $\mathcal{E}_L(u_0)$. Then by uniqueness, we deduce $\mathcal{E}_L(u_0) = \mathcal{E}_{-\Delta}(u_0) + \tilde{v}$ as wanted for (4.7).

One can also verify (4.8) via an analogous argument. Details are left to the reader. This completes the proof. □

## 5. PROPERTIES OF THE LIONS' OPERATOR

We provide weak solutions to Lions' equation (L), as asserted in Theorem 1.6, by studying properties of the Lions' operator $\mathcal{R}_{1/2}^L$ given by (4.4). We state a general result here. Proof of the bounded extension is addressed in this section, while that of trace and continuity properties is in the next one.

### 5.1. Main properties of the Lions' operator.
We use the critical exponent $\tilde{p}_L(\beta)$ defined in (1.8) and (1.9).

**Theorem 5.1** (Extension of $\mathcal{R}_{1/2}^L$). Let $\beta > -1$ and $\tilde{p}_L(\beta) < p \le \infty$. Then $\mathcal{R}_{1/2}^L$ extends to a bounded operator from $T_{\beta+1/2}^p$ to $T_{\beta+1}^p$, also denoted by $\mathcal{R}_{1/2}^L$. Moreover, the following properties hold for any $F \in T_{\beta+1/2}^p$ and $u := \mathcal{R}_{1/2}^L(F)$.

(a) (Regularity) $u$ lies in $T_{\beta+1}^p$ and $\nabla u$ lies in $T_{\beta+1/2}^p$ with
$$ \|u\|_{T_{\beta+1}^p} \lesssim \|F\|_{T_{\beta+1/2}^p}, \quad \|\nabla u\|_{T_{\beta+1/2}^p} \lesssim \|F\|_{T_{\beta+1/2}^p}. $$

(b) (Explicit formulae) Define $\tilde{F} := (A - \mathbb{I})\nabla \mathcal{R}_{1/2}^L(F) + F$. Then

$$ u = \mathcal{R}_{1/2}^{-\Delta}(\tilde{F}) = \operatorname{div} \mathcal{L}_{1}^{-\Delta}(\tilde{F}) \quad \text{in } \mathcal{D}'(\mathbb{R}_{+}^{1+n}), $$

where $\mathcal{L}_{1}^{-\Delta}$ is defined in (4.5).

(c) $u$ is a global weak solution to the equation $\partial_t u - \operatorname{div}(A \nabla u) = \operatorname{div} F$.
with $\nabla \tilde{u} \in L^2(\mathbb{R}_+^{1+n})$. Therefore, $w := u - \tilde{u}$ is a global weak solution to the heat equation with null source term and null initial data. Since $\nabla w \in L^2(\mathbb{R}_+^{1+n})$, we deduce from uniqueness in Proposition 4.4 for the heat equation that $w = 0$, proving the first equality in (4.6). For the last, as $\tilde{F} \in L^2(\mathbb{R}_+^{1+n})$, for a.e. $t > 0$, we have

$$ \begin{aligned} \mathcal{R}_{1/2}^{-\Delta}(\tilde{F})(t) &= \int_0^t e^{(t-s)\Delta} \operatorname{div} \tilde{F}(s) ds \\ &= \operatorname{div} \int_0^t e^{(t-s)\Delta} \tilde{F}(s) ds = \operatorname{div} \mathcal{L}_1^{-\Delta}(\tilde{F})(t) \end{aligned} $$

in the sense of distributions in $\mathbb{R}^n$ and (i) follows.

Next, we consider (ii). Proposition 4.4 asserts that $v := \mathcal{E}_L(u_0)$ is the unique global solution to the homogeneous Cauchy problem

$$ (\text{HC}) \qquad \begin{cases} \partial_t v - \operatorname{div}(A \nabla v) = 0 \\ v(0) = u_0 \end{cases}, $$

with $\nabla v \in L^2(\mathbb{R}_+^{1+n})$. Meanwhile, as $\nabla \mathcal{E}_{-\Delta}(u_0) \in L^2(\mathbb{R}_+^{1+n})$, it also shows that $\tilde{v} := \mathcal{R}_{1/2}^L((A - \mathbb{I})\nabla \mathcal{E}_{-\Delta}(u_0))$ is a global solution to the Cauchy problem

$$ \begin{cases} \partial_t \tilde{v} - \operatorname{div}(A \nabla \tilde{v}) = \operatorname{div}((A - \mathbb{I})\nabla \mathcal{E}_{-\Delta}(u_0)) \\ \tilde{v}(0) = 0 \end{cases}. $$

Notice that $\mathcal{E}_{-\Delta}(u_0) + \tilde{v}$ is a global weak solution to the homogeneous Cauchy problem (HC) in the same class as $\mathcal{E}_L(u_0)$. Then by uniqueness, we deduce $\mathcal{E}_L(u_0) = \mathcal{E}_{-\Delta}(u_0) + \tilde{v}$ as wanted for (4.7).

One can also verify (4.8) via an analogous argument. Details are left to the reader. This completes the proof. $\square$

## 5. PROPERTIES OF THE LIONS' OPERATOR

We provide weak solutions to Lions' equation (L), as asserted in Theorem 1.6, by studying properties of the Lions' operator $\mathcal{R}_{1/2}^L$ given by (4.4). We state a general result here. Proof of the bounded extension is addressed in this section, while that of trace and continuity properties is in the next one.

### 5.1. Main properties of the Lions' operator.
We use the critical exponent $\tilde{p}_L(\beta)$ defined in (1.8) and (1.9).

**Theorem 5.1** (Extension of $\mathcal{R}_{1/2}^L$). Let $\beta > -1$ and $\tilde{p}_L(\beta) < p \le \infty$. Then $\mathcal{R}_{1/2}^L$ extends to a bounded operator from $T_{\beta+1/2}^p$ to $T_{\beta+1}^p$, also denoted by $\mathcal{R}_{1/2}^L$. Moreover, the following properties hold for any $F \in T_{\beta+1/2}^p$ and $u := \mathcal{R}_{1/2}^L(F)$.

(a) (Regularity) $u$ lies in $T_{\beta+1}^p$ and $\nabla u$ lies in $T_{\beta+1/2}^p$ with
$$ \|u\|_{T_{\beta+1}^p} \lesssim \|F\|_{T_{\beta+1/2}^p}, \quad \|\nabla u\|_{T_{\beta+1/2}^p} \lesssim \|F\|_{T_{\beta+1/2}^p}. $$

(b) (Explicit formulae) Define $\tilde{F} := (A - \mathbb{I})\nabla \mathcal{R}_{1/2}^L(F) + F$. Then

$$ u = \mathcal{R}_{1/2}^{-\Delta}(\tilde{F}) = \operatorname{div} \mathcal{L}_1^{-\Delta}(\tilde{F}) \quad \text{in } \mathcal{D}'(\mathbb{R}_+^{1+n}), $$

where $\mathcal{L}_1^{-\Delta}$ is defined in (4.5).

(c) $u$ is a global weak solution to the equation $\partial_t u - \operatorname{div}(A \nabla u) = \operatorname{div} F$.
(d) (Continuity and trace) $u \in C([0, \infty); \mathcal{S}')$ with $u(0) = 0$.

When $t \to 0$, the convergence occurs in the following spaces shown in Table 1, with arbitrary parameters $\delta > 0$, $q \in [p, \infty]$, and $s \in [-1, 2\beta + 1]$, and a fixed one $r := \frac{np}{n-(2\beta+2)p} > 1$.

<table><caption>TABLE 1. Spaces for convergence of $\mathcal{R}_{1/2}^L(F)(t)$ as $t \to 0$.</caption><thead><tr><th>Conditions</th><th>β ≥ -1/2</th><th>-1/2 &lt; β &lt; 1/2</th></tr></thead><tbody><tr><td></td><td>β ≥ -1/2</td><td>β ≥ -1/2</td></tr><tr><td>Conditions</td><td>β ≥ -1/2</td><td>β ≥ -1/2</td></tr><tr><td>Conditions</td><td>β ≥ -1/2</td><td>β ≥ -1/2</td></tr></tbody></table>

Consequently, $u$ is a global weak solution to the Lions' equation (L).

We provide the proof of (a) in this section, and the proof of (b), (c), and (d) is deferred to Section 6. Let us first give some remarks.

**Remark 5.2.** Property (d) allows us to make sense of $u(t)$ for any $t \ge 0$ in $\mathcal{S}'$. Besides, when $\beta \ge -1/2$ and $\tilde{p}_L(\beta) < p < 1$, we also have trace in $L^p$, but this topology is not compatible with that of $\mathcal{S}'$, so one cannot identify the limits.

**Remark 5.3** (Whitney trace). When $\beta > -1/2$, there is another notion of trace (valid for any function $u \in T_{\beta+1}^p$ for $0 < p \le \infty$) in the sense of taking the limit of averages on Whitney cubes as $t \to 0$ for a.e. $x \in \mathbb{R}^n$. More precisely, when $\beta > -1/2$, for a.e. $x \in \mathbb{R}^n$, we have

$$ \lim_{t \to 0+} \left( \int_{W(t,x)} |u(s, y)|^2 dsdy \right)^{1/2} = 0. $$

The reader can refer to [AH23b, Lemma 5.3] for the proof.

**Remark 5.4.** We mention that the bounded extension of $\mathcal{R}_{1/2}^L$ on tent spaces $T_0^p$ was studied in [AMP19, Proposition 2.12]. It was claimed there that $\mathcal{R}_{1/2}^L$ extends to a bounded operator from $T_0^p$ to the Kenig-Pipher space $X^p$ for $0 < p \le \infty$. However, the proof has a gap and we do not know if their full conclusion is correct. Yet, as $T_{1/2}^p$ embeds into $X^p$, our Theorem 5.1 ensures that their proposition is still valid in some range of $p$, which in fact covers the usage of this proposition (only when $L = -\Delta$ or $p = 2$) in that paper.

**Remark 5.5.** As we shall see in Section 6, the proofs of (b), (c), and (d) work for any $\beta > -1$ and $\frac{n}{n+2\beta+2} < p \le \infty$, for which the operators $\mathcal{R}_\kappa^L$ are bounded from $T_{\beta+1/2}^p$ to $T_{\beta+1/2+\kappa}^p$ for $\kappa = 0, 1/2$ ($\mathcal{R}_0^L$ is defined right below).

We prove Theorem 5.1 (a) using the following two lemmas requiring different methods. Define the operator $\mathcal{R}_0^L: L_c^1((0, \infty); W^{2,2}) \to L_{\text{loc}}^\infty((0, \infty); L^2)$ by the $L^2$-valued Bochner integrals

$$ (5.1) \qquad \mathcal{R}_0^L(F)(t) := \int_0^t \nabla e^{-(t-s)L} \operatorname{div} F(s) ds, \quad \forall t > 0. $$
(d) (Continuity and trace) $u \in C([0, \infty); \mathcal{S}')$ with $u(0) = 0$.

When $t \to 0$, the convergence occurs in the following spaces shown in Table 1, with arbitrary parameters $\delta > 0$, $q \in [p, \infty]$, and $s \in [-1, 2\beta + 1]$, and a fixed one $r := \frac{np}{n-(2\beta+2)p} > 1$.

<table><caption>TABLE 1. Spaces for convergence of $\mathcal{R}_{1/2}^L(F)(t)$ as $t \to 0$.</caption><thead><tr><th>Conditions</th><th>tilde p<sub>L</sub>(&beta;) &lt; p &le; 2</th><th>2 &lt; p &le; &infin;</th></tr></thead><tbody><tr><td>&beta; &ge; -1/2</td><td>{ <i>S'</i> if p &lt; 1<br>L<sup>p</sup> if 1 &le; p &le; 2}</td><td>E<sub>&delta;</sub><sup>-1,q</sup></td></tr><tr><td>-1 &lt; &beta; &lt; -1/2</td><td>{ <i>H</i>'<sup>-1,r</sup> if p &le; 1<br><i>H</i>'<sup>s,p</sup> if 1 &lt; p &le; 2}</td><td>E<sub>&delta;</sub><sup>-1,q</sup></td></tr></tbody></table>

Consequently, $u$ is a global weak solution to the Lions' equation (L).

We provide the proof of (a) in this section, and the proof of (b), (c), and (d) is deferred to Section 6. Let us first give some remarks.

**Remark 5.2.** Property (d) allows us to make sense of $u(t)$ for any $t \ge 0$ in $\mathcal{S}'$. Besides, when $\beta \ge -1/2$ and $\tilde{p}_L(\beta) < p < 1$, we also have trace in $L^p$, but this topology is not compatible with that of $\mathcal{S}'$, so one cannot identify the limits.

**Remark 5.3** (Whitney trace). When $\beta > -1/2$, there is another notion of trace (valid for any function $u \in T_{\beta+1}^p$ for $0 < p \le \infty$) in the sense of taking the limit of averages on Whitney cubes as $t \to 0$ for a.e. $x \in \mathbb{R}^n$. More precisely, when $\beta > -1/2$, for a.e. $x \in \mathbb{R}^n$, we have

$$ \lim_{t \to 0+} \left( \int_{W(t,x)} |u(s, y)|^2 ds dy \right)^{1/2} = 0. $$

The reader can refer to [AH23b, Lemma 5.3] for the proof.

**Remark 5.4.** We mention that the bounded extension of $\mathcal{R}_{1/2}^L$ on tent spaces $T_0^p$ was studied in [AMP19, Proposition 2.12]. It was claimed there that $\mathcal{R}_{1/2}^L$ extends to a bounded operator from $T_0^p$ to the Kenig-Pipher space $X^p$ for $0 < p \le \infty$. However, the proof has a gap and we do not know if their full conclusion is correct. Yet, as $T_{1/2}^p$ embeds into $X^p$, our Theorem 5.1 ensures that their proposition is still valid in some range of $p$, which in fact covers the usage of this proposition (only when $L = -\Delta$ or $p = 2$) in that paper.

**Remark 5.5.** As we shall see in Section 6, the proofs of (b), (c), and (d) work for any $\beta > -1$ and $\frac{n}{n+2\beta+2} < p \le \infty$, for which the operators $\mathcal{R}_\kappa^L$ are bounded from $T_{\beta+1/2}^p$ to $T_{\beta+1/2+\kappa}^p$ for $\kappa = 0, 1/2$ ($\mathcal{R}_0^L$ is defined right below).

We prove Theorem 5.1 (a) using the following two lemmas requiring different methods. Define the operator $\mathcal{R}_0^L: L_c^1((0, \infty); W^{2,2}) \to L_{\text{loc}}^\infty((0, \infty); L^2)$ by the $L^2$-valued Bochner integrals

$$ (5.1) \qquad \mathcal{R}_0^L(F)(t) := \int_0^t \nabla e^{-(t-s)L} \operatorname{div} F(s) ds, \quad \forall t > 0. $$
We also use $p_L(\beta)$ defined in (1.7) and set

$$ \tilde{p}_L^b(\beta) := \frac{nq_+(L^*)'}{n + 2(\beta + 1)q_+(L^*)'}. $$

**Lemma 5.6.** Let $\beta > -1$ and $\tilde{p}_L^b(\beta) < p \le \infty$. The operator $\mathcal{R}_{1/2}^L$ (resp. $\mathcal{R}_0^L$) extends to a bounded operator from $T_{\beta+1/2}^p$ to $T_{\beta+1}^p$ (resp. $T_{\beta+1/2}^p$), also denoted by $\mathcal{R}_{1/2}^L$ (resp. $\mathcal{R}_0^L$).

**Lemma 5.7.** (i) Suppose $p_-(L) \ge 1$. Then the properties of $\mathcal{R}_{1/2}^L$ and $\mathcal{R}_0^L$ in Lemma 5.6 are also valid for $\beta > -1/2$ and $p_L(\beta) < p \le \infty$.

(ii) Suppose $p_-(L) < 1$. Then the properties of $\mathcal{R}_{1/2}^L$ and $\mathcal{R}_0^L$ in Lemma 5.6 are also valid for $\beta > \beta(L)$ and $p_L(\beta) < p \le 1$.

The proofs are provided in Sections 5.2 and 5.3, respectively. Let us first show that given these two lemmas, Theorem 5.1 (a) holds.

*Proof of Theorem 5.1 (a) assuming Lemmas 5.6 and 5.7.* Boundedness of $\mathcal{R}_{1/2}^L$ and $\mathcal{R}_0^L$ comes from interpolation of tent spaces, thanks to Lemmas 5.6 and 5.7. The number $\tilde{p}_L(\beta)$ is exactly designed for this interpolation argument. Using Proposition 5.8 (iii), the equality $\nabla \mathcal{R}_{1/2}^L(F) = \mathcal{R}_0^L(F)$ in $\mathcal{D}'(\mathbb{R}_+^{1+n})$ extends to all $F \in T_{\beta+1/2}^p$ by density (or weak*-density if $p = \infty$), so the estimate of $\nabla \mathcal{R}_{1/2}^L(F)$ follows. $\square$

We begin to verify the lemmas. An important tool for the proof of both lemmas is that the semigroup $(e^{-tL})$ satisfies the $L^p - L^q$ off-diagonal estimates, see [Aus07, Proposition 3.15]. More precisely, for $\max\{p_-(L), 1\} < p < q < p_+(L)$, there are constants $c, C > 0$ so that for any $t > 0$, $E, F \subset \mathbb{R}^n$ as Borel sets, and $f \in L^2 \cap L^p$,

$$ \|1_E e^{-tL} 1_F f\|_q \le C t^{-\frac{n}{2}[p,q]} \exp\left(-c \frac{\operatorname{dist}(E, F)^2}{t}\right) \|1_F f\|_p. $$

The same also holds for the family $(t^{1/2}\nabla e^{-tL})_{t>0}$ if replacing $p_\pm(L)$ by $q_\pm(L)$.

## 5.2. Proof of Lemma 5.6.

*Proof of Lemma 5.6.* It is a direct consequence of [AH23b, Proposition 3.3 & Corollary 3.5]. The assumptions there are verified in Proposition 5.8 just below. $\square$

**Proposition 5.8 (SIO properties of $\mathcal{R}_{1/2}^L$ and $\mathcal{R}_0^L$).** Let $\beta > -1$.

(i) The operator $\mathcal{R}_{1/2}^L$ given by (4.4) belongs to $\text{SIO}_{2,q,\infty}^{1/2+}$ when $q_+(L^*)' < q < p_+(L)$. It is bounded from $L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$ to $L_{\beta+1}^2(\mathbb{R}_+^{1+n})$.

(ii) The operator $\mathcal{R}_0^L$ given by (5.1) extends to an operator (also denoted by $\mathcal{R}_0^L$) in $\text{SIO}_{2,q,\infty}^{0+}$ for $q_+(L^*)' < q < q_+(L)$ that is bounded on $L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$.

(iii) For any $F \in L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$,

$$ \nabla \mathcal{R}_{1/2}^L(F) = \mathcal{R}_0^L(F) \quad \text{in } \mathcal{D}'(\mathbb{R}_+^{1+n}). $$

*Proof.* For (i), the off-diagonal estimates of $(t^{1/2}\nabla e^{-tL^*})_{t>0}$ imply that the function $(t, s) \mapsto 1_{\{s>t\}}(t, s)\nabla e^{-(s-t)L^*}$ lies in $\text{SK}_{2,q,\infty}^{1/2}$ for $\max\{q_-(L^*), 1\} < q < q_+(L^*)$ by definition of this class. Using duality of singular kernels (see [AH23b, Corollary 2.5]), we get that the kernel of $\mathcal{R}_{1/2}^L$, $K_{1/2}(t, s) := 1_{\{t>s\}}(t, s)e^{-(t-s)L}$ div belongs
We also use $p_L(\beta)$ defined in (1.7) and set

$$ \tilde{p}_L^b(\beta) := \frac{nq_+(L^*)'}{n + 2(\beta + 1)q_+(L^*)'}. $$

**Lemma 5.6.** Let $\beta > -1$ and $\tilde{p}_L^b(\beta) < p \le \infty$. The operator $\mathcal{R}_{1/2}^L$ (resp. $\mathcal{R}_0^L$) extends to a bounded operator from $T_{\beta+1/2}^p$ to $T_{\beta+1}^p$ (resp. $T_{\beta+1/2}^p$), also denoted by $\mathcal{R}_{1/2}^L$ (resp. $\mathcal{R}_0^L$).

**Lemma 5.7.** (i) Suppose $p_-(L) \ge 1$. Then the properties of $\mathcal{R}_{1/2}^L$ and $\mathcal{R}_0^L$ in Lemma 5.6 are also valid for $\beta > -1/2$ and $p_L(\beta) < p \le \infty$.

(ii) Suppose $p_-(L) < 1$. Then the properties of $\mathcal{R}_{1/2}^L$ and $\mathcal{R}_0^L$ in Lemma 5.6 are also valid for $\beta > \beta(L)$ and $p_L(\beta) < p \le 1$.

The proofs are provided in Sections 5.2 and 5.3, respectively. Let us first show that given these two lemmas, Theorem 5.1 (a) holds.

*Proof of Theorem 5.1 (a) assuming Lemmas 5.6 and 5.7.* Boundedness of $\mathcal{R}_{1/2}^L$ and $\mathcal{R}_0^L$ comes from interpolation of tent spaces, thanks to Lemmas 5.6 and 5.7. The number $\tilde{p}_L(\beta)$ is exactly designed for this interpolation argument. Using Proposition 5.8 (iii), the equality $\nabla \mathcal{R}_{1/2}^L(F) = \mathcal{R}_0^L(F)$ in $\mathcal{D}'(\mathbb{R}_+^{1+n})$ extends to all $F \in T_{\beta+1/2}^p$ by density (or weak*-density if $p = \infty$), so the estimate of $\nabla \mathcal{R}_{1/2}^L(F)$ follows. $\square$

We begin to verify the lemmas. An important tool for the proof of both lemmas is that the semigroup $(e^{-tL})$ satisfies the $L^p - L^q$ off-diagonal estimates, see [Aus07, Proposition 3.15]. More precisely, for $\max\{p_-(L), 1\} < p < q < p_+(L)$, there are constants $c, C > 0$ so that for any $t > 0$, $E, F \subset \mathbb{R}^n$ as Borel sets, and $f \in L^2 \cap L^p$,

$$ \|\mathbb{1}_E e^{-tL} \mathbb{1}_F f\|_q \le C t^{-\frac{n}{2}[p,q]} \exp\left(-c \frac{\operatorname{dist}(E, F)^2}{t}\right) \|\mathbb{1}_F f\|_p. $$

The same also holds for the family $(t^{1/2}\nabla e^{-tL})_{t>0}$ if replacing $p_\pm(L)$ by $q_\pm(L)$.

## 5.2. Proof of Lemma 5.6.

*Proof of Lemma 5.6.* It is a direct consequence of [AH23b, Proposition 3.3 & Corollary 3.5]. The assumptions there are verified in Proposition 5.8 just below. $\square$

**Proposition 5.8 (SIO properties of $\mathcal{R}_{1/2}^L$ and $\mathcal{R}_0^L$).** Let $\beta > -1$.

(i) The operator $\mathcal{R}_{1/2}^L$ given by (4.4) belongs to $\text{SIO}_{2,q,\infty}^{1/2+}$ when $q_+(L^*)' < q < p_+(L)$. It is bounded from $L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$ to $L_{\beta+1}^2(\mathbb{R}_+^{1+n})$.

(ii) The operator $\mathcal{R}_0^L$ given by (5.1) extends to an operator (also denoted by $\mathcal{R}_0^L$) in $\text{SIO}_{2,q,\infty}^{0+}$ for $q_+(L^*)' < q < q_+(L)$ that is bounded on $L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$.

(iii) For any $F \in L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$,

$$ \nabla \mathcal{R}_{1/2}^L(F) = \mathcal{R}_0^L(F) \quad \text{in } \mathcal{D}'(\mathbb{R}_+^{1+n}). $$

*Proof.* For (i), the off-diagonal estimates of $(t^{1/2}\nabla e^{-tL^*})_{t>0}$ imply that the function $(t, s) \mapsto \mathbb{1}_{\{s>t\}}(t, s)\nabla e^{-(s-t)L^*}$ lies in $\text{SK}_{2,q,\infty}^{1/2}$ for $\max\{q_-(L^*), 1\} < q < q_+(L^*)$ by definition of this class. Using duality of singular kernels (see [AH23b, Corollary 2.5]), we get that the kernel of $\mathcal{R}_{1/2}^L$, $K_{1/2}(t, s) := \mathbb{1}_{\{t>s\}}(t, s)e^{-(t-s)L}$ div belongs
to $\mathrm{SK}_{2,q,\infty}^{1/2}$ for $q_+(L^*)' < q < \max\{q_-(L^*), 1\}' = p_+(L)$. Applying [AH23b, Lemma 2.6] yields that $\mathcal{R}_{1/2}^L$ is bounded from $L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$ to $L_{\beta+1}^2(\mathbb{R}_+^{1+n})$ as $\beta + 1/2 > -1/2$. This proves (i).

For (ii), we construct the extension as follows. Define the operator $\mathcal{L}_0$ from $L^2((0, \infty); D(L))$ to $L_{\mathrm{loc}}^{\infty}((0, \infty); L^2)$ by the $L^2$-valued Bochner integrals

$$ \mathcal{L}_0(f)(t) := \int_0^t Le^{-(t-s)L} f(s) ds, \quad \forall t > 0. $$

De Simon's theorem states $\mathcal{L}_0$ extends to a bounded operator $\tilde{\mathcal{L}}_0$ on $L^2(\mathbb{R}_+^{1+n})$ [dS64]. Moreover, [AA11, Theorem 1.3] shows that $\tilde{\mathcal{L}}_0$ is bounded on $L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$ for any $\beta > -1$, and [AMP19, Proposition 2.5] reveals

$$ \mathcal{R}_0^L(F) = \nabla L^{-1/2} \tilde{\mathcal{L}}_0 L^{-1/2} \mathrm{div} F $$

for any $F \in L^1((0, \infty); W^{2,2})$. Recall that $\nabla L^{-1/2}$ and $L^{-1/2} \mathrm{div}$ are bounded on $L^2$ (see [AHL+02]), so we define the bounded extension of $\mathcal{R}_0^L$ on $L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$ by

$$ \mathcal{R}_0^L := \nabla L^{-1/2} \tilde{\mathcal{L}}_0 L^{-1/2} \mathrm{div}. $$

Next, let us verify that its kernel $K_0(t, s) := \mathbb{1}_{\{t>s\}}(t, s) \nabla e^{-(t-s)L^*} \mathrm{div}$ belongs to $\mathrm{SK}_{2,q,\infty}^0$ for $q_+(L^*)' < q < q_+(L)$. Indeed, it can be written as

$$ K_0(t, s) = \mathbb{1}_{\{t>s\}}(t, s) \nabla e^{-\frac{1}{2}(t-s)L} \mathbb{1}_{\{t>s\}}(t, s) e^{-\frac{1}{2}(t-s)L} \mathrm{div}, $$

where $(t, s) \mapsto \mathbb{1}_{\{t>s\}}(t, s) e^{-\frac{1}{2}(t-s)L}$ div lies in $\mathrm{SK}_{2,q,\infty}^{1/2}$ for $q_+(L^*)' < q < p_+(L)$, and $(t, s) \mapsto \mathbb{1}_{\{t>s\}}(t, s) \nabla e^{-\frac{1}{2}(t-s)L}$ lies in $\mathrm{SK}_{2,q,\infty}^{1/2}$ for $\max\{q_-(L), 1\} < q < q_+(L)$. Hence, we obtain $K_0 \in \mathrm{SK}_{2,q,\infty}^0$ for $q_+(L^*)' < q < q_+(L)$ by composition.

Finally, for the representation, we have to show that for any $F \in L_b^2(\mathbb{R}_+^{1+n})$ and a.e. $(t, x) \in (\mathbb{R}_+^{1+n} \setminus \pi(F))$,

$$ (5.2) \qquad \mathcal{R}_0^L(F)(t, x) = \int_0^t (\nabla e^{-(t-s)L} \mathrm{div} F(s))(x) ds. $$

The proof of (5.2), as well as the identity in (iii) follows by a verbatim adaptation of [AH23b, Lemma 5.1]. Details are left to the reader. $\square$

5.3. **Proof of Lemma 5.7.** It consists of two parts. For $p > 1$, we argue by duality. For $p \le 1$, in the spirit of [AP23, Theorem 3.2], we use atomic decomposition in weighted tent spaces. Recall that for $0 < p \le 1$, a measurable function $a$ on $\mathbb{R}_+^{1+n}$ is called a $T_\beta^p$-atom, if there exists a ball $B \subset \mathbb{R}^n$ so that $\mathrm{supp}(a) \subset [0, r(B)^2] \times B$, and

$$ \|a\|_{L_{\beta}^{2}(\mathbb{R}_{+}^{1+n})} \le |B|^{-[p,2]}. $$

Such a ball $B$ is said to be associated to $a$. Write $r := r(B)$, $C_0 := 2B$, and $C_j := 2^{j+1}B \setminus 2^j B$ for $j \ge 1$. For $j \ge 4$, define

$$ M_j^{(1)} := (0, (2^3 r)^2] \times C_j $$

$$ M_j^{(2)} := ((2^3 r)^2, (2^j r)^2) \times C_j $$

$$ M_j^{(3)} := [(2^j r)^2, (2^{j+1} r)^2) \times 2^{j+1} B. $$

The next lemma shows molecular decay for $\mathcal{R}_{1/2}^L$ acting on atoms.
to $\mathrm{SK}_{2,q,\infty}^{1/2}$ for $q_+(L^*)' < q < \max\{q_-(L^*), 1\}' = p_+(L)$. Applying [AH23b, Lemma 2.6] yields that $\mathcal{R}_{1/2}^L$ is bounded from $L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$ to $L_{\beta+1}^2(\mathbb{R}_+^{1+n})$ as $\beta + 1/2 > -1/2$. This proves (i).

For (ii), we construct the extension as follows. Define the operator $\mathcal{L}_0$ from $L^2((0, \infty); D(L))$ to $L_{\mathrm{loc}}^{\infty}((0, \infty); L^2)$ by the $L^2$-valued Bochner integrals

$$ \mathcal{L}_0(f)(t) := \int_0^t L e^{-(t-s)L} f(s) ds, \quad \forall t > 0. $$

De Simon's theorem states $\mathcal{L}_0$ extends to a bounded operator $\tilde{\mathcal{L}}_0$ on $L^2(\mathbb{R}_+^{1+n})$ [dS64]. Moreover, [AA11, Theorem 1.3] shows that $\tilde{\mathcal{L}}_0$ is bounded on $L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$ for any $\beta > -1$, and [AMP19, Proposition 2.5] reveals

$$ \mathcal{R}_0^L(F) = \nabla L^{-1/2} \tilde{\mathcal{L}}_0 L^{-1/2} \mathrm{div} F $$

for any $F \in L^1((0, \infty); W^{2,2})$. Recall that $\nabla L^{-1/2}$ and $L^{-1/2} \mathrm{div}$ are bounded on $L^2$ (see [AHL+02]), so we define the bounded extension of $\mathcal{R}_0^L$ on $L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$ by

$$ \mathcal{R}_0^L := \nabla L^{-1/2} \tilde{\mathcal{L}}_0 L^{-1/2} \mathrm{div}. $$

Next, let us verify that its kernel $K_0(t, s) := \mathbb{1}_{\{t>s\}}(t, s) \nabla e^{-(t-s)L^*} \mathrm{div}$ belongs to $\mathrm{SK}_{2,q,\infty}^0$ for $q_+(L^*)' < q < q_+(L)$. Indeed, it can be written as

$$ K_0(t, s) = \mathbb{1}_{\{t>s\}}(t, s) \nabla e^{-\frac{1}{2}(t-s)L} \mathbb{1}_{\{t>s\}}(t, s) e^{-\frac{1}{2}(t-s)L} \mathrm{div}, $$

where $(t, s) \mapsto \mathbb{1}_{\{t>s\}}(t, s) e^{-\frac{1}{2}(t-s)L}$ div lies in $\mathrm{SK}_{2,q,\infty}^{1/2}$ for $q_+(L^*)' < q < p_+(L)$, and $(t, s) \mapsto \mathbb{1}_{\{t>s\}}(t, s) \nabla e^{-\frac{1}{2}(t-s)L}$ lies in $\mathrm{SK}_{2,q,\infty}^{1/2}$ for $\max\{q_-(L), 1\} < q < q_+(L)$. Hence, we obtain $K_0 \in \mathrm{SK}_{2,q,\infty}^0$ for $q_+(L^*)' < q < q_+(L)$ by composition.

Finally, for the representation, we have to show that for any $F \in L_b^2(\mathbb{R}_+^{1+n})$ and a.e. $(t, x) \in (\mathbb{R}_+^{1+n} \setminus \pi(F))$,

$$ (5.2) \qquad \mathcal{R}_0^L(F)(t, x) = \int_0^t (\nabla e^{-(t-s)L} \mathrm{div} F(s))(x) ds. $$

The proof of (5.2), as well as the identity in (iii) follows by a verbatim adaptation of [AH23b, Lemma 5.1]. Details are left to the reader. $\square$

5.3. **Proof of Lemma 5.7.** It consists of two parts. For $p > 1$, we argue by duality. For $p \le 1$, in the spirit of [AP23, Theorem 3.2], we use atomic decomposition in weighted tent spaces. Recall that for $0 < p \le 1$, a measurable function $a$ on $\mathbb{R}_+^{1+n}$ is called a $T_\beta^p$-atom, if there exists a ball $B \subset \mathbb{R}^n$ so that $\mathrm{supp}(a) \subset [0, r(B)^2] \times B$, and

$$ \|a\|_{L_{\beta}^{2}(\mathbb{R}_{+}^{1+n})} \le |B|^{-[p,2]}. $$

Such a ball $B$ is said to be associated to $a$. Write $r := r(B)$, $C_0 := 2B$, and $C_j := 2^{j+1}B \setminus 2^j B$ for $j \ge 1$. For $j \ge 4$, define

$$ M_j^{(1)} := (0, (2^3 r)^2] \times C_j $$

$$ M_j^{(2)} := ((2^3 r)^2, (2^j r)^2) \times C_j $$

$$ M_j^{(3)} := [(2^j r)^2, (2^{j+1} r)^2) \times 2^{j+1} B. $$

The next lemma shows molecular decay for $\mathcal{R}_{1/2}^L$ acting on atoms.
**Lemma 5.9** (Molecular decay when $p_-(L) \ge 1$). Assume $p_-(L) \ge 1$. Let $\beta > -1/2$, $0 < p \le 1$, and $p_-(L) < q < 2$. There exists a constant $c > 0$ depending on $L$ and $q$, so that for any $T_{\beta+1/2}^p$-atom $a$ with an associated ball $B \subset \mathbb{R}^n$, the following estimates hold for $u := \mathcal{R}_{1/2}^L(a)$ and any $j \ge 4$,

$$ (5.3) \qquad \|u\|_{L_{\beta+1}^2(M_j^{(1)})} \lesssim 2^{jn[p,2]} e^{-c2^{2j}} |2^{j+1}B|^{[2,p]}, $$

$$ (5.4) \qquad \|u\|_{L_{\beta+1}^2(M_j^{(2)})} \lesssim 2^{-j(2\beta+1+n[q,p])} |2^{j+1}B|^{[2,p]}, $$

$$ (5.5) \qquad \|u\|_{L_{\beta+1}^2(M_j^{(3)})} \lesssim 2^{-j(2\beta+1+n[q,p])} |2^{j+1}B|^{[2,p]}. $$

The implicit constants are independent of $j$ and $B$.

*Proof.* Note that since $a \in L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$, Proposition 5.8 (i) ensures that $u := \mathcal{R}_{1/2}^L(a)$ lies in $L_{\beta+1}^2(\mathbb{R}_+^{1+n})$, and for a.e. $(t, x) \in \mathbb{R}_+^{1+n}$,

$$ u(t, x) = \int_0^t (e^{-(t-s)L} \operatorname{div} a(s))(x) \, ds. $$

This allows us to use duality to get estimates of $u(t)$ for a.e. $t > 0$. To this end, we fix $\phi \in C_c^\infty(\mathbb{R}^n)$ and set $v_t(s, y) := (e^{-(t-s)L^*} \phi)(y)$. We get

$$ \langle u(t), \phi \rangle = \int_0^t \int_{\mathbb{R}^n} a(s, y) \cdot \overline{\nabla v_t(s, y)} \, ds dy. $$

Using the properties of $a$ and Cauchy-Schwarz inequality, we have

$$ (5.6) \qquad |\langle u(t), \phi \rangle| \le |B|^{[2,p]} \left( \int_0^{\min\{r^2, t\}} \int_B s^{2\beta+1} |\nabla v_t(s, y)|^2 ds dy \right)^{1/2}. $$

The first inequality (5.3) follows from the estimate

$$ (5.7) \qquad \|u(t)\|_{L^2(C_j)} \lesssim |B|^{[2,p]} t^{\beta+1/2} e^{-\frac{c(2^j r)^2}{t}} \quad 0 < t \le (2^3 r)^2, $$

by integrating the square of both sides over $0 < t \le (2^3r)^2$. To prove (5.7), one applies (5.6) for $\phi \in C_c^\infty(\mathbb{R}^n)$ and $\operatorname{supp}(\phi) \subset C_j$. The $L^2-L^2$ off-diagonal estimates of $(t^{1/2}\nabla e^{-tL^*})$ yield that there exists $c_1 > 0$ only depending on $L$ so that for any $c \in (0, c_1)$,

$$ \int_0^t \int_B s^{2\beta+1} |\nabla v_t(s, y)|^2 ds dy \lesssim t^{2\beta+1} e^{-\frac{2c(2^j r)^2}{t}} \|\phi\|_2^2. $$

Thus, one obtains (5.7).

The second inequality (5.4) is obtained similarly from

$$ \|u(t)\|_{L^2(C_j)} \lesssim |B|^{[q,p]} r^{2\beta+1} t^{-\frac{n}{2}[2,q]} e^{-\frac{c(2^j r)^2}{t}}, \quad (2^3 r)^2 < t < (2^j r)^2. $$

To this end, we use again (5.6) for $\phi \in C_c^\infty(\mathbb{R}^n)$ and $\operatorname{supp}(\phi) \subset C_j$. Note that $v_t$ is a weak solution to the backward equation $-\partial_s v - \operatorname{div}(A^*\nabla v) = 0$ on $(-\infty, t) \times \mathbb{R}^n$. As $\beta > -1/2$, we infer from Caccioppoli's inequality (cf. Lemma 4.2) that

$$ \begin{aligned} \int_0^{r^2} \int_B s^{2\beta+1} |\nabla e^{-(t-s)L^*} \phi(y)|^2 ds dy &\le (r^2)^{2\beta+1} \int_0^{r^2} \int_B |\nabla e^{-(t-s)L^*} \phi(y)|^2 ds dy \\ &\lesssim (r^2)^{2\beta} \int_0^{2r^2} \int_{2B} |e^{-(t-s)L^*} \phi(y)|^2 ds dy. \end{aligned} $$
**Lemma 5.9** (Molecular decay when $p_-(L) \ge 1$). Assume $p_-(L) \ge 1$. Let $\beta > -1/2$, $0 < p \le 1$, and $p_-(L) < q < 2$. There exists a constant $c > 0$ depending on $L$ and $q$, so that for any $T_{\beta+1/2}^p$-atom $a$ with an associated ball $B \subset \mathbb{R}^n$, the following estimates hold for $u := \mathcal{R}_{1/2}^L(a)$ and any $j \ge 4$,

$$ (5.3) \qquad \|u\|_{L_{\beta+1}^2(M_j^{(1)})} \lesssim 2^{jn[p,2]} e^{-c2^{2j}} |2^{j+1}B|^{[2,p]}, $$

$$ (5.4) \qquad \|u\|_{L_{\beta+1}^2(M_j^{(2)})} \lesssim 2^{-j(2\beta+1+n[q,p])} |2^{j+1}B|^{[2,p]}, $$

$$ (5.5) \qquad \|u\|_{L_{\beta+1}^2(M_j^{(3)})} \lesssim 2^{-j(2\beta+1+n[q,p])} |2^{j+1}B|^{[2,p]}. $$

The implicit constants are independent of $j$ and $B$.

*Proof.* Note that since $a \in L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})$, Proposition 5.8 (i) ensures that $u := \mathcal{R}_{1/2}^L(a)$ lies in $L_{\beta+1}^2(\mathbb{R}_+^{1+n})$, and for a.e. $(t, x) \in \mathbb{R}_+^{1+n}$,

$$ u(t, x) = \int_0^t (e^{-(t-s)L} \operatorname{div} a(s))(x) \, ds. $$

This allows us to use duality to get estimates of $u(t)$ for a.e. $t > 0$. To this end, we fix $\phi \in C_c^\infty(\mathbb{R}^n)$ and set $v_t(s, y) := (e^{-(t-s)L^*} \phi)(y)$. We get

$$ \langle u(t), \phi \rangle = \int_0^t \int_{\mathbb{R}^n} a(s, y) \cdot \overline{\nabla v_t(s, y)} \, ds dy. $$

Using the properties of $a$ and Cauchy-Schwarz inequality, we have

$$ (5.6) \qquad |\langle u(t), \phi \rangle| \le |B|^{[2,p]} \left( \int_0^{\min\{r^2, t\}} \int_B s^{2\beta+1} |\nabla v_t(s, y)|^2 ds dy \right)^{1/2}. $$

The first inequality (5.3) follows from the estimate

$$ (5.7) \qquad \|u(t)\|_{L^2(C_j)} \lesssim |B|^{[2,p]} t^{\beta+1/2} e^{-\frac{c(2^j r)^2}{t}} \quad 0 < t \le (2^3 r)^2, $$

by integrating the square of both sides over $0 < t \le (2^3r)^2$. To prove (5.7), one applies (5.6) for $\phi \in C_c^\infty(\mathbb{R}^n)$ and $\operatorname{supp}(\phi) \subset C_j$. The $L^2-L^2$ off-diagonal estimates of $(t^{1/2}\nabla e^{-tL^*})$ yield that there exists $c_1 > 0$ only depending on $L$ so that for any $c \in (0, c_1)$,

$$ \int_0^t \int_B s^{2\beta+1} |\nabla v_t(s, y)|^2 ds dy \lesssim t^{2\beta+1} e^{-\frac{2c(2^j r)^2}{t}} \|\phi\|_2^2. $$

Thus, one obtains (5.7).

The second inequality (5.4) is obtained similarly from

$$ \|u(t)\|_{L^2(C_j)} \lesssim |B|^{[q,p]} r^{2\beta+1} t^{-\frac{n}{2}[2,q]} e^{-\frac{c(2^j r)^2}{t}}, \quad (2^3 r)^2 < t < (2^j r)^2. $$

To this end, we use again (5.6) for $\phi \in C_c^\infty(\mathbb{R}^n)$ and $\operatorname{supp}(\phi) \subset C_j$. Note that $v_t$ is a weak solution to the backward equation $-\partial_s v - \operatorname{div}(A^*\nabla v) = 0$ on $(-\infty, t) \times \mathbb{R}^n$. As $\beta > -1/2$, we infer from Caccioppoli's inequality (cf. Lemma 4.2) that

$$ \begin{aligned} \int_0^{r^2} \int_B s^{2\beta+1} |\nabla e^{-(t-s)L^*} \phi(y)|^2 ds dy &\le (r^2)^{2\beta+1} \int_0^{r^2} \int_B |\nabla e^{-(t-s)L^*} \phi(y)|^2 ds dy \\ &\lesssim (r^2)^{2\beta} \int_0^{2r^2} \int_{2B} |e^{-(t-s)L^*} \phi(y)|^2 ds dy. \end{aligned} $$
Using Hölder's inequality and $L^2 - L^{q'}$ off-diagonal estimates of $(e^{-tL^*})$, we get $c_2 > 0$, depending on $L$ and $q$, so that for any $c \in (0, c_2)$,

$$ \begin{aligned} \|\mathbf{1}_{2B} e^{-(t-s)L^*} \phi\|_2 &\lesssim |B|^{[2,q']}|\mathbf{1}_{2B} e^{-(t-s)L^*} \phi\|_q \\ &\lesssim |B|^{[2,q']} t^{-\frac{n}{2}[2,q']} e^{-\frac{c(2^j r)^2}{t}} \|\phi\|_2. \end{aligned} $$

Hence, if $\|\phi\|_2 = 1$, then plugging the above estimates in (5.6) implies

$$ \begin{aligned} |\langle u(t), \phi \rangle|^2 &\le |B|^{2[2,p]} \int_0^{r^2} \int_B s^{2\beta+1} |\nabla v_t(s, y)|^2 ds dy \\ &\lesssim r^{4\beta} \int_0^{2r^2} \|\mathbf{1}_{2B} e^{-(t-s)L^*} \phi\|_2^2 ds \lesssim |B|^{2[q,p]} r^{2(2\beta+1)} t^{-n[2,q']} e^{-\frac{2c(2^j r)^2}{t}} \end{aligned} $$

as desired.

The third inequality (5.5) follows from a similar argument as that of (5.4) with the same constant $c_2$. Details are left to the reader. The assertions hence hold for $0 < c < \min\{c_1, c_2\}$, which only depends on $L$ and $q$. $\square$

When $p_-(L) < 1$, we have even better decay for all $\beta > -1$, using pointwise estimates and Hölder continuity of weak solutions to the backward equation.

**Corollary 5.10** (Molecular decay when $p_-(L) < 1$). Suppose $p_-(L) < 1$. Let $\beta > -1$, $0 < p \le 1$, and $0 < \eta < n(\frac{1}{p_-(L)} - 1)$. There exists a constant $c > 0$ only depending on $L$, so that for any $T_{\beta+1/2}^p$-atom $a$ with an associated ball $B \subset \mathbb{R}^n$, the following estimates hold for $u := \mathcal{R}_{1/2}^L(a)$ and any $j \ge 4$,

$$ \begin{aligned} \|u\|_{L_{\beta+1}^2(M_j^{(1)})} &\lesssim 2^{jn[p,2]} e^{-c2^{2j}} |2^{j+1}B|^{[2,p]}, \\ \|u\|_{L_{\beta+1}^2(M_j^{(2)})} &\lesssim 2^{-j(2\beta+1+\eta-n[p,1])} |2^{j+1}B|^{[2,p]}, \\ \|u\|_{L_{\beta+1}^2(M_j^{(3)})} &\lesssim 2^{-j(2\beta+1+\eta-n[p,1])} |2^{j+1}B|^{[2,p]}. \end{aligned} $$

The implicit constants are independent of $j$ and $B$.

**Proof.** Fix $c > 0$ as the constant given by the $L^2 - L^2$ off-diagonal estimates of $(t^{1/2}\nabla e^{-tL^*})$, which only depends on $L$. The first inequality is the same as in Lemma 5.9. We only need to show the second, and the third follows analogously. The second inequality is also obtained by integrating the square of

$$ (5.8) \quad \|u(t)\|_{L^2(C_j)} \lesssim |B|^{[2,p]} r^{\frac{n}{2}+2\beta+1+\eta} t^{-\left(\frac{n}{4}+\frac{\eta}{2}\right)} e^{-\frac{c(2^j r)^2}{t}}, \quad (2^3 r)^2 < t < (2^j r)^2. $$

To prove it, we use (5.6) again for $\phi \in C_c^\infty(\mathbb{R}^n)$ with $\|\phi\|_2 = 1$ and $\text{supp}(\phi) \subset C_j$. When $p_-(L) < 1$, we know from [AE23, Chapter 14] that $(e^{-tL^*})$ satisfies $L^2 - L^\infty$ off-diagonal estimates, and it is uniformly bounded from $L^2$ to $\dot{\Lambda}^\eta$ for $0 < \eta < n(\frac{1}{p_-(L)} - 1)$, where $\dot{\Lambda}^\eta$ is the homogeneous $\eta$-Hölder space. In particular, $v_t(s, y) := (e^{-(t-s)L^*} \phi)(y)$ has pointwise values. Set $\tilde{v}_t(s, y) := v_t(s, y) - v_t(0, 0)$. Write $\tilde{v}_t(s, y) = (v_t(s, y) - v_t(s, 0)) + (v_t(s, 0) - v_t(0, 0))$ and we estimate each term separately. For the first term, we use Hölder continuity. For the second, we write it as $\int_0^s (L^* e^{-(t-\tau)L^*} \phi)(0) d\tau$ and use $L^2 - L^\infty$ boundedness of $(tL^* e^{-tL^*})$. In summary, as $\text{dist}(\text{supp}(\phi), B) \sim 2^j r$, we obtain that for any $0 < \eta < n(\frac{1}{p_-(L)} - 1)$,

$$ (5.9) \quad \sup_{0<s<2r^2, y \in 2B} |\tilde{v}_t(s, y)| \lesssim t^{-\left(\frac{n}{4}+\frac{\eta}{2}\right)} e^{-\frac{c(2^j r)^2}{t}} r^\eta \|\phi\|_2. $$
Using Hölder's inequality and $L^2 - L^{q'}$ off-diagonal estimates of $(e^{-tL^*})$, we get $c_2 > 0$, depending on $L$ and $q$, so that for any $c \in (0, c_2)$,

$$ \begin{aligned} \|\mathbf{1}_{2B} e^{-(t-s)L^*} \phi\|_2 &\lesssim |B|^{[2,q']}|\mathbf{1}_{2B} e^{-(t-s)L^*} \phi\|_q \\ &\lesssim |B|^{[2,q']} t^{-\frac{n}{2}[2,q']} e^{-\frac{c(2^j r)^2}{t}} \|\phi\|_2. \end{aligned} $$

Hence, if $\|\phi\|_2 = 1$, then plugging the above estimates in (5.6) implies

$$ \begin{aligned} |\langle u(t), \phi \rangle|^2 &\le |B|^{2[2,p]} \int_0^{r^2} \int_B s^{2\beta+1} |\nabla v_t(s, y)|^2 ds dy \\ &\lesssim r^{4\beta} \int_0^{2r^2} \|\mathbf{1}_{2B} e^{-(t-s)L^*} \phi\|_2^2 ds \lesssim |B|^{2[q,p]} r^{2(2\beta+1)} t^{-n[2,q']} e^{-\frac{2c(2^j r)^2}{t}} \end{aligned} $$

as desired.

The third inequality (5.5) follows from a similar argument as that of (5.4) with the same constant $c_2$. Details are left to the reader. The assertions hence hold for $0 < c < \min\{c_1, c_2\}$, which only depends on $L$ and $q$. $\square$

When $p_-(L) < 1$, we have even better decay for all $\beta > -1$, using pointwise estimates and Hölder continuity of weak solutions to the backward equation.

**Corollary 5.10** (Molecular decay when $p_-(L) < 1$). Suppose $p_-(L) < 1$. Let $\beta > -1$, $0 < p \le 1$, and $0 < \eta < n(\frac{1}{p_-(L)} - 1)$. There exists a constant $c > 0$ only depending on $L$, so that for any $T_{\beta+1/2}^p$-atom $a$ with an associated ball $B \subset \mathbb{R}^n$, the following estimates hold for $u := \mathcal{R}_{1/2}^L(a)$ and any $j \ge 4$,

$$ \begin{aligned} \|u\|_{L_{\beta+1}^2(M_j^{(1)})} &\lesssim 2^{jn[p,2]} e^{-c2^{2j}} |2^{j+1}B|^{[2,p]}, \\ \|u\|_{L_{\beta+1}^2(M_j^{(2)})} &\lesssim 2^{-j(2\beta+1+\eta-n[p,1])} |2^{j+1}B|^{[2,p]}, \\ \|u\|_{L_{\beta+1}^2(M_j^{(3)})} &\lesssim 2^{-j(2\beta+1+\eta-n[p,1])} |2^{j+1}B|^{[2,p]}. \end{aligned} $$

The implicit constants are independent of $j$ and $B$.

**Proof.** Fix $c > 0$ as the constant given by the $L^2 - L^2$ off-diagonal estimates of $(t^{1/2}\nabla e^{-tL^*})$, which only depends on $L$. The first inequality is the same as in Lemma 5.9. We only need to show the second, and the third follows analogously. The second inequality is also obtained by integrating the square of

$$ (5.8) \quad \|u(t)\|_{L^2(C_j)} \lesssim |B|^{[2,p]} r^{\frac{n}{2}+2\beta+1+\eta} t^{-\left(\frac{n}{4}+\frac{\eta}{2}\right)} e^{-\frac{c(2^j r)^2}{t}}, \quad (2^3 r)^2 < t < (2^j r)^2. $$

To prove it, we use (5.6) again for $\phi \in C_c^\infty(\mathbb{R}^n)$ with $\|\phi\|_2 = 1$ and $\text{supp}(\phi) \subset C_j$. When $p_-(L) < 1$, we know from [AE23, Chapter 14] that $(e^{-tL^*})$ satisfies $L^2 - L^\infty$ off-diagonal estimates, and it is uniformly bounded from $L^2$ to $\dot{\Lambda}^\eta$ for $0 < \eta < n(\frac{1}{p_-(L)} - 1)$, where $\dot{\Lambda}^\eta$ is the homogeneous $\eta$-Hölder space. In particular, $v_t(s, y) := (e^{-(t-s)L^*} \phi)(y)$ has pointwise values. Set $\tilde{v}_t(s, y) := v_t(s, y) - v_t(0, 0)$. Write $\tilde{v}_t(s, y) = (v_t(s, y) - v_t(s, 0)) + (v_t(s, 0) - v_t(0, 0))$ and we estimate each term separately. For the first term, we use Hölder continuity. For the second, we write it as $\int_0^s (L^* e^{-(t-\tau)L^*} \phi)(0) d\tau$ and use $L^2 - L^\infty$ boundedness of $(tL^* e^{-tL^*})$. In summary, as $\text{dist}(\text{supp}(\phi), B) \sim 2^j r$, we obtain that for any $0 < \eta < n(\frac{1}{p_-(L)} - 1)$,

$$ (5.9) \quad \sup_{0<s<2r^2, y \in 2B} |\tilde{v}_t(s, y)| \lesssim t^{-\left(\frac{n}{4}+\frac{\eta}{2}\right)} e^{-\frac{c(2^j r)^2}{t}} r^\eta \|\phi\|_2. $$
By analyticity of the semigroup again, one finds that $\partial_s \tilde{v}_t(s, y)$ satisfies the similar estimate with an extra factor $(t-s)^{-1} \sim t^{-1}$.

Let $\chi$ be in $C^\infty([0, \infty))$ with $\text{supp}(\chi) \subset [0, \frac{3}{2}r^2]$, $0 \le \chi \le 1$, $\chi = 1$ on $[0, r^2]$, and $\|\chi'\|_\infty \lesssim r^{-2}$. Using integration by parts in variable $s$, we get

$$ \begin{aligned} \int_0^{r^2} \int_B |\nabla \tilde{v}_t(s, y)|^2 s^{2\beta+1} ds dy &\le \int_0^{\frac{3}{2}r^2} \int_B \chi(s) |\nabla \tilde{v}_t(s, y)|^2 s^{2\beta+1} ds dy \\ &\lesssim \int_0^{\frac{3}{2}r^2} \int_B |\chi'(s)| |\nabla \tilde{v}_t(s, y)|^2 s^{2\beta+2} ds dy \\ &\quad + \int_0^{\frac{3}{2}r^2} \int_B \chi(s) |\nabla \tilde{v}_t(s, y)| |\nabla \partial_s \tilde{v}_t(s, y)| s^{2\beta+2} ds dy =: I_1 + I_2. \end{aligned} $$

For $I_1$, note that $\tilde{v}_t$ is also a weak solution to the backward equation $-\partial_s v - \text{div}(A^*\nabla v) = 0$ on $(-\infty, t) \times \mathbb{R}^n$. As $|\chi'| \lesssim r^{-2}$ and $2\beta + 2 > 0$, we get from Caccioppoli's inequality and (5.9) that

$$ \begin{aligned} I_1 &\lesssim r^{4\beta+2} \int_0^{\frac{3}{2}r^2} \int_B |\nabla \tilde{v}_t|^2 \lesssim r^{4\beta} \int_0^{2r^2} \int_{2B} |\tilde{v}_t|^2 \\ &\lesssim |B| r^{4\beta+2+2\eta} t^{-(\frac{n}{2}+\eta)} e^{-\frac{2c(2^j r)^2}{t}} \|\phi\|_2^2. \end{aligned} $$

For $I_2$, using again Caccioppoli's inequality and the above estimate for $\partial_s \tilde{v}_t(s, y)$, we find

$$ \begin{aligned} I_2 &\lesssim r^{4\beta+4} \int_0^{\frac{3}{2}r^2} \int_B |\nabla \partial_s \tilde{v}_t|^2 \lesssim r^{4\beta+2} \int_0^{2r^2} \int_{2B} |\partial_s \tilde{v}_t|^2 \\ &\lesssim |B| r^{4\beta+4+2\eta} t^{-(\frac{n}{2}+1+\eta)} e^{-\frac{2c(2^j r)^2}{t}} \|\phi\|_2^2. \end{aligned} $$

Gathering these estimates, we easily obtain (5.8). $\square$

<figure><img src="image_3.png" /></figure>

*Proof of Lemma 5.7.* We begin with (i) and split the discussion into two cases.

Case 1: $\max\{p_L(\beta), 1\} < p \le \infty$. First consider $\mathcal{R}_{1/2}^L$. By density (or weak*-density when $p = \infty$), it suffices to show that for any $F \in T_{\beta+1/2}^p \cap L^2(\mathbb{R}_+^{1+n})$,

$$ (5.10) \qquad \|\mathcal{R}_{1/2}^L(F)\|_{T_{\beta+1}^p} \lesssim \|F\|_{T_{\beta+1/2}^p}. $$

To prove it, we use duality. Fix $\phi \in C_c^\infty(\mathbb{R}_+^{1+n})$ and Fubini's theorem ensures that

$$ \langle \mathcal{R}_{1/2}^L(F), \phi \rangle_{L^2(\mathbb{R}_+^{1+n})} = \int_0^\infty \int_{\mathbb{R}^n} F(s, y) \left( \int_s^\infty \overline{\nabla e^{-(t-s)L^*} \phi(t)} \right) (y) dt \, ds dy. $$

Define

$$ v(s, y) := \int_s^\infty e^{-(t-s)L^*} \phi(t) dt. $$

As $1 \le p' < \infty$, by duality of tent spaces, (5.10) follows the claim

$$ (5.11) \qquad \|\nabla v\|_{T_{-(\beta+1/2)}^{p'}} \lesssim \|\phi\|_{T_{-(\beta+1)}^{p'}}. $$

Indeed, observe that $v \in L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$ is a weak solution to the backward equation $-\partial_s v - \text{div}(A^*\nabla v) = \phi$ on $\mathbb{R}^{1+n}$. Applying Corollary 4.3 leads to

$$ \|\nabla v\|_{T_{-(\beta+1/2)}^{p'}} \lesssim \|v\|_{T_{-\beta}^{p'}} + \|\phi\|_{T_{-\beta-1}^{p'}}. $$
By analyticity of the semigroup again, one finds that $\partial_s \tilde{v}_t(s, y)$ satisfies the similar estimate with an extra factor $(t-s)^{-1} \sim t^{-1}$.

Let $\chi$ be in $C^\infty([0, \infty))$ with $\text{supp}(\chi) \subset [0, \frac{3}{2}r^2]$, $0 \le \chi \le 1$, $\chi = 1$ on $[0, r^2]$, and $\|\chi'\|_\infty \lesssim r^{-2}$. Using integration by parts in variable $s$, we get

$$ \begin{aligned} \int_0^{r^2} \int_B |\nabla \tilde{v}_t(s, y)|^2 s^{2\beta+1} ds dy &\le \int_0^{\frac{3}{2}r^2} \int_B \chi(s) |\nabla \tilde{v}_t(s, y)|^2 s^{2\beta+1} ds dy \\ &\lesssim \int_0^{\frac{3}{2}r^2} \int_B |\chi'(s)| |\nabla \tilde{v}_t(s, y)|^2 s^{2\beta+2} ds dy \\ &\quad + \int_0^{\frac{3}{2}r^2} \int_B \chi(s) |\nabla \tilde{v}_t(s, y)| |\nabla \partial_s \tilde{v}_t(s, y)| s^{2\beta+2} ds dy =: I_1 + I_2. \end{aligned} $$

For $I_1$, note that $\tilde{v}_t$ is also a weak solution to the backward equation $-\partial_s v - \text{div}(A^*\nabla v) = 0$ on $(-\infty, t) \times \mathbb{R}^n$. As $|\chi'| \lesssim r^{-2}$ and $2\beta + 2 > 0$, we get from Caccioppoli's inequality and (5.9) that

$$ \begin{aligned} I_1 &\lesssim r^{4\beta+2} \int_0^{\frac{3}{2}r^2} \int_B |\nabla \tilde{v}_t|^2 \lesssim r^{4\beta} \int_0^{2r^2} \int_{2B} |\tilde{v}_t|^2 \\ &\lesssim |B| r^{4\beta+2+2\eta} t^{-(\frac{n}{2}+\eta)} e^{-\frac{2c(2^j r)^2}{t}} \|\phi\|_2^2. \end{aligned} $$

For $I_2$, using again Caccioppoli's inequality and the above estimate for $\partial_s \tilde{v}_t(s, y)$, we find

$$ \begin{aligned} I_2 &\lesssim r^{4\beta+4} \int_0^{\frac{3}{2}r^2} \int_B |\nabla \partial_s \tilde{v}_t|^2 \lesssim r^{4\beta+2} \int_0^{2r^2} \int_{2B} |\partial_s \tilde{v}_t|^2 \\ &\lesssim |B| r^{4\beta+4+2\eta} t^{-(\frac{n}{2}+1+\eta)} e^{-\frac{2c(2^j r)^2}{t}} \|\phi\|_2^2. \end{aligned} $$

Gathering these estimates, we easily obtain (5.8). □

*Proof of Lemma 5.7.* We begin with (i) and split the discussion into two cases.

Case 1: $\max\{p_L(\beta), 1\} < p \le \infty$. First consider $\mathcal{R}_{1/2}^L$. By density (or weak*-density when $p = \infty$), it suffices to show that for any $F \in T_{\beta+1/2}^p \cap L^2(\mathbb{R}_+^{1+n})$,

$$ (5.10) \qquad \|\mathcal{R}_{1/2}^L(F)\|_{T_{\beta+1}^p} \lesssim \|F\|_{T_{\beta+1/2}^p}. $$

To prove it, we use duality. Fix $\phi \in C_c^\infty(\mathbb{R}_+^{1+n})$ and Fubini's theorem ensures that

$$ \langle \mathcal{R}_{1/2}^L(F), \phi \rangle_{L^2(\mathbb{R}_+^{1+n})} = \int_0^\infty \int_{\mathbb{R}^n} F(s, y) \left( \int_s^\infty \overline{\nabla e^{-(t-s)L^*} \phi(t)} \right) (y) dt \, ds dy. $$

Define

$$ v(s, y) := \int_s^\infty e^{-(t-s)L^*} \phi(t) dt. $$

As $1 \le p' < \infty$, by duality of tent spaces, (5.10) follows the claim

$$ (5.11) \qquad \|\nabla v\|_{T_{-(\beta+1/2)}^{p'}} \lesssim \|\phi\|_{T_{-(\beta+1)}^{p'}}. $$

Indeed, observe that $v \in L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$ is a weak solution to the backward equation $-\partial_s v - \text{div}(A^*\nabla v) = \phi$ on $\mathbb{R}^{1+n}$. Applying Corollary 4.3 leads to

$$ \|\nabla v\|_{T_{-(\beta+1/2)}^{p'}} \lesssim \|v\|_{T_{-\beta}^{p'}} + \|\phi\|_{T_{-\beta-1}^{p'}}. $$
As we assume $p_-(L) \ge 1$, we claim that for $\beta > -1/2$ and $0 < p' < \infty$,

$$ \|v\|_{T_{-\beta}^{p'}} \lesssim \|\phi\|_{T_{-\beta-1}^{p'}}. $$

Namely, apply the results in [AH23b]: first, the operator $\phi \mapsto v$ lies in $\text{SIO}_{2,q,\infty}^{1-}$ for $\max\{p_-(L^*), 1\} < q < p_+(L^*)$ by (4.14) and Proposition 2.11, then apply Proposition 3.4 and Corollary 3.6. This, together with the previous inequality, proves the claim (5.11). The bounded extension of $\mathcal{R}_{1/2}^L$ hence follows.

Next, consider $\mathcal{R}_0^L$. Fix $F \in T_{\beta+1/2}^p \cap L^2(\mathbb{R}_+^{1+n})$. Proposition 5.8 (iii) says $\mathcal{R}_0^L(F) = \nabla \mathcal{R}_{1/2}^L(F)$ in $L^2(\mathbb{R}_+^{1+n})$. Meanwhile, by Proposition 4.4, $u = \mathcal{R}_{1/2}^L(F)$ is a global weak solution to the equation $\partial_t u - \text{div}(A\nabla u) = \text{div} F$. Applying Corollary 4.3 and (5.10) yields that for any $F \in T_{\beta+1/2}^p \cap L^2(\mathbb{R}_+^{1+n})$,

$$ \|\mathcal{R}_0^L(F)\|_{T_{\beta+1/2}^p} = \|\nabla \mathcal{R}_{1/2}^L(F)\|_{T_{\beta+1/2}^p} \lesssim \|\mathcal{R}_{1/2}^L(F)\|_{T_{\beta+1}^p} + \|F\|_{T_{\beta+1/2}^p} \lesssim \|F\|_{T_{\beta+1/2}^p}. $$

Then the bounded extension of $\mathcal{R}_0^L$ follows by density.

Case 2: $p_L(\beta) < p \le 1$. Note that by definition of $\beta(L)$ (cf. (1.6)), this case only occurs when $\beta > \beta(L)$. We first consider the extension of $\mathcal{R}_{1/2}^L$. Thanks to [AH23b, Lemma 3.2], it is enough to show $\mathcal{R}_{1/2}^L$ is uniformly bounded on $T_{\beta+1/2}^p$-atoms.

Let us verify this. Let $a$ be a $T_{\beta+1/2}^p$-atom, $B \subset \mathbb{R}^n$ be a ball associated to $a$, and $u := \mathcal{R}_{1/2}^L(a)$. Denote by $Q_0 := (0, (2^3r(B))^2) \times 2^3B$. As $\beta + 1/2 > -1/2$, Proposition 5.8 (i) implies

$$ (5.12) \qquad \|u\|_{L_{\beta+1}^2(Q_0)} \lesssim \|a\|_{L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})} \lesssim |2^3B|^{[2,p]}. $$

Pick $q < 2$ sufficiently close to $p_-(L)$ so that

$$ (5.13) \qquad \frac{np_-(L)}{n + (2\beta + 1)p_-(L)} < \frac{nq}{n + (2\beta + 1)q} < p \le 1. $$

As $\beta > \beta(L) \ge -1/2$, we may apply Lemma 5.9 as well as the estimates given by (5.12) to get

$$ \|u\|_{T_{\beta+1}^p} \lesssim 1 + \sum_{j \ge 4} 2^{-jp(2\beta+1+n[q,p])} \le C $$

for some constant $C < \infty$ by (5.13). The assertion hence holds in this case.

Next, we consider the bounded extension of $\mathcal{R}_0^L$. In fact, one can apply Caccioppoli's inequality as in Corollary 4.3 to establish the same molecular decay estimates as in Lemma 5.9 for $\nabla u$ in $L_{\beta+1/2}^2(M_j^{(i)})$ ($1 \le i \le 3$). Meanwhile, using Proposition 5.8 (ii), one can also get as an alternative of (5.12)

$$ \|\nabla u\|_{L_{\beta+1/2}^2(Q_0)} \lesssim \|a\|_{L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})} \lesssim |2^3B|^{[2,p]}. $$

It is then enough to repeat the above computation. Details are left to the reader. This proves (i).

To prove (ii), as $p \le 1$, we can use the argument in Case 2, using (5.12) directly and replacing the estimates of Lemma 5.9 by those in Corollary 5.10. Picking $\eta > 0$ sufficiently close to $n(\frac{1}{p_-(L)} - 1)$ so that

$$ \frac{np_-(L)}{n + (2\beta + 1)p_-(L)} < \frac{n}{n + 2\beta + 1 + \eta} < p \le 1, $$
As we assume $p_-(L) \ge 1$, we claim that for $\beta > -1/2$ and $0 < p' < \infty$,

$$ \|v\|_{T_{-\beta}^{p'}} \lesssim \|\phi\|_{T_{-\beta-1}^{p'}}. $$

Namely, apply the results in [AH23b]: first, the operator $\phi \mapsto v$ lies in $\text{SIO}_{2,q,\infty}^{1-}$ for $\max\{p_-(L^*), 1\} < q < p_+(L^*)$ by (4.14) and Proposition 2.11, then apply Proposition 3.4 and Corollary 3.6. This, together with the previous inequality, proves the claim (5.11). The bounded extension of $\mathcal{R}_{1/2}^L$ hence follows.

Next, consider $\mathcal{R}_0^L$. Fix $F \in T_{\beta+1/2}^p \cap L^2(\mathbb{R}_+^{1+n})$. Proposition 5.8 (iii) says $\mathcal{R}_0^L(F) = \nabla \mathcal{R}_{1/2}^L(F)$ in $L^2(\mathbb{R}_+^{1+n})$. Meanwhile, by Proposition 4.4, $u = \mathcal{R}_{1/2}^L(F)$ is a global weak solution to the equation $\partial_t u - \text{div}(A\nabla u) = \text{div} F$. Applying Corollary 4.3 and (5.10) yields that for any $F \in T_{\beta+1/2}^p \cap L^2(\mathbb{R}_+^{1+n})$,

$$ \|\mathcal{R}_0^L(F)\|_{T_{\beta+1/2}^p} = \|\nabla \mathcal{R}_{1/2}^L(F)\|_{T_{\beta+1/2}^p} \lesssim \|\mathcal{R}_{1/2}^L(F)\|_{T_{\beta+1}^p} + \|F\|_{T_{\beta+1/2}^p} \lesssim \|F\|_{T_{\beta+1/2}^p}. $$

Then the bounded extension of $\mathcal{R}_0^L$ follows by density.

Case 2: $p_L(\beta) < p \le 1$. Note that by definition of $\beta(L)$ (cf. (1.6)), this case only occurs when $\beta > \beta(L)$. We first consider the extension of $\mathcal{R}_{1/2}^L$. Thanks to [AH23b, Lemma 3.2], it is enough to show $\mathcal{R}_{1/2}^L$ is uniformly bounded on $T_{\beta+1/2}^p$-atoms.

Let us verify this. Let $a$ be a $T_{\beta+1/2}^p$-atom, $B \subset \mathbb{R}^n$ be a ball associated to $a$, and $u := \mathcal{R}_{1/2}^L(a)$. Denote by $Q_0 := (0, (2^3r(B))^2) \times 2^3B$. As $\beta + 1/2 > -1/2$, Proposition 5.8 (i) implies

$$ (5.12) \qquad \|u\|_{L_{\beta+1}^2(Q_0)} \lesssim \|a\|_{L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})} \lesssim |2^3B|^{[2,p]}. $$

Pick $q < 2$ sufficiently close to $p_-(L)$ so that

$$ (5.13) \qquad \frac{np_-(L)}{n + (2\beta + 1)p_-(L)} < \frac{nq}{n + (2\beta + 1)q} < p \le 1. $$

As $\beta > \beta(L) \ge -1/2$, we may apply Lemma 5.9 as well as the estimates given by (5.12) to get

$$ \|u\|_{T_{\beta+1}^p} \lesssim 1 + \sum_{j \ge 4} 2^{-jp(2\beta+1+n[q,p])} \le C $$

for some constant $C < \infty$ by (5.13). The assertion hence holds in this case.

Next, we consider the bounded extension of $\mathcal{R}_0^L$. In fact, one can apply Caccioppoli's inequality as in Corollary 4.3 to establish the same molecular decay estimates as in Lemma 5.9 for $\nabla u$ in $L_{\beta+1/2}^2(M_j^{(i)})$ ($1 \le i \le 3$). Meanwhile, using Proposition 5.8 (ii), one can also get as an alternative of (5.12)

$$ \|\nabla u\|_{L_{\beta+1/2}^2(Q_0)} \lesssim \|a\|_{L_{\beta+1/2}^2(\mathbb{R}_+^{1+n})} \lesssim |2^3 B|^{[2,p]}. $$

It is then enough to repeat the above computation. Details are left to the reader. This proves (i).

To prove (ii), as $p \le 1$, we can use the argument in Case 2, using (5.12) directly and replacing the estimates of Lemma 5.9 by those in Corollary 5.10. Picking $\eta > 0$ sufficiently close to $n(\frac{1}{p_-(L)} - 1)$ so that

$$ \frac{np_-(L)}{n + (2\beta + 1)p_-(L)} < \frac{n}{n + 2\beta + 1 + \eta} < p \le 1, $$
we obtain some constant $C < \infty$ so that

$$ \|u\|_{T_{\beta+1}^p} \lesssim 1 + \sum_{j \ge 4} 2^{-jp(2\beta+1+\eta-n[p,1])} \le C. $$

This proves bounded extension of $\mathcal{R}_{1/2}^L$.

Bounded extension of $\mathcal{R}_0^L$ follows by the same argument as in the above Case 2, using Corollary 5.10. This completes the proof. $\square$

## 6. TRACES AND CONTINUITY

In this section, we first prove Theorem 5.1 (b), (c), and (d), and then obtain a trace for arbitrary weak solutions with gradient controlled in weighted tent spaces in the largest possible range of exponents $\beta$ and $p$ (as it is the same as the one for $L = -\Delta$).

*Proof of Theorem 5.1, (b) and (c).* First, Corollary 4.5 (i) says that (b) holds for $F \in T_{\beta+1/2}^p \cap L^2(\mathbb{R}_+^{1+n})$. Also note that all the operators involved have bounded extensions to $T_{\beta+1/2}^p$. Indeed, for $\mathcal{R}_{1/2}^L$, $\nabla \mathcal{R}_{1/2}^L$, and $\mathcal{R}_{1/2}^{-\Delta}$, it has been shown in (a); and we invoke [AH23b, Theorem 4.2] to get boundedness of $\mathcal{L}_1^{-\Delta}$ from $T_{\beta+1/2}^p$ to $T_{\beta+3/2}^p$. We conclude by density (or weak*-density if $p = \infty$), since all the weighted tent spaces embed into $L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$, hence into $\mathcal{D}'(\mathbb{R}_+^{1+n})$.

For (c), we infer from (a) that for any $F \in T_{\beta+1/2}^p$, both $u = \mathcal{R}_{1/2}^L(F)$ and $\nabla u$ lie in $L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$, so $u \in L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$. Recall that Proposition 4.4 says for any $F \in T_{\beta+1/2}^p \cap L^2(\mathbb{R}_+^{1+n})$, $u$ is a global weak solution to $\partial_t u - \text{div}(A\nabla u) = \text{div} F$. The same density argument concludes the verification for any $F \in T_{\beta+1/2}^p$. $\square$

To prove (d), we need a lemma allowing us to reduce the proof to the case $p > 1$.

**Lemma 6.1.** Let $\beta > -1$ and $\tilde{p}_L(\beta) < p \le 1$. There exist $\beta_0 > -1$ and $\max\{\tilde{p}_L(\beta_0), 1\} < p_0 < 2$ so that $T_{\beta+1/2}^p$ embeds into $T_{\beta_0+1/2}^{p_0}$.

*Proof.* The crucial point is the embedding theorem for weighted tent spaces, recalled in Section 2.2. In parabolic scaling it asserts that $T_{\beta+1/2}^p$ embeds into $T_{\beta_0+1/2}^{p_0}$ if $\beta_0 < \beta$, $p < p_0$, and $2\beta_0 + 1 - \frac{n}{p_0} = 2\beta + 1 - \frac{n}{p}$. Let $\gamma < \beta$ be the number so that $2\gamma + 1 - n = 2\beta + 1 - \frac{n}{p}$. We claim that $\gamma > -1$ and $\tilde{p}_L(\gamma) < 1$. If it holds, then by perturbation, we can take $\beta_0 > -1$ sufficiently close to $\gamma$ and the corresponding $p_0 > 1$. The claim is elementary using the definitions of the exponents in the introduction but we provide an argument for the sake of completeness.

By definition (see also Figure 1), we already observe that:

(i) $\beta(L) \ge -1/2 \iff p_-(L) \ge 1$;

(ii) $p_L(\beta) \le \tilde{p}_L(\beta)$, and the equality holds if and only if $\beta \ge \min\{\beta(L), -1/2\}$.

This implies

$$ \beta > -1 \text{ and } \tilde{p}_L(\beta) < 1 \iff \beta > \beta(L). $$

Indeed, if the right-hand side holds, then $\beta > \beta(L) \ge -1$ and $\tilde{p}_L(\beta) = p_L(\beta) < p_L(\beta(L)) = 1$. Conversely, assume the left-hand side. When $p_-(L) < 1$, $\tilde{p}_L(\beta) < 1$ implies $\beta > \beta(L)$. When $p_-(L) \ge 1$, by construction, $\tilde{p}_L(\beta) < 1$ implies $\beta > -1/2 = \min\{\beta(L), -1/2\}$, so $\tilde{p}_L(\beta) = p_L(\beta) < 1$. We thus infer $\beta > \beta(L)$.
we obtain some constant $C < \infty$ so that

$$ \|u\|_{T_{\beta+1}^p} \lesssim 1 + \sum_{j \ge 4} 2^{-jp(2\beta+1+\eta-n[p,1])} \le C. $$

This proves bounded extension of $\mathcal{R}_{1/2}^L$.

Bounded extension of $\mathcal{R}_0^L$ follows by the same argument as in the above Case 2, using Corollary 5.10. This completes the proof. $\square$

## 6. TRACES AND CONTINUITY

In this section, we first prove Theorem 5.1 (b), (c), and (d), and then obtain a trace for arbitrary weak solutions with gradient controlled in weighted tent spaces in the largest possible range of exponents $\beta$ and $p$ (as it is the same as the one for $L = -\Delta$).

*Proof of Theorem 5.1, (b) and (c).* First, Corollary 4.5 (i) says that (b) holds for $F \in T_{\beta+1/2}^p \cap L^2(\mathbb{R}_+^{1+n})$. Also note that all the operators involved have bounded extensions to $T_{\beta+1/2}^p$. Indeed, for $\mathcal{R}_{1/2}^L$, $\nabla \mathcal{R}_{1/2}^L$, and $\mathcal{R}_{1/2}^{-\Delta}$, it has been shown in (a); and we invoke [AH23b, Theorem 4.2] to get boundedness of $\mathcal{L}_1^{-\Delta}$ from $T_{\beta+1/2}^p$ to $T_{\beta+3/2}^p$. We conclude by density (or weak*-density if $p = \infty$), since all the weighted tent spaces embed into $L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$, hence into $\mathcal{D}'(\mathbb{R}_+^{1+n})$.

For (c), we infer from (a) that for any $F \in T_{\beta+1/2}^p$, both $u = \mathcal{R}_{1/2}^L(F)$ and $\nabla u$ lie in $L_{\text{loc}}^2(\mathbb{R}_+^{1+n})$, so $u \in L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$. Recall that Proposition 4.4 says for any $F \in T_{\beta+1/2}^p \cap L^2(\mathbb{R}_+^{1+n})$, $u$ is a global weak solution to $\partial_t u - \text{div}(A\nabla u) = \text{div} F$. The same density argument concludes the verification for any $F \in T_{\beta+1/2}^p$. $\square$

To prove (d), we need a lemma allowing us to reduce the proof to the case $p > 1$.

**Lemma 6.1.** Let $\beta > -1$ and $\tilde{p}_L(\beta) < p \le 1$. There exist $\beta_0 > -1$ and $\max\{\tilde{p}_L(\beta_0), 1\} < p_0 < 2$ so that $T_{\beta+1/2}^p$ embeds into $T_{\beta_0+1/2}^{p_0}$.

*Proof.* The crucial point is the embedding theorem for weighted tent spaces, recalled in Section 2.2. In parabolic scaling it asserts that $T_{\beta+1/2}^p$ embeds into $T_{\beta_0+1/2}^{p_0}$ if $\beta_0 < \beta$, $p < p_0$, and $2\beta_0 + 1 - \frac{n}{p_0} = 2\beta + 1 - \frac{n}{p}$. Let $\gamma < \beta$ be the number so that $2\gamma + 1 - n = 2\beta + 1 - \frac{n}{p}$. We claim that $\gamma > -1$ and $\tilde{p}_L(\gamma) < 1$. If it holds, then by perturbation, we can take $\beta_0 > -1$ sufficiently close to $\gamma$ and the corresponding $p_0 > 1$. The claim is elementary using the definitions of the exponents in the introduction but we provide an argument for the sake of completeness.

By definition (see also Figure 1), we already observe that:

(i) $\beta(L) \ge -1/2 \iff p_-(L) \ge 1$;

(ii) $p_L(\beta) \le \tilde{p}_L(\beta)$, and the equality holds if and only if $\beta \ge \min\{\beta(L), -1/2\}$.

This implies

$$ \beta > -1 \text{ and } \tilde{p}_L(\beta) < 1 \iff \beta > \beta(L). $$

Indeed, if the right-hand side holds, then $\beta > \beta(L) \ge -1$ and $\tilde{p}_L(\beta) = p_L(\beta) < p_L(\beta(L)) = 1$. Conversely, assume the left-hand side. When $p_-(L) < 1$, $\tilde{p}_L(\beta) < 1$ implies $\beta > \beta(L)$. When $p_-(L) \ge 1$, by construction, $\tilde{p}_L(\beta) < 1$ implies $\beta > -1/2 = \min\{\beta(L), -1/2\}$, so $\tilde{p}_L(\beta) = p_L(\beta) < 1$. We thus infer $\beta > \beta(L)$.
It remains to verify that $\gamma > \beta(L)$. Remark that for any $\delta > -1$ and $r > 0$,

$$r > \frac{np_{-}(L)}{n + (2\delta + 1)p_{-}(L)} \iff 2\delta + 1 - \frac{n}{r} > -\frac{n}{p_{-}(L)}.$$

As $\frac{np_{-}(L)}{n+(2\beta+1)p_{-}(L)} \le \tilde{p}_L(\beta) < p \le 1$, we have $2\gamma + 1 - n = 2\beta + 1 - \frac{n}{p} > -\frac{n}{p_{-}(L)}$. By definition of $\beta(L)$, this exactly says $\gamma > \beta(L)$ as desired. $\square$

**Proof of Theorem 5.1 (d).** We first show that $u \in C((0, \infty); \mathcal{S}')$. Since $u$ lies in $L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$, using the equation (by (c)), we get $\partial_t u \in L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{-1,2})$. Next, fix $t > 0$, $\phi \in \mathcal{S}$, and pick $a, b$ so that $0 < a < t < b$. As $p > \tilde{p}_L(\beta) \ge \frac{n}{n+2\beta+2}$, one can infer from the proof of (3.9) in Proposition 3.5 that

$$(6.1) \qquad \|\mathbf{1}_{(a,b)}\nabla\phi\|_{(T_{\beta+1/2}^p)} \lesssim_{a,b} \mathcal{P}_N(\phi)$$

for some sufficiently large $N > 0$, where $\mathcal{P}_N$ is the semi-norm on $\mathcal{S}$ given by (3.5). Then for $s \in (a, t)$, we apply the equation to get

$$\begin{aligned} |\langle u(t) - u(s), \phi \rangle| & \le \left| \int_s^t \langle \partial_\tau u(\tau), \phi \rangle d\tau \right| \\ & \le \int_s^t \int_{\mathbb{R}^n} |A \nabla u| |\nabla \phi| + \int_s^t \int_{\mathbb{R}^n} |F| |\nabla \phi| \\ & \lesssim (\|\mathbf{1}_{(s,t)} \nabla u\|_{T_{\beta+1/2}^p} + \|\mathbf{1}_{(s,t)} F\|_{T_{\beta+1/2}^p}) \mathcal{P}_N(\phi). \end{aligned}$$

For $p < \infty$, this proves the continuity as desired, since both $\|\mathbf{1}_{(s,t)}\nabla u\|_{T_{\beta+1/2}^p}$ and $\|\mathbf{1}_{(s,t)}F\|_{T_{\beta+1/2}^p}$ tend to 0 as $s \to t$. For $p = \infty$, we need finer estimates. Note that in this case, (6.1) can be refined as

$$\|\mathbf{1}_{(s,t)} \nabla \phi\|_{T_{-(\beta+1/2)}^1} \lesssim \left( \int_s^t \tau^{-(2\beta+1)} d\tau \right)^{1/2} \mathcal{P}_N(\phi).$$

Repeating the above computation yields

$$|\langle u(t) - u(s), \phi \rangle| \lesssim \left( \int_s^t \tau^{-(2\beta+1)} d\tau \right)^{1/2} (\|\nabla u\|_{T_{\beta+1/2}^\infty} + \|F\|_{T_{\beta+1/2}^\infty}) \mathcal{P}_N(\phi),$$

which also converges to 0 as $s \to t$.

Next, we consider the trace. We split the discussion in 4 cases.

**Case 1:** $\beta > -1$ and $2 < p \le \infty$. We use the formula in (b). Write $\tilde{F} := (A-I)\nabla u + F$ so that $u = \text{div } \mathcal{L}_1^{-\Delta}(\tilde{F})$. Since $\tilde{F} \in T_{\beta+1/2}^p$, we invoke [AH23b, Theorem 4.2(e)] to get that $\mathcal{L}_1^{-\Delta}(\tilde{F})(t)$ tends to 0 in $E_\delta^q$ as $t \to 0$, so $u(t) = \text{div } \mathcal{L}_1^{-\Delta}(\tilde{F})(t)$ tends to 0 in $E_\delta^{-1,q}$ as $t \to 0$.

**Case 2:** $-1 < \beta < -1/2$ and $\max\{\tilde{p}_L(\beta), 1\} < p \le 2$. By interpolation of Hardy-Sobolev spaces, it is enough to show that $u(t)$ tends to 0 as $t \to 0$ in $\dot{H}^{2\beta+1,p}$ and $\dot{H}^{-1,p}$.

For $\dot{H}^{-1,p}$, we also apply the formula in (b). Again, as $\tilde{F} \in T_{\beta+1/2}^p$, [AH23b, Theorem 4.2(e)] says that $\mathcal{L}_1^{-\Delta}(\tilde{F})(t)$ tends to 0 in $L^p$ when $t \to 0$, so $u(t)$ tends to 0 in $\dot{W}^{-1,p} = \dot{H}^{-1,p}$ as $t \to 0$.
It remains to verify that $\gamma > \beta(L)$. Remark that for any $\delta > -1$ and $r > 0$,

$$r > \frac{np_{-}(L)}{n + (2\delta + 1)p_{-}(L)} \iff 2\delta + 1 - \frac{n}{r} > -\frac{n}{p_{-}(L)}.$$

As $\frac{np_{-}(L)}{n+(2\beta+1)p_{-}(L)} \le \tilde{p}_L(\beta) < p \le 1$, we have $2\gamma + 1 - n = 2\beta + 1 - \frac{n}{p} > -\frac{n}{p_{-}(L)}$. By definition of $\beta(L)$, this exactly says $\gamma > \beta(L)$ as desired. $\square$

**Proof of Theorem 5.1 (d).** We first show that $u \in C((0, \infty); \mathcal{S}')$. Since $u$ lies in $L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$, using the equation (by (c)), we get $\partial_t u \in L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{-1,2})$. Next, fix $t > 0$, $\phi \in \mathcal{S}$, and pick $a, b$ so that $0 < a < t < b$. As $p > \tilde{p}_L(\beta) \ge \frac{n}{n+2\beta+2}$, one can infer from the proof of (3.9) in Proposition 3.5 that

$$(6.1) \qquad \|\mathbf{1}_{(a,b)} \nabla \phi\|_{(T_{\beta+1/2}^p)} \lesssim_{a,b} \mathcal{P}_N(\phi)$$

for some sufficiently large $N > 0$, where $\mathcal{P}_N$ is the semi-norm on $\mathcal{S}$ given by (3.5). Then for $s \in (a, t)$, we apply the equation to get

$$\begin{aligned} |\langle u(t) - u(s), \phi \rangle| & \le \left| \int_s^t \langle \partial_\tau u(\tau), \phi \rangle d\tau \right| \\ & \le \int_s^t \int_{\mathbb{R}^n} |A \nabla u| |\nabla \phi| + \int_s^t \int_{\mathbb{R}^n} |F| |\nabla \phi| \\ & \lesssim (\|\mathbf{1}_{(s,t)} \nabla u\|_{T_{\beta+1/2}^p} + \|\mathbf{1}_{(s,t)} F\|_{T_{\beta+1/2}^p}) \mathcal{P}_N(\phi). \end{aligned}$$

For $p < \infty$, this proves the continuity as desired, since both $\|\mathbf{1}_{(s,t)} \nabla u\|_{T_{\beta+1/2}^p}$ and $\|\mathbf{1}_{(s,t)} F\|_{T_{\beta+1/2}^p}$ tend to 0 as $s \to t$. For $p = \infty$, we need finer estimates. Note that in this case, (6.1) can be refined as

$$\|\mathbf{1}_{(s,t)} \nabla \phi\|_{T_{-(\beta+1/2)}^1} \lesssim \left( \int_s^t \tau^{-(2\beta+1)} d\tau \right)^{1/2} \mathcal{P}_N(\phi).$$

Repeating the above computation yields

$$|\langle u(t) - u(s), \phi \rangle| \lesssim \left( \int_s^t \tau^{-(2\beta+1)} d\tau \right)^{1/2} (\|\nabla u\|_{T_{\beta+1/2}^\infty} + \|F\|_{T_{\beta+1/2}^\infty}) \mathcal{P}_N(\phi),$$

which also converges to 0 as $s \to t$.

Next, we consider the trace. We split the discussion in 4 cases.

**Case 1:** $\beta > -1$ and $2 < p \le \infty$. We use the formula in (b). Write $\tilde{F} := (A-I)\nabla u + F$ so that $u = \text{div } \mathcal{L}_1^{-\Delta}(\tilde{F})$. Since $\tilde{F} \in T_{\beta+1/2}^p$, we invoke [AH23b, Theorem 4.2(e)] to get that $\mathcal{L}_1^{-\Delta}(\tilde{F})(t)$ tends to 0 in $E_\delta^q$ as $t \to 0$, so $u(t) = \text{div } \mathcal{L}_1^{-\Delta}(\tilde{F})(t)$ tends to 0 in $E_\delta^{-1,q}$ as $t \to 0$.

**Case 2:** $-1 < \beta < -1/2$ and $\max\{\tilde{p}_L(\beta), 1\} < p \le 2$. By interpolation of Hardy-Sobolev spaces, it is enough to show that $u(t)$ tends to 0 as $t \to 0$ in $\dot{H}^{2\beta+1,p}$ and $\dot{H}^{-1,p}$.

For $\dot{H}^{-1,p}$, we also apply the formula in (b). Again, as $\tilde{F} \in T_{\beta+1/2}^p$, [AH23b, Theorem 4.2(e)] says that $\mathcal{L}_1^{-\Delta}(\tilde{F})(t)$ tends to 0 in $L^p$ when $t \to 0$, so $u(t)$ tends to 0 in $\dot{W}^{-1,p} = \dot{H}^{-1,p}$ as $t \to 0$.
For $\dot{H}^{2\beta+1,p}$, we first claim that for any $t > 0$,

$$ (6.2) \qquad \|u(t)\|_{\dot{H}^{2\beta+1,p}} \lesssim \|\mathbf{1}_{(0,t)} \tilde{F}\|_{T_{\beta+1/2}^p}. $$

If it holds, then it is clear that $u(t)$ tends to 0 in $\dot{H}^{2\beta+1,p}$ as $t \to 0$, since $\|\mathbf{1}_{(0,t)} \tilde{F}\|_{T_{\beta+1/2}^p}$ tends to 0 as $t \to 0$.

Let us prove the claim (6.2) by duality. Fix $\varphi \in \mathcal{S}_\infty$. Define

$$ g(s, x) := \mathbf{1}_{\{s<t\}}(s) \nabla e^{-(t-s)\Delta} \varphi(x). $$

Observe that

$$ (6.3) \qquad \|g\|_{T_{-(\beta+1/2)}^{p'}} \lesssim \|\varphi\|_{\dot{H}^{-(2\beta+1),p'}}. $$

Indeed, for $\tilde{p}_L(\beta) < p < 2$, we have

$$ \|g\|_{T_{-(\beta+1/2)}^{p'}} \le \left( \int_{\mathbb{R}^n} \left( \int_0^t \mathcal{M}(|\nabla e^{(t-s)\Delta} \varphi|^2)(x) s^{2\beta+1} ds \right)^{p'/2} dx \right)^{1/p'}, $$

where $\mathcal{M}$ is the Hardy–Littlewood maximal operator on $\mathbb{R}^n$. By Minkowski’s inequality and boundedness of $\mathcal{M}$ on $L^{p'/2}$, we further have

$$ \begin{aligned} \|g\|_{T_{-(\beta+1/2)}^{p'}} &\lesssim \left( \int_0^t \|\nabla e^{(t-s)\Delta} \varphi\|_{p'}^2 s^{2\beta+1} ds \right)^{1/2} \\ &\lesssim \left( \int_0^t (t-s)^{-2(\beta+1)} \|\varphi\|_{\dot{H}^{-(2\beta+1),p'}}^2 s^{2\beta+1} ds \right)^{1/2} \lesssim \|\varphi\|_{\dot{H}^{-(2\beta+1),p'}}. \end{aligned} $$

proving (6.3). Here, as $0 < -(2\beta + 1) < 1$, the second inequality comes by interpolating the boundedness of $(e^{t\Delta})$ on $\dot{H}^{1,p'}$ (see Proposition 3.1) and that of $(t^{1/2}\nabla e^{t\Delta})$ on $L^{p'} \simeq \dot{H}^{0,p'}$. For $p = 2$, the same interpolation argument also yields

$$ \|g\|_{T_{-(\beta+1/2)}^2} \approx \left( \int_0^t \|\nabla e^{(t-s)\Delta} \varphi\|_2^2 s^{2\beta+1} ds \right)^{1/2} \lesssim \|\varphi\|_{\dot{H}^{-(2\beta+1),2}}. $$

This shows (6.3). Then we apply the formula (b), duality of tent spaces, and the estimate (6.3) to get

$$ \begin{aligned} |\langle u(t), \varphi \rangle| &= |\langle \operatorname{div} \mathcal{L}_1^{-\Delta}(\tilde{F})(t), \varphi \rangle| \le \int_0^t \int_{\mathbb{R}^n} |\tilde{F}(s, x)| |\nabla e^{-(t-s)\Delta} \varphi(x)| ds dx \\ &\lesssim \|\mathbf{1}_{(0,t)} \tilde{F}\|_{T_{\beta+1/2}^p} \|g\|_{T_{-(\beta+1/2)}^{p'}} \lesssim \|\mathbf{1}_{(0,t)} \tilde{F}\|_{T_{\beta+1/2}^p} \|\varphi\|_{\dot{H}^{-(2\beta+1),p'}}. \end{aligned} $$

Therefore, the claim (6.2) follows since for any $t > 0$,

$$ \|u(t)\|_{\dot{H}^{2\beta+1,p}} = \sup_{\varphi} |\langle u(t), \varphi \rangle| \lesssim \|\mathbf{1}_{(0,t)} \tilde{F}\|_{T_{\beta+1/2}^p}, $$

where the supremum is taken among $\varphi \in \mathcal{S}_\infty$ with $\|\varphi\|_{\dot{H}^{-(2\beta+1),p'}} = 1$, noting that $\mathcal{S}_\infty$ is dense in $\dot{H}^{-(2\beta+1),p'}$.
For $\dot{H}^{2\beta+1,p}$, we first claim that for any $t > 0$,

$$ (6.2) \qquad \|u(t)\|_{\dot{H}^{2\beta+1,p}} \lesssim \|\mathbf{1}_{(0,t)} \tilde{F}\|_{T_{\beta+1/2}^p}. $$

If it holds, then it is clear that $u(t)$ tends to 0 in $\dot{H}^{2\beta+1,p}$ as $t \to 0$, since $\|\mathbf{1}_{(0,t)} \tilde{F}\|_{T_{\beta+1/2}^p}$ tends to 0 as $t \to 0$.

Let us prove the claim (6.2) by duality. Fix $\varphi \in \mathcal{S}_\infty$. Define

$$ g(s, x) := \mathbf{1}_{\{s<t\}}(s) \nabla e^{-(t-s)\Delta} \varphi(x). $$

Observe that

$$ (6.3) \qquad \|g\|_{T_{-(\beta+1/2)}^{p'}} \lesssim \|\varphi\|_{\dot{H}^{-(2\beta+1),p'}}. $$

Indeed, for $\tilde{p}_L(\beta) < p < 2$, we have

$$ \|g\|_{T_{-(\beta+1/2)}^{p'}} \le \left( \int_{\mathbb{R}^n} \left( \int_0^t \mathcal{M}(|\nabla e^{(t-s)\Delta} \varphi|^2)(x) s^{2\beta+1} ds \right)^{p'/2} dx \right)^{1/p'}, $$

where $\mathcal{M}$ is the Hardy–Littlewood maximal operator on $\mathbb{R}^n$. By Minkowski’s inequality and boundedness of $\mathcal{M}$ on $L^{p'/2}$, we further have

$$ \begin{aligned} \|g\|_{T_{-(\beta+1/2)}^{p'}} & \lesssim \left( \int_0^t \|\nabla e^{(t-s)\Delta} \varphi\|_{p'}^2 s^{2\beta+1} ds \right)^{1/2} \\ & \lesssim \left( \int_0^t (t-s)^{-2(\beta+1)} \|\varphi\|_{\dot{H}^{-(2\beta+1),p'}}^2 s^{2\beta+1} ds \right)^{1/2} \lesssim \|\varphi\|_{\dot{H}^{-(2\beta+1),p'}}. \end{aligned} $$

proving (6.3). Here, as $0 < -(2\beta + 1) < 1$, the second inequality comes by interpolating the boundedness of $(e^{t\Delta})$ on $\dot{H}^{1,p'}$ (see Proposition 3.1) and that of $(t^{1/2}\nabla e^{t\Delta})$ on $L^{p'} \simeq \dot{H}^{0,p'}$. For $p = 2$, the same interpolation argument also yields

$$ \|g\|_{T_{-(\beta+1/2)}^2} \approx \left( \int_0^t \|\nabla e^{(t-s)\Delta} \varphi\|_2^2 s^{2\beta+1} ds \right)^{1/2} \lesssim \|\varphi\|_{\dot{H}^{-(2\beta+1),2}}. $$

This shows (6.3). Then we apply the formula (b), duality of tent spaces, and the estimate (6.3) to get

$$ \begin{aligned} |\langle u(t), \varphi \rangle| &= |\langle \operatorname{div} \mathcal{L}_1^{-\Delta}(\tilde{F})(t), \varphi \rangle| \le \int_0^t \int_{\mathbb{R}^n} |\tilde{F}(s, x)| |\nabla e^{-(t-s)\Delta} \varphi(x)| ds dx \\ &\lesssim \|\mathbf{1}_{(0,t)} \tilde{F}\|_{T_{\beta+1/2}^p} \|g\|_{T_{-(\beta+1/2)}^{p'}} \lesssim \|\mathbf{1}_{(0,t)} \tilde{F}\|_{T_{\beta+1/2}^p} \|\varphi\|_{\dot{H}^{-(2\beta+1),p'}}. \end{aligned} $$

Therefore, the claim (6.2) follows since for any $t > 0$,

$$ \|u(t)\|_{\dot{H}^{2\beta+1,p}} = \sup_{\varphi} |\langle u(t), \varphi \rangle| \lesssim \|\mathbf{1}_{(0,t)} \tilde{F}\|_{T_{\beta+1/2}^p}, $$

where the supremum is taken among $\varphi \in \mathcal{S}_\infty$ with $\|\varphi\|_{\dot{H}^{-(2\beta+1),p'}} = 1$, noting that $\mathcal{S}_\infty$ is dense in $\dot{H}^{-(2\beta+1),p'}$.
Case 3: $-1 < \beta < -1/2$ and $\tilde{p}_L(\beta) < p \le 1$. We use embedding of tent spaces. Pick $-1 < \beta_0 < \beta < -1/2$ and $\max\{\tilde{p}_L(\beta_0), 1\} < p_0 < 2$ as in Lemma 6.1 so that $T_{\beta+1/2}^p$ embeds into $T_{\beta_0+1/2}^{p_0}$. In particular, as $F \in T_{\beta+1/2}^p$, we have $F \in T_{\beta_0+1/2}^{p_0}$, for which we get back to Case 2. Hence, $u(t)$ tends to 0 as $t \to 0$ in $\dot{H}^{2\beta_0+1, p_0}$, and therefore in $\dot{H}^{-1, r}$ by Sobolev embedding and definition of $r$.

Case 4: $\beta \ge -1/2$ and $\tilde{p}_L(\beta) < p \le 2$. Lemma 4.2 yields

$$ \begin{aligned} \|u(t)\|_p^p &\le \int_{\mathbb{R}^n} \left( \int_{B(x, \frac{\sqrt{t}}{4})} |u(t, y)|^2 dy \right)^{p/2} dx \\ &\lesssim \int_{\mathbb{R}^n} \left( \int_{t/2}^t \int_{B(x, \frac{\sqrt{t}}{2})} |u|^2 dy \right)^{p/2} dx + \int_{\mathbb{R}^n} \left( \int_{t/2}^t \int_{B(x, \frac{\sqrt{t}}{2})} |F|^2 dy \right)^{p/2} dx \end{aligned} $$

When $\beta = -1/2$, Since $u \in T_{1/2}^p$ and $F \in T_0^p$, Lebesgue's dominated convergence theorem yields the integrals on the right-hand side tend to 0 as $t \to 0$. Thus, $u(t)$ tends to 0 in $L^p$ as $t \to 0$. When $\beta > -1/2$, the right-hand side can be controlled by $t^{p(\beta+1/2)}(\|u\|_{T_{\beta+1}^p} + \|F\|_{T_{\beta+1/2}^p})^p$, which also tends to 0 as $t \to 0$ as $\beta > -1/2$.

It remains to prove that for $\beta \ge -1/2$ and $\tilde{p}_L(\beta) < p < 1$, $u(t)$ tends to 0 in $\mathcal{S}'$, because in this case, $L^p$ does not embed into $\mathcal{S}'$. To this end, we also use embedding of tent spaces. Pick $\beta_0 \in (-1, \beta]$ and $p_0 > 1$ as defined in Lemma 6.1, so $F \in T_{\beta_0+1/2}^{p_0}$. Then we apply the above discussion to get $u(t)$ tends to 0 as $t \to 0$ in $L^{p_0}$ if $\beta_0 \ge -1/2$, or in $\dot{H}^{-1, r}$ if $-1 < \beta_0 < -1/2$, see Case 2. But both spaces of course embed into $\mathcal{S}'$. We hence obtain the trace in $\mathcal{S}'$ as desired.

This completes the proof. $\square$

<figure><img src="image_4.png" /></figure>

As an application, we establish distributional traces for weak solutions to the equation $\partial_t u - \text{div}(A \nabla u) = 0$, which is useful when studying uniqueness.

**Proposition 6.2** (Trace). Let $\beta > -1$ and $\frac{n}{n+2\beta+2} < p \le \infty$. Let $u$ be a global weak solution to $\partial_t u - \text{div}(A \nabla u) = 0$ with $\nabla u \in T_{\beta+1/2}^p$. Then there exists a unique $u_0 \in \mathcal{S}'$ such that $u(t)$ converges to $u_0$ in $\mathcal{S}'$ as $t \to 0$, and

$$ (6.4) \qquad u = \mathcal{E}_{-\Delta}(u_0) + \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u). $$

In addition,

(i) If $\beta \ge 0$ and $\frac{n}{n+2\beta} \le p \le \infty$, then $u_0$ is a constant.

(ii) If $-1 < \beta < 0$, then there exist $g \in \dot{H}^{2\beta+1, p}$ and $c \in \mathbb{C}$ so that $u_0 = g + c$ with

$$ \|g\|_{\dot{H}^{2\beta+1, p}} \lesssim \|\nabla u\|_{T_{\beta+1/2}^p}. $$

*Proof.* Note that $\frac{n}{n+2\beta+2} = \tilde{p}_{-\Delta}(\beta)$, so by Theorem 5.1 (a), as $\nabla u \in T_{\beta+1/2}^p$, we get $\mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u) \in T_{\beta+1}^p$ and $\nabla \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u) \in T_{\beta+1/2}^p$. Besides, Theorem 5.1 (d) says that $\mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u)(t)$ vanishes in $\mathcal{S}'$ as $t \to 0$.

Define $\tilde{u} \in L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$ by

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

Observe that $\tilde{u}$ is a global weak (hence distributional) solution to the heat equation with $\nabla \tilde{u} \in T_{\beta+1/2}^p$. Then we invoke Proposition 3.5 to get $u_0 \in \mathcal{S}'$ so that $\tilde{u} =$
Case 3: $-1 < \beta < -1/2$ and $\tilde{p}_L(\beta) < p \le 1$. We use embedding of tent spaces. Pick $-1 < \beta_0 < \beta < -1/2$ and $\max\{\tilde{p}_L(\beta_0), 1\} < p_0 < 2$ as in Lemma 6.1 so that $T_{\beta+1/2}^p$ embeds into $T_{\beta_0+1/2}^{p_0}$. In particular, as $F \in T_{\beta+1/2}^p$, we have $F \in T_{\beta_0+1/2}^{p_0}$, for which we get back to Case 2. Hence, $u(t)$ tends to 0 as $t \to 0$ in $\dot{H}^{2\beta_0+1, p_0}$, and therefore in $\dot{H}^{-1, r}$ by Sobolev embedding and definition of $r$.

Case 4: $\beta \ge -1/2$ and $\tilde{p}_L(\beta) < p \le 2$. Lemma 4.2 yields

$$ \begin{aligned} \|u(t)\|_p^p &\le \int_{\mathbb{R}^n} \left( \int_{B(x, \frac{\sqrt{t}}{4})} |u(t, y)|^2 dy \right)^{p/2} dx \\ &\lesssim \int_{\mathbb{R}^n} \left( \int_{t/2}^t \int_{B(x, \frac{\sqrt{t}}{2})} |u|^2 dy \right)^{p/2} dx + \int_{\mathbb{R}^n} \left( \int_{t/2}^t \int_{B(x, \frac{\sqrt{t}}{2})} |F|^2 dy \right)^{p/2} dx \end{aligned} $$

When $\beta = -1/2$, Since $u \in T_{1/2}^p$ and $F \in T_0^p$, Lebesgue's dominated convergence theorem yields the integrals on the right-hand side tend to 0 as $t \to 0$. Thus, $u(t)$ tends to 0 in $L^p$ as $t \to 0$. When $\beta > -1/2$, the right-hand side can be controlled by $t^{p(\beta+1/2)}(\|u\|_{T_{\beta+1}^p} + \|F\|_{T_{\beta+1/2}^p})^p$, which also tends to 0 as $t \to 0$ as $\beta > -1/2$.

It remains to prove that for $\beta \ge -1/2$ and $\tilde{p}_L(\beta) < p < 1$, $u(t)$ tends to 0 in $\mathcal{S}'$, because in this case, $L^p$ does not embed into $\mathcal{S}'$. To this end, we also use embedding of tent spaces. Pick $\beta_0 \in (-1, \beta]$ and $p_0 > 1$ as defined in Lemma 6.1, so $F \in T_{\beta_0+1/2}^{p_0}$. Then we apply the above discussion to get $u(t)$ tends to 0 as $t \to 0$ in $L^{p_0}$ if $\beta_0 \ge -1/2$, or in $\dot{H}^{-1, r}$ if $-1 < \beta_0 < -1/2$, see Case 2. But both spaces of course embed into $\mathcal{S}'$. We hence obtain the trace in $\mathcal{S}'$ as desired.

This completes the proof. $\square$

<figure><img src="image_4.png" /></figure>

As an application, we establish distributional traces for weak solutions to the equation $\partial_t u - \text{div}(A\nabla u) = 0$, which is useful when studying uniqueness.

**Proposition 6.2** (Trace). Let $\beta > -1$ and $\frac{n}{n+2\beta+2} < p \le \infty$. Let $u$ be a global weak solution to $\partial_t u - \text{div}(A\nabla u) = 0$ with $\nabla u \in T_{\beta+1/2}^p$. Then there exists a unique $u_0 \in \mathcal{S}'$ such that $u(t)$ converges to $u_0$ in $\mathcal{S}'$ as $t \to 0$, and

$$ (6.4) \qquad u = \mathcal{E}_{-\Delta}(u_0) + \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u). $$

In addition,

(i) If $\beta \ge 0$ and $\frac{n}{n+2\beta} \le p \le \infty$, then $u_0$ is a constant.

(ii) If $-1 < \beta < 0$, then there exist $g \in \dot{H}^{2\beta+1, p}$ and $c \in \mathbb{C}$ so that $u_0 = g + c$ with

$$ \|g\|_{\dot{H}^{2\beta+1, p}} \lesssim \|\nabla u\|_{T_{\beta+1/2}^p}. $$

*Proof.* Note that $\frac{n}{n+2\beta+2} = \tilde{p}_{-\Delta}(\beta)$, so by Theorem 5.1 (a), as $\nabla u \in T_{\beta+1/2}^p$, we get $\mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u) \in T_{\beta+1}^p$ and $\nabla \mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u) \in T_{\beta+1/2}^p$. Besides, Theorem 5.1 (d) says that $\mathcal{R}_{1/2}^{-\Delta}((A - \mathbb{I})\nabla u)(t)$ vanishes in $\mathcal{S}'$ as $t \to 0$.

Define $\tilde{u} \in L_{\text{loc}}^2((0, \infty); W_{\text{loc}}^{1,2})$ by

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

Observe that $\tilde{u}$ is a global weak (hence distributional) solution to the heat equation with $\nabla \tilde{u} \in T_{\beta+1/2}^p$. Then we invoke Proposition 3.5 to get $u_0 \in \mathcal{S}'$ so that $\tilde{u} =$