# On Hardy-Littlewood-Sobolev estimates for degenerate Laplacians

Pascal Auscher, Khalid Baadi

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Pascal Auscher, Khalid Baadi. On Hardy-Littlewood-Sobolev estimates for degenerate Laplacians. 2025. hal-05131096

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# ON HARDY-LITTLEWOOD-SOBOLEV ESTIMATES FOR DEGENERATE LAPLACIANS

PASCAL AUSCHER AND KHALID BAADI

**ABSTRACT.** We establish norm inequalities for fractional powers of degenerate Laplacians, with degeneracy being determined by weights in the Muckenhoupt class $A_2(\mathbb{R}^n)$, accompanied by specific additional reverse Hölder assumptions. This extends the known results for classical Riesz potentials. The approach is based on size estimates for the degenerate heat kernels. The approach also applies to more general weighted degenerate operators.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Preliminaries and basic assumptions</td><td>3</td></tr><tr><td>2.1. Muckenhoupt weights</td><td>3</td></tr><tr><td>2.2. Function spaces</td><td>4</td></tr><tr><td>2.3. The degenerate Laplacian</td><td>4</td></tr><tr><td>3. Hardy-Littlewood-Sobolev estimates</td><td>6</td></tr><tr><td>3.1. Key lemmas</td><td>6</td></tr><tr><td>3.2. Proof of Theorem 1.1</td><td>8</td></tr><tr><td>3.3. Extension to weighted Lorentz spaces</td><td>10</td></tr><tr><td>3.4. Extension to the functional calculus</td><td>12</td></tr><tr><td>3.5. Sharpness of the condition on the weight</td><td>12</td></tr><tr><td>4. Comparison with related results</td><td>14</td></tr><tr><td>5. Case of degenerate elliptic operators with coefficients</td><td>16</td></tr><tr><td>References</td><td>17</td></tr></table>

## 1. INTRODUCTION

Degenerate elliptic operators where the degeneracy is controlled by weights in the Muckenhoupt class have been introduced in [FKS82]. Since then, they have attracted attention with extension to parabolic equations [CS84a, CS84b, CS85, Ish99]. For $A_2(\mathbb{R}^n)$ weights $\omega$, $-\Delta_\omega$ can be defined as a maximal accretive operator on the weighted space $L_\omega^2(\mathbb{R}^n)$. Much is known. For example, the associated semigroup is bounded on $L_\omega^p(\mathbb{R}^n)$ for all $1 \le p \le \infty$, it has $H^\infty$ functional calculus (equivalently, Borel functional calculus) if $1 < p < \infty$. The particular study of its square root can be found in [CUR15, CUMR18] with a proof of the Kato problem on $L_\omega^2(\mathbb{R}^n)$ and extension to some range of $p$, and even a "weighted theory" with weights in the Muckenhoupt classes for the measure $w(x) dx$.

It is not known whether there are $L^p - L^q$ bounds for the fractional powers $(-\Delta_\omega)^{-\alpha}$ in the spirit of the Hardy-Littlewood-Sobolev estimates as in the case where $\omega = 1$. Recall that a possible way to obtain such estimates when $\omega = 1$ is to start from the heat semigroup $L^p - L^q$ bounds for $q > p$, and use this in a Calderón's representation formula to prove weak type $(p, q)$ bounds for the fractional powers $(-\Delta)^{-\alpha}$ with the relation between $\alpha, p, q$ and dimension obeying the rule imposed by scaling, and then proceed by Marcinkiewicz interpolation. However, this strategy does

<footnote>Date: June 25, 2025.</footnote>

<footnote n="2010">2010 Mathematics Subject Classification. Primary: 46E35, 35K08, 42B37 Secondary: 35K65, 42B20.</footnote>

<footnote>Key words and phrases. Degenerate Laplacians, Sobolev embeddings, heat kernel, Muckenhoupt weights, weighted Lorentz spaces.</footnote>
not apply as such in the weighted case as the semigroup is not bounded from $L_{\omega}^{p}(\mathbb{R}^{n})$ to $L_{\omega}^{q}(\mathbb{R}^{n})$ if $q > p$. Our observation is that when $\omega$ additionally belongs to the reverse Hölder class $RH_{\frac{q}{p}}(\mathbb{R}^{n})$ then the correct boundedness property is from $L_{\omega}^{p}(\mathbb{R}^{n})$ to $L_{\omega q/p}^{q}(\mathbb{R}^{n})$. Hence, the above strategy can be developed for the fractional powers and this allows us to prove the following result. Definitions of weighted spaces and operators are in the next section.

**Theorem 1.1.** Let $n \ge 1$ and $\omega \in A_2(\mathbb{R}^n)$. Let $\Delta_{\omega}$ be the weighted Laplacian defined by the method of forms. Let $1 < p < q < \infty$. Assume that $\omega \in RH_{\frac{q}{p}}(\mathbb{R}^n)$ and set $\alpha = \frac{n}{2}(\frac{1}{p} - \frac{1}{q})$. Then, there exists a constant $C = C([\omega]_{A_2}, [\omega]_{RH_{\frac{q}{p}}}, n, p, q) < \infty$ such that

$$ (1.2) \qquad \|\omega^{1/p}f\|_{L_{dx}^q(\mathbb{R}^n)} = \|f\|_{L_{\omega q/p}^q(\mathbb{R}^n)} \le C\|(-\Delta_{\omega})^{\alpha}f\|_{L_{\omega}^p(\mathbb{R}^n)}, \quad \forall f \in D((-\Delta_{\omega})^{\alpha}). $$

In particular, $(-\Delta_{\omega})^{-\alpha}$ defines a bounded operator from $L_{\omega}^{p}(\mathbb{R}^{n})$ to $L_{\omega q/p}^{q}(\mathbb{R}^{n})$ or, equivalently, $w^{1/p}(-\Delta_{\omega})^{-\alpha}$ is bounded from $L_{\omega}^{p}(\mathbb{R}^{n})$ to $L_{dx}^{q}(\mathbb{R}^{n})$. If $p = 1$, then there exists a constant $C = C([\omega]_{A_2}, [\omega]_{RH_{\frac{q}{p}}}, n, q) < \infty$ such that

$$ (1.3) \qquad \|\omega f\|_{L_{dx}^{q,\infty}(\mathbb{R}^n)} \le C\|(-\Delta_{\omega})^{\alpha}f\|_{L_{\omega}^1(\mathbb{R}^n)}, \quad \forall f \in D((-\Delta_{\omega})^{\alpha}). $$

In particular, $\omega(-\Delta_{\omega})^{-\alpha}$ defines a bounded operator from $L_{\omega}^{1}(\mathbb{R}^{n})$ to $L_{dx}^{q,\infty}(\mathbb{R}^{n})$.

The expert reader may find the statement for $(-\Delta_{\omega})^{-\alpha}$ alike to the famous Muckenhoupt-Wheeden weighted estimates [MW74]: under the same conditions on $p, q, \alpha$ and dimension, the unweighted fractional Laplacian $(-\Delta)^{-\alpha}$ is bounded from $L_{\tilde{\omega}^p}^p(\mathbb{R}^n)$ to $L_{\tilde{\omega}^q}^q(\mathbb{R}^n)$ if and only if $\tilde{\omega} \in A_{p,q}(\mathbb{R}^n)$. There is indeed the same rule for the change of weight. However, aside from the $A_2$ condition required to define our degenerate Laplacian, we have a weaker condition on $\omega$ compared to $\tilde{\omega}^p$. Indeed, the reverse Hölder condition $\tilde{w}^p \in RH_{\frac{q}{p}}(\mathbb{R}^n)$ holds when $\tilde{\omega} \in A_{p,q}(\mathbb{R}^n)$, but the converse is not true as we shall see. Besides, since $\omega$ is used to construct the weighted Laplacian $\Delta_{\omega}$ on $L_{\omega}^2(\mathbb{R}^n)$, this is not really a weighted estimate that we are proving.

Inequalities (1.2) and (1.3) are a priori weighted Sobolev embeddings of a functional nature. Some more geometric weighted Sobolev-Gagliardo-Nirenberg inequalities were obtained in [LMPT10]. Namely, for all $1 \le p < q < \infty$ with $\frac{1}{2} = \frac{n}{2}(\frac{1}{p} - \frac{1}{q})$, $\|f\|_{L_{\tilde{\omega}^q}^q(\mathbb{R}^n)} \lesssim \|\nabla f\|_{L_{\tilde{\omega}^p}^p(\mathbb{R}^n)}$ when $\tilde{\omega} \in A_{p,q}(\mathbb{R}^n)$ (which, again, requires $\tilde{\omega}^p \in RH_{\frac{q}{p}}(\mathbb{R}^n)$).

We can compare these results with ours only when $\alpha = 1/2$, for $p = 2$ where $\|(-\Delta_{\omega})^{1/2}f\|_{L_{\tilde{\omega}}^2(\mathbb{R}^n)} = \|\nabla f\|_{L_{\tilde{\omega}}^2(\mathbb{R}^n)}$ by construction and in the ranges of $p$ where $\|\nabla f\|_{L_{\tilde{\omega}}^p(\mathbb{R}^n)}$ dominates or is dominated by $\|(-\Delta_{\omega})^{1/2}f\|_{L_{\tilde{\omega}}^p(\mathbb{R}^n)}$. Deducing our results from them or vice-versa also require some further restrictions on the class of weights.

Coming back to our result, we will show that the reverse Hölder assumption is sharp in the sense that it is implied by (1.2). At the level of the semigroup, we shall see that with the assumptions of the theorem, then $\omega \in RH_{\frac{q}{p}}(\mathbb{R}^n)$ is equivalent to $e^{t\Delta_{\omega}} : L_{\omega}^p(\mathbb{R}^n) \to L_{\omega q/p}^q(\mathbb{R}^n)$ with bound on the order $t^{-\alpha}$ for all $t > 0$. This polynomial bound is the key point of our method.

Our method also allows for various generalizations. In [O’N63], O’Neil showed that the classical unweighted $L^p-L^q$ estimates for Riesz potentials (that is, the fractional Laplacians of negative order) can be improved to $L^p-L^{q,p}$ estimates, where $L^{q,p}$ is a Lorentz space, using the translation invariance of the operators. Later, it was observed by Tartar (see [Tar98]) that this is a consequence of real interpolation more than translation invariance. Here too, real interpolation applies and yields two apparently distinct improvements for weighted fractional Laplacian, see Theorem 3.15 and Theorem 3.17, in agreement with observations made in [CMM22]. In particular, we shall show that $w^{\frac{1}{p}}(-\Delta_{\omega})^{-\alpha}$ is bounded from $L_{\omega}^p(\mathbb{R}^n)$ to the unweigthed Lorentz space $L^{q,p}(\mathbb{R}^n)$ under the same assumption as for (1.2).

Incorporating results on bounded holomorphic functional calculus, fractional powers $z^{-\alpha}$ can be replaced by holomorphic functions in sectors with the same behavior at the origin and infinity.

Given that our methods rely solely on functional calculus and upper bounds for the kernel of the semigroup $e^{t\Delta_{\omega}}$, our results extend to degenerate second-order elliptic operators whose semigroups satisfy Gaussian upper bounds, including those with real coefficients (see Section 5). For non real
coefficients, our results can be extended in limited ranges of exponents for $p, q$ up to changing pointwise upper bounds for the heat kernel to off-diagonal estimates for the heat semigroup.

As a matter of motivation for this work, bounds for fractional weighted Laplacians are needed to study second order degenerate parabolic equations with unbounded lower order terms. The second author will do it in a forthcoming work.

**Notation.** Throughout this paper, we fix an integer $n \ge 1$. We use the notation $dx$ for the Lebesgue measure on $\mathbb{R}^n$. For any Lebesgue measurable set $E \subset \mathbb{R}^n$, we denote its Lebesgue measure by $|E|$. By convention, the notation $C(a, b, \dots)$ denotes a constant depending only on the parameters $(a, b, \dots)$. All cubes $Q \subset \mathbb{R}^n$ considered in this paper are assumed to have sides parallel to the coordinate axes.

**Acknowledgements.** We want to thank Chema Martell for bringing our attention to [CMM22] concerning Lorentz spaces after we completed this work. We would also like to thank Carlos Pérez for enlightening discussions.

## 2. PRELIMINARIES AND BASIC ASSUMPTIONS

### 2.1. Muckenhoupt weights.
We say that an almost everywhere (for Lebesgue measure) defined function $\omega : \mathbb{R}^n \to (0, \infty]$ is a weight if it is locally integrable on $\mathbb{R}^n$. The Muckenhoupt class $A_p(\mathbb{R}^n)$, $p \in (1, \infty)$, is defined as the set of all weights $\omega$ verifying

$$ (2.1) \qquad [\omega]_{A_p} := \sup_{Q \subset \mathbb{R}^n} \left( \frac{1}{|Q|} \int_Q \omega(x) \, dx \right) \left( \frac{1}{|Q|} \int_Q \omega^{1-p'}(x) \, dx \right)^{p-1} < \infty, $$

where the supremum is taken with respect to all cubes $Q \subset \mathbb{R}^n$. For $p = 1$, the Muckenhoupt class $A_1(\mathbb{R}^n)$ is defined as the set of all weights $\omega$ verifying

$$ \frac{1}{|Q|} \int_Q \omega(x) \, dx \le C\omega(y), \quad \text{for almost every } y \in Q, $$

for some constant $C < \infty$ and for all cubes $Q \subset \mathbb{R}^n$. Finally, we define $A_\infty(\mathbb{R}^n) := \bigcup_{p \ge 1} A_p(\mathbb{R}^n)$.

The reverse Hölder class $RH_q(\mathbb{R}^n)$, $q \in (1, \infty)$, is defined as the set of all weights $\omega$ verifying

$$ \left( \frac{1}{|Q|} \int_Q \omega^q(x) \, dx \right)^{\frac{1}{q}} \le C \frac{1}{|Q|} \int_Q \omega(x) \, dx, $$

for some constant $C < \infty$ and for all cubes $Q \subset \mathbb{R}^n$. The infimum of such constants $C$ is what we denote by $[\omega]_{RH_q}$. For $q = \infty$, the reverse Hölder class $RH_\infty(\mathbb{R}^n)$ is defined as the set of all weights $\omega$ verifying

$$ \omega(y) \le C \frac{1}{|Q|} \int_Q \omega(x) \, dx, \quad \text{for almost every } y \in Q, $$

for some constant $C < \infty$ and for all cubes $Q \subset \mathbb{R}^n$.

The Muckenhoupt-Wheeden class $A_{p,q}(\mathbb{R}^n)$, $1 \le p \le q < \infty$, consists of all weights $\omega$ satisfying

$$ [\omega]_{A_{p,q}} := \sup_{Q \subset \mathbb{R}^n} \left( \frac{1}{|Q|} \int_Q \omega^q(x) \, dx \right)^{\frac{1}{q}} \left( \frac{1}{|Q|} \int_Q \omega^{-p'}(x) \, dx \right)^{\frac{1}{p'}} < \infty, $$

when $1 < p$, and

$$ \left( \frac{1}{|Q|} \int_Q \omega^q(x) \, dx \right)^{\frac{1}{q}} \le C\omega(y), \quad \text{for almost every } y \in Q, $$

for some constant $C < \infty$ and for all cubes $Q \subset \mathbb{R}^n$ if $p = 1$.

In the following proposition, we summarize several properties of Muckenhoupt weights. For the proofs, except for (8) which is a new formulation that is more convenient for our purpose, refer to [GF85] and [Gra08, Chapter 7].

**Proposition 2.2.** We have the following properties.

(1) $A_p(\mathbb{R}^n) \subset A_q(\mathbb{R}^n)$ for all $1 \le p \le q \le \infty$.

(2) $RH_q(\mathbb{R}^n) \subset RH_p(\mathbb{R}^n)$ for all $1 < p \le q \le \infty$.
(3) If $\omega \in A_p(\mathbb{R}^n)$, $1 < p < \infty$, then there exists $q \in (1, p)$ such that $\omega \in A_q(\mathbb{R}^n)$.

(4) If $\omega \in RH_q(\mathbb{R}^n)$, $1 < q < \infty$, then there exists $p \in (q, \infty)$ such that $\omega \in RH_p(\mathbb{R}^n)$.

(5) $A_\infty(\mathbb{R}^n) = \bigcup_{1 \le p < \infty} A_p(\mathbb{R}^n) = \bigcup_{1 < q \le \infty} RH_q(\mathbb{R}^n)$.

(6) If $1 \le p \le \infty$ and $1 < q < \infty$, then $\omega \in A_p(\mathbb{R}^n) \cap RH_q(\mathbb{R}^n)$ if and only if $\omega^q \in A_{q(p-1)+1}(\mathbb{R}^n)$.

(7) If $1 \le p \le q < \infty$, then $\omega \in A_{p,q}(\mathbb{R}^n)$ if and only if $\omega^q \in A_{1+q/p'}(\mathbb{R}^n)$ if and only if $\omega \in A_{1+1/p'}(\mathbb{R}^n) \cap RH_q(\mathbb{R}^n)$.

(8) If $1 \le p < q < \infty$, then $\omega^{1/p} \in A_{p,q}(\mathbb{R}^n)$ if and only if $\omega \in A_p(\mathbb{R}^n) \cap RH_{q/p}(\mathbb{R}^n)$.

*Proof.* As we said, we only prove (8). By (6), the condition $\omega \in A_p(\mathbb{R}^n) \cap RH_{q/p}(\mathbb{R}^n)$ is equivalent to $\omega^{q/p} \in A_{q(p-1)+1}(\mathbb{R}^n)$, which can be rewritten as $(\omega^{1/p})^q \in A_{1+q/p'}(\mathbb{R}^n)$. By (7), this is equivalent to the fact that $\omega^{1/p} \in A_{p,q}(\mathbb{R}^n)$. $\square$

*Remark 2.3.* It is well known (see, for instance, [Gra08, Example 7.1.7]) that a weight of the form $\omega(x) = |x|^\beta$ belongs to the Muckenhoupt class $A_p(\mathbb{R}^n)$ for $p > 1$ if and only if $-n < \beta < n(p-1)$. Furthermore, according to point (6) of Proposition 2.2 above, a weight $\omega$ belongs to the intersection $A_2(\mathbb{R}^n) \cap RH_q(\mathbb{R}^n)$ for $p < q$ if and only if $\omega^q \in A_{1+q/p}(\mathbb{R}^n)$. As a result, a typical example of a weight $\omega$ considered in the statements of the results in this paper, satisfying $\omega \in A_2(\mathbb{R}^n) \cap RH_q(\mathbb{R}^n)$ with $p < q$, is given by

$$ \omega(x) = |x|^\beta, \quad \forall x \in \mathbb{R}^n, \quad \text{with} \quad -n \frac{p}{q} < \beta < n. $$

## 2.2. Function spaces.
We refer to [Ste93, Ch. V], [GF85] and [Gra08, Chapter 7] for general background and for the proofs of all the results concerning weights that we will cite below.

**During all this paper, $\omega$ denotes a fixed weight belonging to the Muckenhoupt class $A_2(\mathbb{R}^n)$.** By definition (2.1) with $p=2$, the weight $\omega^{-1}$ is also in the Muckenhoupt class $A_2(\mathbb{R}^n)$ with $[\omega^{-1}]_{A_2} = [\omega]_{A_2}$. We introduce the measure $d\omega := \omega(x)dx$ and if $E \subset \mathbb{R}^n$ a Lebesgue measurable set, we write $\omega(E)$ instead of $\int_E d\omega$. For all $x \in \mathbb{R}^n$ and $r > 0$, we set $\omega_r(x) := \omega(B(x, \sqrt{r}))$.

It follows from (2.1) with $p=2$ that there exists constants $\eta \in (0,1)$ and $\beta > 0$, depending only on $n$ and $[\omega]_{A_2}$, such that

$$ (2.4) \qquad \beta^{-1} \left( \frac{|E|}{|Q|} \right)^{\frac{1}{2\eta}} \le \frac{\omega(E)}{\omega(Q)} \le \beta \left( \frac{|E|}{|Q|} \right)^{2\eta}, $$

whenever $Q \subset \mathbb{R}^n$ is a cube and for all measurable sets $E \subset Q$. In particular, there exists a constant $D$, depending only on $n$ and $[\omega]_{A_2}$, called the doubling constant for $\omega$ such that

$$ (2.5) \qquad \omega(2Q) \le D\omega(Q), $$

for all cubes $Q \subset \mathbb{R}^n$. We may replace cubes by Euclidean balls. For simplicity, we keep using the same notation and constants.

For every $p \ge 1$ and $K \subset \mathbb{R}^n$ a measurable set, we let $L_\omega^p(K)$ be the space of all measurable functions $f: K \to \mathbb{C}$ such that

$$ \|f\|_{L_{\omega}^{p}(K)} := \left( \int_{K} |f|^{p} d\omega \right)^{1/p} < \infty. $$

In particular, $L_\omega^2(\mathbb{R}^n)$ is the Hilbert space of square-integrable functions on $\mathbb{R}^n$ with respect to $d\omega$. We denote its norm by $\|\cdot\|_{2,\omega}$ and its inner product by $\langle \cdot, \cdot \rangle_{2,\omega}$. The class $D(\mathbb{R}^n)$ use of smooth $(C^\infty)$, compactly supported test functions on $\mathbb{R}^n$, is dense in $L_\omega^2(\mathbb{R}^n)$ as it is dense in $C_c(\mathbb{R}^n)$, the space of continuous functions on $\mathbb{R}^n$ with compact support, and this latter space is dense in $L_\omega^2(\mathbb{R}^n)$ as $d\omega$ is a Radon measure on $\mathbb{R}^n$. Moreover, using the condition (2.1) with $p=2$, we have

$$ L_{\omega}^{2}(\mathbb{R}^{n}) \subset L_{\text{loc}}^{1}(\mathbb{R}^{n}, dx). $$

## 2.3. The degenerate Laplacian.
We define $H_\omega^1(\mathbb{R}^n)$ as the space of functions $f \in L_\omega^2(\mathbb{R}^n)$ for which the distributional gradient $\nabla_x f$ belongs to $L_\omega^2(\mathbb{R}^n)^n$, and equip this space with the norm $\|f\|_{H_\omega^1} := (\|f\|_{2,\omega}^2 + \|\nabla_x f\|_{2,\omega}^2)^{1/2}$ making it a Hilbert space. The class $D(\mathbb{R}^n)$ is dense in $H_\omega^1(\mathbb{R}^n)$ and
this follows from standard truncation and convolution techniques combined with the boundedness of the maximal operator on $L_{\omega}^2(\mathbb{R}^n)$. For a proof, see [Kil94, Thm. 2.5].

We define $-\Delta_{\omega}$ as the unbounded self-adjoint operator on $L_{\omega}^2(\mathbb{R}^n)$ associated to the positive symmetric sesquilinear form on $H_{\omega}^1(\mathbb{R}^n) \times H_{\omega}^1(\mathbb{R}^n)$ defined by

$$ (u, v) \mapsto \int_{\mathbb{R}^n} \nabla_x u \cdot \overline{\nabla_x v} \, d\omega. $$

The operator $-\Delta_{\omega}$ is injective as the measure $d\omega$ has infinite mass by (2.4). For all $\beta \in \mathbb{R}$, we define $(-\Delta_{\omega})^{\beta}$ as the self-adjoint operator $\mathbf{t}^{\beta}(-\Delta_{\omega})$, which is constructed via the functional calculus for sectorial operators, or equivalently, through the Borel functional calculus, since $-\Delta_{\omega}$ is self-adjoint. For further details, we refer to [RS80, McI86, Haa06]. Let the domain of $(-\Delta_{\omega})^{\beta}$ be denoted by $D((-\Delta_{\omega})^{\beta})$. Note that if $\beta > 0$, then $D((-\Delta_{\omega})^{-\beta})$ is the range of $(-\Delta_{\omega})^{\beta}$ from its domain.

The operator $-\Delta_{\omega}$ generates a contractive holomorphic semi-group $(e^{z\Delta_{\omega}})_{z \in S_{\frac{\pi}{2}}}$ on the half-plane $S_{\frac{\pi}{2}} := \{z \in \mathbb{C} \setminus \{0\} : |\arg(z)| < \frac{\pi}{2}\} = \{z \in \mathbb{C} : \mathrm{Re}(z) > 0\}$. We now present the following lemma, which provides the gaussian upper and lower bounds for the positive-time heat kernels. These estimates are well known and we refer to [CUR14, AN25, Baa25] by setting $A = I_n$ in those references.

**Lemma 2.6.** For all $t > 0$ and $f \in L_{\omega}^2(\mathbb{R}^n)$, we have for almost every $x \in \mathbb{R}^n$,

$$ e^{t\Delta_{\omega}} f(x) = \int_{\mathbb{R}^n} K_t(x, y) f(y) \, d\omega(y), $$

where $K_t(x, y) = \Gamma(t, x; 0, y)$ and $\Gamma(t, x; s, y)$ is the generalized fundamental solution for the weighted heat operator $\partial_t - \Delta_{\omega}$. Moreover, there exist two constants $1 < C = C([\omega]_{A_2}, n) < \infty$ and $1 < c = c([\omega]_{A_2}, n) < \infty$ such that

$$ (2.7) \qquad \frac{C^{-1}}{\min(\omega_t(x), \omega_t(y))} e^{-\frac{|x-y|^2}{t}} \le K_t(x, y) \le \frac{C}{\max(\omega_t(x), \omega_t(y))} e^{-\frac{|x-y|^2}{ct}}, $$

for all $t > 0$ and for (almost) all $(x, y) \in \mathbb{R}^{2n}$.

Note that if $c > 0$, $(t, x, y) \to \frac{\max(\omega_t(x), \omega_t(y))}{\min(\omega_t(x), \omega_t(y))} e^{-\frac{|x-y|^2}{t}}$ is uniformly bounded, hence having the mininum, the maximum or a multiplicative convex combination $\omega_t(x)^{\gamma}\omega_t(y)^{1-\gamma}$ at the denominator can be used interchangeably with different constants $C, c$.

We remark that there are also Hölder continuity bounds for $K_t(x, y)$ but we do not need them.

Following the standard argument in [AT98, Lemma 19, p. 48] (more precisely its adaptation to the weighted setting in [CUR14, Theorem 4]), one derives the following Gaussian estimates for complex-time and time derivatives of the heat kernels.

**Corollary 2.8.** For all $z \in \mathbb{C}$ with $\mathrm{Re}(z) > 0$ and $f \in L_{\omega}^2(\mathbb{R}^n)$, we have for almost every $x \in \mathbb{R}^n$,

$$ e^{z\Delta_{\omega}} f(x) = \int_{\mathbb{R}^n} K_z(x, y) f(y) \, d\omega(y). $$

Moreover, for all $\mu \in (0, \frac{\pi}{2})$, there are constants $C = C([\omega]_{A_2}, n, \mu) < \infty$ and $c = c([\omega]_{A_2}, n, \mu) > 0$ such that

$$ (2.9) \qquad |K_z(x, y)| \le \frac{C}{\max(\omega_{|z|}(x), \omega_{|z|}(y))} e^{-\frac{|x-y|^2}{c|z|}}. $$

for all $z \in S_{\mu} := \{z \in \mathbb{C} \setminus \{0\} : |\arg(z)| < \mu\}$ and (almost) all $(x, y) \in \mathbb{R}^{2n}$. Furthermore, for all $k \ge 1$ and all $t > 0$, the operator $(t\Delta_{\omega})^k e^{t\Delta_{\omega}} = t^k \frac{d^k}{dt^k} e^{t\Delta_{\omega}}$ is an integral operator whose kernel satisfies Gaussian upper bounds of the same form as in (2.9), with $t$ replacing $|z|$.

With these results, we can deduce the following (classical) consequences.

**Proposition 2.10.** We have the following statements

(1) Let $1 \le p \le \infty$ and $k \in \mathbb{N}$. Then $t^k \frac{d^k}{dt^k} e^{t\Delta_{\omega}}$ extends to a bounded operator on $L_{\omega}^p(\mathbb{R}^n)$ with uniform bound with respect to $t > 0$.
(2) Let $1 \le p < \infty$ and $k \in \mathbb{N}$. Then for all $f \in L_{\omega}^{p}(\mathbb{R}^{n})$, $t^k \frac{d^k}{dt^k} e^{t\Delta_{\omega}} f$ converges in $L_{\omega}^{p}(\mathbb{R}^{n})$ to $f$ when $t \to 0$ and $k = 0$, to $0$ when $t \to 0$ and $k \ge 1$ and to $0$ when $t \to \infty$ and $k \ge 0$.

*Proof.* The first item is a classical consequence of the Schur Lemma together with

$$ \esssup_x \int_{\mathbb{R}^n} \left| t^k \frac{d^k}{dt^k} K_t(x, y) \right| d\omega(y) + \esssup_y \int_{\mathbb{R}^n} \left| t^k \frac{d^k}{dt^k} K_t(x, y) \right| d\omega(x) \le C_k $$

that follow from the Gaussian upper bounds. The convergences come from the construction of the semigroup when $p = 2$. They extend to all $1 \le p < \infty$ by standard arguments using the Gaussian upper bounds. $\square$

More properties hold but we point out that this is all we need for our proof of Theorem 1.1.

### 3. HARDY-LITTLEWOOD-SOBOLEV ESTIMATES

In this section, we begin with key lemmas that play a central role in the proof of Theorem 1.1 that we present next. We then discuss several extensions and refinements: first, we extend the result to weighted Lorentz spaces; second, to the functional calculus; and eventually, we show that the condition imposed on the weight is sharp.

**3.1. Key lemmas.** Recall that the semigroup extends to a bounded operator on $L_{\omega}^{p}(\mathbb{R}^{n})$ for all $1 \le p \le \infty$. We wish to obtain "$L^{p} - L^{q}$ bounds" when $p < q$. We record the following key lemma on the boundedness of the semigroup, with a bound involving a suitable power of $t$, provided the weight $\omega$ lies in a reverse Hölder class. We shall see that the open ended property of the reverse Hölder condition assumed in this proof will be used in an essential way.

**Lemma 3.1.** Let $1 < p < q < \infty$. Assume that $\omega \in RH_{\frac{q}{p}}(\mathbb{R}^n)$. Then, there exist two constants $C = C([\omega]_{A_2}, [\omega]_{RH_{\frac{q}{p}}}, n, p, q) < \infty$ and $\varepsilon = \varepsilon([\omega]_{RH_{\frac{q}{p}}}, p, q, n) > 0$ such that

$$ \|e^{t\Delta_{\omega}} f\|_{L_{\omega^{q/p+\varepsilon}}^{q}(\mathbb{R}^{n})} \le C t^{-\frac{n}{2}(\frac{1}{p}-\frac{1}{q})} \|f\|_{L_{\omega^{1+(p/q)\varepsilon}}^{p}(\mathbb{R}^{n})}, $$

for all $t > 0$, $\varepsilon \in (-\varepsilon_{\omega,p,q}, \varepsilon_{\omega,p,q})$ and all $f \in L_{\omega}^{2}(\mathbb{R}^{n}) \cap L_{\omega^{1+(p/q)\varepsilon}}^{p}(\mathbb{R}^{n})$. In particular, $e^{t\Delta_{\omega}}$ extends to a bounded operator from $L_{\omega^{1+(p/q)\varepsilon}}^{p}(\mathbb{R}^{n})$ to $L_{\omega^{q/p+\varepsilon}}^{q}(\mathbb{R}^{n})$.

*Proof.* The last point follows by density once the inequality is established and we prove it next.

Given that $\omega \in RH_{\frac{q}{p}}(\mathbb{R}^n)$, point (4) of Proposition 2.2 and Hölder's inequality imply that there exist $\varepsilon_{\omega,p,q} = \varepsilon([\omega]_{RH_{\frac{q}{p}}}, p, q, n) \in (0, \min(\frac{q}{p}, \frac{q}{p}))$ and a constant $C = C([\omega]_{RH_{\frac{q}{p}}}, p, q, n) < \infty$ such that, for all $\varepsilon \in (-\varepsilon_{\omega,p,q}, \varepsilon_{\omega,p,q})$ and all cubes $Q \subset \mathbb{R}^n$, the following inequalities hold

$$ (3.2) \qquad \left( \frac{1}{|Q|} \int_Q \omega^{\frac{q}{p}+\varepsilon}(x) \, dx \right)^{\frac{1}{q}+\varepsilon} \le C \frac{1}{|Q|} \int_Q \omega(x) \, dx, $$

$$ (3.3) \qquad \left( \frac{1}{|Q|} \int_Q \omega^{1-\frac{p'}{q}\varepsilon}(x) \, dx \right)^{\frac{1}{1-\frac{p'}{q}\varepsilon}} \le \frac{1}{|Q|} \int_Q \omega(x) \, dx. $$

We fix $\varepsilon \in (-\varepsilon_{\omega,p,q}, \varepsilon_{\omega,p,q})$ and set $\gamma = \frac{1}{p} + \frac{\varepsilon}{q} \in (0, 1)$. We fix $f \in L_{\omega^{1+(p/q)\varepsilon}}^{p}(\mathbb{R}^{n}) \cap L_{\omega}^{2}(\mathbb{R}^{n})$. We fix $t > 0$ and let $m \in \mathbb{Z}$ be the integer verifying $\frac{\sqrt{t}}{2} < \frac{1}{2^m} \le \sqrt{t}$. Let $(Q_k)_{k \in \mathbb{Z}^n}$ the family of dyadic cubes with side lengths $\frac{1}{2^m}$. Using the upper bound for the heat kernel in Lemma 2.6, we have

$$ \begin{aligned} \|e^{t\Delta_{\omega}} f\|_{L_{\omega^{q/p+\varepsilon}}^{q}(\mathbb{R}^{n})} &= \int_{\mathbb{R}^{n}} |e^{t\Delta_{\omega}} f(x)|^{q} \omega^{q/p+\varepsilon}(x) \, dx \\ &\le C \int_{\mathbb{R}^{n}} \left( \int_{\mathbb{R}^{n}} \frac{e^{-\frac{|x-y|^2}{ct}}}{\omega_t(x)^\gamma \omega_t(y)^{1-\gamma}} |f(y)| \, d\omega(y) \right)^q \omega^{q/p+\varepsilon}(x) \, dx \\ &\le C \sum_{k \in \mathbb{Z}^n} \int_{Q_k} \left( \sum_{\ell \in \mathbb{Z}^n} \int_{Q_\ell} \frac{e^{-\frac{|x-y|^2}{ct}}}{\omega_t(y)^{1-\gamma}} |f(y)| \frac{\omega_q^{\frac{\varepsilon}{q}}(y)}{\omega_q^{\frac{\varepsilon}{q}}(y)} \, d\omega(y) \right)^q \frac{\omega^{q/p+\varepsilon}(x)}{\omega_t(x)^{q\gamma}} \, dx \end{aligned} $$
$$
\begin{align*}
&\le C \sum_{k \in \mathbb{Z}^n} \int_{Q_k} \left( \sum_{\ell \in \mathbb{Z}^n} \left( \int_{Q_\ell} \frac{e^{-p' \frac{|x-y|^2}{ct}}}{\omega_t(y)^{p'(1-\gamma)}} \omega(y)^{-\frac{p'}{q}\epsilon} \, d\omega(y) \right)^{\frac{1}{p'}} \|\omega^{\frac{\epsilon}{q}} f\|_{L^p_\omega(Q_\ell)} \right)^q \frac{\omega^{q/p+\epsilon}(x)}{\omega_t(x)^{q\gamma}} \, dx \\
&\le C \sum_{k \in \mathbb{Z}^n} \int_{Q_k} \left( \sum_{\ell \in \mathbb{Z}^n} e^{-\frac{d(Q_k, Q_\ell)^2}{ct}} \left( \int_{Q_\ell} \frac{\omega(y)^{1-\frac{p'}{q}\epsilon}}{\omega_t(y)^{p'(1-\gamma)}} \, dy \right)^{\frac{1}{p'}} \|f\|_{L^p_{\omega^{1+(p/q)\epsilon}(Q_\ell)}} \right)^q \frac{\omega^{q/p+\epsilon}(x)}{\omega_t(x)^{q\gamma}} \, dx \\
&= C \sum_{k \in \mathbb{Z}^n} \left( \sum_{\ell \in \mathbb{Z}^n} e^{-\frac{d(Q_k, Q_\ell)^2}{ct}} \left( \int_{Q_\ell} \frac{\omega(y)^{1-\frac{p'}{q}\epsilon}}{\omega_t(y)^{p'(1-\gamma)}} \, dy \right)^{\frac{1}{p'}} \|f\|_{L^p_{\omega^{1+(p/q)\epsilon}(Q_\ell)}} \right)^q \int_{Q_k} \frac{\omega^{q/p+\epsilon}(x)}{\omega_t(x)^{q\gamma}} \, dx.
\end{align*}
$$

Using the doubling property (2.5) with the equivalence of norms on $\mathbb{R}^n$, we see easily that there
exists a constant $C = C(n, D) \ge 1$ such that

$$
\frac{1}{C}\omega(Q_k) \le \omega_t(x) \le C\omega(Q_k), \quad \text{for all } k \in \mathbb{Z}^n \text{ and all } x \in Q_k.
$$

Furthermore, we have for all $k, \ell \in \mathbb{Z}^n$,

$$
d(Q_k, Q_\ell) \ge C(n) \left( \frac{|k-\ell|}{2^m} - \frac{1}{2^m} \right) \ge C(n) (|k-\ell|-1) \frac{\sqrt{t}}{2}.
$$

Thus, we obtain
(3.4)

$$
\|e^{t\Delta_{\omega}} f\|_{L^{q}_{\omega^{q/p+\epsilon}(\mathbb{R}^n)}} \le C \sum_{k \in \mathbb{Z}^n} \left( \sum_{\ell \in \mathbb{Z}^n} e^{-\frac{|k-\ell|^2}{c}} \left( \int_{Q_\ell} \frac{\omega(y)^{1-\frac{p'}{q}\epsilon}}{\omega(Q_\ell)^{1-\gamma}} \, dy \right)^{\frac{1}{p'}} \|f\|_{L^{p}_{\omega^{1+(p/q)\epsilon}(Q_\ell)}} \right)^q \frac{\int_{Q_k} \frac{\omega^{\frac{q}{p}+\epsilon}(x)}{\omega(Q_k)^{q\gamma}} \, dx}{\omega(Q_k)^{q\gamma}}.
$$

In addition, applying inequality (3.3) yields:

$$
(3.5) \quad \left( \int_{Q_\ell} \omega(y)^{1-\frac{p'}{q}\epsilon} \, dy \right)^{\frac{1}{p'}} \le |Q_\ell|^{\frac{1}{p'}} \left( \frac{\omega(Q_\ell)}{|Q_\ell|} \right)^{\frac{1-\frac{p'}{q}\epsilon}{p'}} = |Q_\ell|^{\frac{\epsilon}{q}} \omega(Q_\ell)^{\frac{1}{p'} - \frac{\epsilon}{q}} = |Q_\ell|^{\frac{\epsilon}{q}} \omega(Q_\ell)^{1-\gamma}.
$$

Likewise, since $\frac{q}{p} + \epsilon = q\gamma$, it follows from (3.2) that:

$$
(3.6) \qquad \frac{\int_{Q_k} \omega^{q/p+\epsilon}(x) \, dx}{\omega(Q_k)^{q\gamma}} \le C |Q_k|^{1-(\frac{q}{p}+\epsilon)}.
$$

Since $|Q_k| = |Q_\ell| \sim t^{n/2}$, combining (3.5) with (3.6), inequality (3.4) takes the form:

$$
\begin{align*}
\|e^{t\Delta_{\omega}} f\|_{L^{q}_{\omega^{q/p+\epsilon}(\mathbb{R}^n)}}^q &\le Ct^{\frac{n}{2}\epsilon} \times t^{\frac{n}{2}(1-(\frac{q}{p}+\epsilon))} \sum_{k \in \mathbb{Z}^n} \left( \sum_{\ell \in \mathbb{Z}^n} e^{-\frac{|k-\ell|^2}{c}} \|f\|_{L^{p}_{\omega^{1+(p/q)\epsilon}(Q_\ell)}} \right)^q \\
&= C \times t^{\frac{n}{2}(1-\frac{q}{p})} \sum_{k \in \mathbb{Z}^n} \left( \sum_{\ell \in \mathbb{Z}^n} e^{-\frac{|k-\ell|^2}{c}} \|f\|_{L^{p}_{\omega^{1+(p/q)\epsilon}(Q_\ell)}} \right)^q.
\end{align*}
$$

Using Young's discrete convolution inequality with $s > 1$ such that $\frac{1}{p} + \frac{1}{s} = 1 + \frac{1}{q}$, we deduce that

$$
\begin{align*}
\|e^{t\Delta_{\omega}} f\|_{L^{q}_{\omega^{q/p+\epsilon}(\mathbb{R}^n)}}^q &\le Ct^{\frac{n}{2}(1-\frac{q}{p})} \left( \sum_{k \in \mathbb{Z}^n} e^{-s \frac{|k|^2}{c}} \right)^{q/s} \left( \sum_{k \in \mathbb{Z}^n} \|f\|_{L^{p}_{\omega^{1+(p/q)\epsilon}(Q_k)}} \right)^{q/p} \\
&= Ct^{\frac{n}{2}(1-\frac{q}{p})} \left( \sum_{k \in \mathbb{Z}^n} e^{-s \frac{|k|^2}{c}} \right)^{q/s} \|f\|_{L^{p}_{\omega^{1+(p/q)\epsilon}(\mathbb{R}^n)}}^q.
\end{align*}
$$

Thus, we obtain

$$
\|e^{t\Delta_{\omega}} f\|_{L^{q}_{\omega^{q/p+\epsilon}(\mathbb{R}^n)}} \le C t^{-\frac{n}{2}(1-\frac{1}{q})} \|f\|_{L^{p}_{\omega^{1+(p/q)\epsilon}(\mathbb{R}^n)}}.
$$

□
When $1 < p = q < \infty$, since $\omega \in RH_{s\omega}(\mathbb{R}^n)$ for some $s_\omega = s([\omega]_{A_2}) > 1$, the proof of Lemma 3.1 still applies with $\varepsilon \neq 0$.

It is not clear what happens when $p = 1$ or $q = \infty$ when $\varepsilon \neq 0$. In the case $\varepsilon = 0$, we obtain the following lemma when $1 \le p < q \le \infty$ by adapting the above proof.

**Lemma 3.7.** Let $1 \le p < q \le \infty$. Assume that $\omega \in RH_{\frac{q}{p}}(\mathbb{R}^n)$. Then, there exists a constant $C = C([\omega]_{A_2}, [\omega]_{RH_{\frac{q}{p}}}, n, p, q) < \infty$ such that

$$ \|e^{t\Delta_{\omega}} f\|_{L_{\omega q/p}^{q}(\mathbb{R}^n)} \le C t^{-\frac{n}{2}(\frac{1}{p}-\frac{1}{q})} \|f\|_{L_{\omega}^{p}(\mathbb{R}^n)}, $$

for all $t > 0$ and all $f \in L_{\omega}^2(\mathbb{R}^n) \cap L_{\omega}^p(\mathbb{R}^n)$. In particular, $e^{t\Delta_{\omega}}$ extends to a bounded operator from $L_{\omega}^p(\mathbb{R}^n)$ to $L_{\omega q/p}^q(\mathbb{R}^n)$. Here, $L_{\omega q/p}^q(\mathbb{R}^n)$ when $q = \infty$ is interpreted as the space of measurable functions $f$ such that $\omega^{1/p}f \in L_{\infty dx}^{\infty}(\mathbb{R}^n)$.

**3.2. Proof of Theorem 1.1.** We adapt [Aus07, Proposition 5.3]. By the Calderón reproducing formula, we have for all $f \in L_{\omega}^2(\mathbb{R}^n)$ and $\alpha > 0$,

$$ f = \lim_{\substack{\delta \to 0^+ \\ R \to \infty}} \frac{1}{\Gamma(\alpha)} \int_{\delta}^{R} (-t\Delta_{\omega})^{\alpha} e^{t\Delta_{\omega}} f \frac{dt}{t} =: \frac{1}{\Gamma(\alpha)} \int_{0}^{\infty} (-t\Delta_{\omega})^{\alpha} e^{t\Delta_{\omega}} f \frac{dt}{t}, $$

with convergence in $L_{\omega}^2(\mathbb{R}^n)$. For all $0 < \delta < R$, we set $T_{\delta,R} := \frac{1}{\Gamma(\alpha)} \int_{\delta}^{R} t^{\alpha} e^{t\Delta_{\omega}} \frac{dt}{t}$, which is defined on $L_{\omega}^2(\mathbb{R}^n)$, into $L_{\omega}^2(\mathbb{R}^n)$ with non uniform bounds.

**Step 1.** We first show weak type bounds: namely, $\omega^{\frac{1}{p}+\frac{\varepsilon}{q}} T_{\delta,R}$ extends to a bounded operator from $L_{\omega^{1+(p/q)\varepsilon}}^p(\mathbb{R}^n)$ to the Lorentz space $L_{\infty dx}^{q,\infty}(\mathbb{R}^n)$ with bounds independent of $\delta, R$ for all $1 < p < q < \infty$ with $\alpha = \frac{n}{2}(\frac{1}{p} - \frac{1}{q})$ and $\varepsilon$ in a small interval $(-\tilde{\varepsilon}_{\omega,p,q}, \tilde{\varepsilon}_{\omega,p,q})$ depending on $[\omega]_{RH_{\frac{q}{p}}}$, $p, q$ and $n$. For $\varepsilon = 0$, this also holds when $p = 1$.

Indeed, fix $(p, q)$ as above and select $q_0$ and $q_1$ such that $p < q_0 < q < q_1$ so that $\omega \in RH_{\frac{q_0}{p}}(\mathbb{R}^n) \cap RH_{\frac{q_1}{p}}(\mathbb{R}^n)$ (see point (4) of Proposition 2.2). If $|\varepsilon| < \tilde{\varepsilon}_{\omega,p,q} := q \min(\frac{\tilde{\varepsilon}_{\omega,p,q_0}}{q_0}, \frac{\tilde{\varepsilon}_{\omega,p,q_1}}{q_1})$, one can select $\varepsilon_0, \varepsilon_1$ with $\frac{\varepsilon_0}{q_0} = \frac{\varepsilon}{q} = \frac{\varepsilon_1}{q_1}$, so that Lemma 3.1 applies to each triplet $(p, q_0, \varepsilon_0)$ and $(p, q_1, \varepsilon_1)$. Hence, Minkowski inequality with the relation between $p, q$ and $\alpha$ imply that for any $\delta < a < R$

$$ \|T_{\delta,a}f\|_{L_{\omega^{q_0/p+\varepsilon_0}}^{q_0}(\mathbb{R}^n)} \le \frac{1}{\Gamma(\alpha)} \int_{\delta}^{a} t^{\alpha-1} \|e^{t\Delta_{\omega}} f\|_{L_{\omega^{q_0/p+\varepsilon_0}}^{q_0}(\mathbb{R}^n)} dt \le C a^{\frac{n}{2}(\frac{1}{q_0}-\frac{1}{q})} \|f\|_{L_{\omega^{1+(p/q_0)\varepsilon_0}}^{p}(\mathbb{R}^n)}, $$

and

$$ \|T_{a,R}f\|_{L_{\omega^{q_1/p+\varepsilon_1}}^{q_1}(\mathbb{R}^n)} \le \frac{1}{\Gamma(\alpha)} \int_{a}^{R} t^{\alpha-1} \|e^{t\Delta_{\omega}} f\|_{L_{\omega^{q_1/p+\varepsilon_1}}^{q_1}(\mathbb{R}^n)} dt \le C a^{\frac{n}{2}(\frac{1}{q_1}-\frac{1}{q})} \|f\|_{L_{\omega^{1+(p/q_1)\varepsilon_1}}^{p}(\mathbb{R}^n)}. $$

We fix $f \in L_{\omega^{1+(p/q)\varepsilon}}^p(\mathbb{R}^n) \cap L_{\omega}^2(\mathbb{R}^n)$ such that $\|f\|_{L_{\omega^{1+(p/q)\varepsilon}}^p(\mathbb{R}^n)} = 1$. As $T_{\delta,R} = T_{\delta,a} + T_{a,R}$, then for all $\lambda > 0$, Markov inequality for the Lebesgue measure implies that

$$ \begin{aligned} |\{\omega^{\frac{1}{p}+\frac{\varepsilon}{q}}|T_{\delta,R}f| > \lambda\}| & \le |\{\omega^{\frac{1}{p}+\frac{\varepsilon_0}{q_0}}|T_{\delta,a}f| > \frac{\lambda}{2}\}| + |\{\omega^{\frac{1}{p}+\frac{\varepsilon_1}{q_1}}|T_{a,R}f| > \frac{\lambda}{2}\}| \\ & \le C\lambda^{-q_0}a^{\frac{nq_0}{2}(\frac{1}{q_0}-\frac{1}{q})} + C\lambda^{-q_1}a^{\frac{nq_1}{2}(\frac{1}{q_1}-\frac{1}{q})}. \end{aligned} $$

Taking $a = \lambda^{-\frac{2q}{n}}$, we obtain the unweighted weak type bound

$$ (3.8) \qquad |\{\omega^{\frac{1}{p}+\frac{\varepsilon}{q}}|T_{\delta,R}f| > \lambda\}| \le 2C \lambda^{-q}. $$

Remark that if $\lambda^{-\frac{2q}{n}} \le \delta$ or $\lambda^{-\frac{2q}{n}} \ge R$, then we obtain the same estimate by using one of the two terms.

According to Lemma 3.7, the weak-type estimate (3.8) remains valid in the endpoint case $p = 1$ with $\varepsilon = 0$, that is

$$ (3.9) \qquad |\{\omega|T_{\delta,R}f| > \lambda\}| \le 2C \lambda^{-q}. $$
**Step 2.** The second step, relevant only for $p > 1$, consists in interpolating the above weak type inequalities obtained for different triplets $(p_i, q_i, \varepsilon_i)$ as follows. We fix again $(p, q)$ with $\alpha = \frac{n}{2}(\frac{1}{p} - \frac{1}{q})$. We may choose $1 < p_0 < p < p_1$, $q_0 < q < q_1 < \infty$ and $|\varepsilon_0| < |\tilde{\varepsilon}_{\omega, p_0, q_0}|, |\varepsilon_1| < |\tilde{\varepsilon}_{\omega, p_1, q_1}|$ such that

$$ \omega \in RH_{\frac{q_1}{p_0}}(\mathbb{R}^n) \subset RH_{\frac{q_1}{p_1}}(\mathbb{R}^n) \cap RH_{\frac{q_0}{p_0}}(\mathbb{R}^n), \\ \frac{1}{p_0} - \frac{1}{q_0} = \frac{1}{p} - \frac{1}{q} = \frac{1}{p_1} - \frac{1}{q_1}, $$

and

$$ \frac{1}{p_0} + \frac{\varepsilon_0}{q_0} = \frac{1}{p_1} = \frac{\varepsilon_1}{q_1}. $$

Note that $\varepsilon_0$ and $\varepsilon_1$ must have different signs. By the first step, we have

$$ (3.10) \qquad \omega^{\frac{1}{p}} T_{\delta,R} = \omega^{\frac{1}{p_0} + \frac{\varepsilon_0}{q_0}} T_{\delta,R} : L_{\omega^{1+(p_0/q_0)}\varepsilon_0}^{p_0} (\mathbb{R}^n) = L_{\omega^{p_0/p}}^{p_0} (\mathbb{R}^n) \to L_{dx}^{q_0,\infty} (\mathbb{R}^n), $$

$$ (3.11) \qquad \omega^{\frac{1}{p}} T_{\delta,R} = \omega^{\frac{1}{p_1} + \frac{\varepsilon_1}{q_1}} T_{\delta,R} : L_{\omega^{1+(p_1/q_1)}\varepsilon_1}^{p_1} (\mathbb{R}^n) = L_{\omega^{p_1/p}}^{p_1} (\mathbb{R}^n) \to L_{dx}^{q_1,\infty} (\mathbb{R}^n). $$

Since $\frac{1}{p} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$ and $\frac{1}{q} = \frac{1-\theta}{q_0} + \frac{\theta}{q_1}$ for some $\theta \in (0,1)$, by Stein-Weiss real interpolation theorem with change of measures [SW58, Theorem 2.9],

$$ \omega^{\frac{1}{p}} T_{\delta,R} : L_{\tilde{\omega}}^{p}(\mathbb{R}^n) \to L_{dx}^{q}(\mathbb{R}^n) $$

with

$$ \tilde{\omega} = \left(\omega^{\frac{p_0}{p}}\right)^{(1-\theta)\frac{p}{p_0}} \times \left(\omega^{\frac{p_1}{p}}\right)^{\theta\frac{p}{p_1}} = \omega. $$

Thus, we have proved that

$$ (3.12) \qquad \|T_{\delta,R} f\|_{L_{\omega^{q/p}}^q(\mathbb{R}^n)} = \|\omega^{\frac{1}{p}} T_{\delta,R} f\|_{L^q(\mathbb{R}^n)} \le C \|f\|_{L_{\omega}^p(\mathbb{R}^n)}, \quad \text{for all } f \in L_{\omega}^2(\mathbb{R}^n) \cap L_{\omega}^p(\mathbb{R}^n), $$

with C independent of $f$, $\delta$ and $R$.

**Step 3.** The third step consists in passing to the limit to obtain (1.2) and (1.3). We begin with the case $p > 1$. The inequality (3.12) remains valid for all $f \in L_{\omega}^2(\mathbb{R}^n)$ (as the inequality is obvious if $\|f\|_{L_{\omega}^p(\mathbb{R}^n)} = \infty$). Now, we consider $f \in D((-\Delta_{\omega})^{\alpha})$. Using this, we have

$$ \|T_{\delta,R}(-\Delta_{\omega})^{\alpha} f\|_{L_{\omega^{q/p}}^q(\mathbb{R}^n)} \le C \|(-\Delta_{\omega})^{\alpha} f\|_{L_{\omega}^p(\mathbb{R}^n)}. $$

As $T_{\delta,R}(-\Delta_{\omega})^{\alpha}f$ tends to $f$ in $L_{\omega}^2(\mathbb{R}^n)$ when $\delta \to 0^+$ and $R \to \infty$ hence almost everywhere in $\mathbb{R}^n$ up to extracting a sub-sequence, then by Fatou's lemma,

$$ (3.13) \qquad \|f\|_{L_{\omega^{q/p}}^q(\mathbb{R}^n)} \le \liminf_{\substack{\delta \to 0^+ \\ R \to \infty}} \|T_{\delta,R}(-\Delta_{\omega})^{\alpha} f\|_{L_{\omega^{q/p}}^q(\mathbb{R}^n)} \le C \|(-\Delta_{\omega})^{\alpha} f\|_{L_{\omega}^p(\mathbb{R}^n)}. $$

When $p = 1$, using the weak-type estimate (3.9) and convergence in measure, we obtain

$$ \|\omega f\|_{L_{dx}^{q,\infty}(\mathbb{R}^n)} \le C \|(-\Delta_{\omega})^{\alpha} f\|_{L_{\omega}^1(\mathbb{R}^n)}, \quad \forall f \in D((-\Delta_{\omega})^{\alpha}). $$

**Step 4.** To obtain the inequalities for $(-\Delta_{\omega})^{-\alpha}$, one can observe that $T_{\delta,R}f \to (-\Delta_{\omega})^{-\alpha}f$ in $L_{\omega}^2(\mathbb{R}^n)$ when $f \in D((-\Delta_{\omega})^{-\alpha})$. Hence by the same arguments starting either from (3.12) or (3.9), we obtain the desired inequalities on $D((-\Delta_{\omega})^{-\alpha}) \cap L_{\omega}^p(\mathbb{R}^n)$. It remains to prove that, for all $1 \le p < \infty$, $D((-\Delta_{\omega})^{-\alpha}) \cap L_{\omega}^p(\mathbb{R}^n)$ is dense in $L_{\omega}^p(\mathbb{R}^n)$. Since the domains form an increasing sequence of spaces in $-\alpha$, it suffices to prove it when $\alpha$ is an integer, say $N \in \mathbb{N}$, $N \ge 1$. We define $\Phi(z) = cz^N e^{-z}$, where $c > 0$ is chosen so that

$$ \int_0^\infty \Phi(t) \frac{dt}{t} = 1. $$

For every $\ell \in \mathbb{N}^*$ and $z \in \mathbb{C}$, we define

$$ \varphi_{\ell}(z) := \int_{\frac{1}{\ell}}^{\ell} \Phi(tz) \frac{dt}{t}. $$
Fix $f \in L_{\omega}^{p}(\mathbb{R}^{n}) \cap L_{\omega}^{2}(\mathbb{R}^{n})$. Clearly it follows that for all $\ell \in \mathbb{N}$, $\ell \ge 1$,

$$ \varphi_{\ell}(-\Delta_{\omega})f \in D((-\Delta_{\omega})^{-N}) \cap L_{\omega}^{p}(\mathbb{R}^{n}). $$

Using integration by parts, one can verify by induction on $N$ that

$$ \varphi_{\ell}(z) = -P_{N}(-\ell z)e^{-\ell z} + P_{N}(-\ell^{-1}z)e^{-\ell^{-1}z} $$

where $P_N$ is a polynomial of degree $N-1$ with $P_N(0) = 1$. Therefore, $\varphi_{\ell}(-\Delta_{\omega}) = -P_N(\ell\Delta_{\omega})e^{\ell\Delta_{\omega}} + P_N(\ell^{-1}\Delta_{\omega})e^{\ell^{-1}\Delta_{\omega}}$. By Proposition 2.10, we have $\varphi_{\ell}(-\Delta_{\omega})f \to f$ in $L_{\omega}^{p}(\mathbb{R}^{n})$ as $\ell \to \infty$. Thus, $D((-\Delta_{\omega})^{-\alpha}) \cap L_{\omega}^{p}(\mathbb{R}^{n})$ is dense in $L_{\omega}^{2}(\mathbb{R}^{n}) \cap L_{\omega}^{p}(\mathbb{R}^{n})$, and hence dense in $L_{\omega}^{p}(\mathbb{R}^{n})$. $\square$

**Remark 3.14.** Having the freedom of choosing small $\varepsilon$ from Lemma 3.1 is what allows us to have the same operator in (3.10) and (3.11). Otherwise, we could not interpolate using the Stein-Weiss theorem.

**3.3. Extension to weighted Lorentz spaces.** Let $\mu$ be a measure on $\mathbb{R}^n$. The Lorentz space $L_{\mu}^{p,r}(\mathbb{R}^n)$ is defined as the set of all measurable functions $f$ on $\mathbb{R}^n$ such that $\|f\|_{L_{\mu}^{p,r}(\mathbb{R}^n)} < \infty$, with

$$ \|f\|_{L_{\mu}^{p,r}(\mathbb{R}^n)} = \begin{cases} (r \int_0^{\infty} s^{r-1} \lambda_f(s)^{r/p} ds)^{1/r}, & 1 \le p, r < \infty, \\ \sup_{s>0} s \lambda_f(s)^{1/p}, & 1 \le p < \infty, r = \infty, \end{cases} $$

where

$$ \lambda_f(s) = \mu(\{x \in \mathbb{R}^n : |f(x)| > s\}). $$

It is known that $L_{\mu}^{p,p}(\mathbb{R}^n) = L_{\mu}^{p}(\mathbb{R}^n)$ with $\|f\|_{L_{\mu}^{p,p}(\mathbb{R}^n)} = \|f\|_{L_{\mu}^{p}(\mathbb{R}^n)}$ and $\|f\|_{L_{\mu}^{p,r}(\mathbb{R}^n)}$ is non-increasing as a function of $r$, so that $L_{\mu}^{p,r_2}(\mathbb{R}^n) \subset L_{\mu}^{p,r_1}(\mathbb{R}^n)$ if $r_1 \le r_2$. We refer to [SW71, Ch. V, §3] for more details. If $\tilde{\omega}$ is a weight on $\mathbb{R}^n$, then $L_{\mu}^{p,r}(\tilde{\omega}, \mathbb{R}^n)$ is the space of all measurable functions $f$ on $\mathbb{R}^n$ such that $\tilde{\omega}f$ belongs to the Lorentz space $L_{\mu}^{p,r}(\mathbb{R}^n)$ and we set

$$ \|f\|_{L_{\mu}^{p,r}(\tilde{\omega}, \mathbb{R}^n)} := \|\tilde{\omega}f\|_{L_{\mu}^{p,r}(\mathbb{R}^n)}. $$

**Theorem 1.1** can be extended to obtain boundedness into Lorentz spaces. There are two possible results, and it is not clear that they are connected.

**Theorem 3.15.** Let $1 < p < q < \infty$. Assume that $\omega \in RH_{\frac{q}{p}}(\mathbb{R}^n)$ and set $\alpha = \frac{n}{2}(\frac{1}{p} - \frac{1}{q})$. Then, there exists a constant $C = C([\omega]_{A_2}, [\omega]_{RH_{\frac{q}{p}}}, n, p, q) < \infty$ such that

$$ \|f\|_{L_{\omega}^{q,p}(\omega^{\frac{1}{p}-\frac{1}{q}}, \mathbb{R}^n)} \le C \|(-\Delta_{\omega})^{\alpha} f\|_{L_{\omega}^{p}(\mathbb{R}^n)}, \quad \forall f \in D((-\Delta_{\omega})^{\alpha}). $$

In particular, $(-\Delta_{\omega})^{-\alpha}$ defines a bounded operator from $L_{\omega}^{p}(\mathbb{R}^n)$ to $L_{\omega}^{q,p}(\omega^{\frac{1}{p}-\frac{1}{q}}, \mathbb{R}^n)$.

**Proof.** We resume the proof of Theorem 1.1 from equation (3.12), where we have proved that $T_{\delta,R}$ extends to a bounded operator from $L_{\omega}^{p}(\mathbb{R}^n)$ to $L_{\omega^{q/p}}^{q}(\mathbb{R}^n)$. Now, to interpolate, we fix exponents $1 < p_0 < p < p_1$ and $q_0 < q < q_1$ such that $\omega \in RH_{\frac{q_1}{p_0}}(\mathbb{R}^n)$ and

$$ \frac{1}{p_0} - \frac{1}{q_0} = \frac{1}{p} - \frac{1}{q} = \frac{1}{p_1} - \frac{1}{q_1}. $$

One can thus choose $\theta \in (0, 1)$ such that $\frac{1}{p} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$ and $\frac{1}{q} = \frac{1-\theta}{q_0} + \frac{\theta}{q_1}$. Using the real interpolation method, for all $1 \le r \le \infty$, we have

$$ (L_{\omega}^{p_0}(\mathbb{R}^n), L_{\omega}^{p_1}(\mathbb{R}^n))_{\theta,r} = L_{\omega}^{p,r}(\mathbb{R}^n) \quad \text{and} \quad (L_{\omega^{q_0/p_0}}^{q_0}(\mathbb{R}^n), L_{\omega^{q_1/p_1}}^{q_1}(\mathbb{R}^n))_{\theta,r} = L_{\omega}^{q,r}(\omega^{\frac{1}{p}-\frac{1}{q}}, \mathbb{R}^n), $$

The first interpolation result, without a change of measure, is well known Hunt interpolation theorem and follows as a special case of [BL76, Theorem 5.3.1]. The second, involving a change of measure, is known as the Lizorkin–Freitag formula and is due to Lizorkin and Freitag (see [Fre78, Theorem 2] and [Liz75]). Thus, for all $1 \le r \le \infty$, $T_{\delta,R}: L_{\omega}^{p,r}(\mathbb{R}^n) \to L_{\omega}^{q,r}(\omega^{\frac{1}{p}-\frac{1}{q}}, \mathbb{R}^n)$ is bounded. We take $r = p$, and conclude the argument by following the same final steps as in the proof of Theorem 1.1. $\square$
Remark 3.16. Theorem 3.15 improves Theorem 1.1, since $p < q$, so for every $f \in L_{\omega}^{q,p}(\omega^{\frac{1}{p}-\frac{1}{q}}, \mathbb{R}^n)$:

$$ \|f\|_{L_{\omega}^{q}(\mathbb{R}^n)} = \|f\|_{L_{\omega}^{q,q}(\omega^{\frac{1}{p}-\frac{1}{q}}, \mathbb{R}^n)} \le \|f\|_{L_{\omega}^{q,p}(\omega^{\frac{1}{p}-\frac{1}{q}}, \mathbb{R}^n)}. $$

The second result, still improving Theorem 1.1, is the following.

**Theorem 3.17.** Let $1 < p < q < \infty$. Assume that $\omega \in RH_q(\mathbb{R}^n)$ and set $\alpha = \frac{n}{2}(\frac{1}{p} - \frac{1}{q})$. Then, there exists a constant $C = C([\omega]_{A_2}, [\omega]_{RH_q}, n, p, q) < \infty$ such that

$$ \|\omega^{\frac{1}{p}} f\|_{L_{\omega}^{q,p}(\mathbb{R}^n)} \le C \|(-\Delta_{\omega})^{\alpha} f\|_{L_{\omega}^{p}(\mathbb{R}^n)}, \quad \forall f \in D((-\Delta_{\omega})^{\alpha}). $$

In particular, $\omega^{\frac{1}{p}}(-\Delta_{\omega})^{-\alpha}$ defines a bounded operator from $L_{\omega}^{p}(\mathbb{R}^n)$ to $L_{dx}^{q,p}(\mathbb{R}^n)$.

*Proof.* The proof can also be seen by resuming the one of Theorem 1.1 from equations (3.10) and (3.11) with the same notation, where it was shown that the operator $\omega^{\frac{1}{p}} T_{\delta,R}$ extends boundedly from $L_{\omega^{p_0/p}}^{p_0}(\mathbb{R}^n)$ to $L_{dx}^{q_0,\infty}(\mathbb{R}^n)$ and from $L_{\omega^{p_1/p}}^{p_1}(\mathbb{R}^n)$ to $L_{dx}^{q_1,\infty}(\mathbb{R}^n)$. With the definition of these exponents, one can choose $\theta \in (0,1)$ such that $\frac{1}{p} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$ and $\frac{1}{q} = \frac{1-\theta}{q_0} + \frac{\theta}{q_1}$. By applying the real interpolation method, we know that

$$ \left( L_{\omega^{p_0/p}}^{p_0}(\mathbb{R}^n), L_{\omega^{p_1/p}}^{p_1}(\mathbb{R}^n) \right)_{\theta, p} = L_{\omega}^{p}(\mathbb{R}^n) \quad \text{and} \quad \left( L_{dx}^{q_0,\infty}(\mathbb{R}^n), L_{dx}^{q_1,\infty}(\mathbb{R}^n) \right)_{\theta, p} = L_{dx}^{q,p}(\mathbb{R}^n). $$

The first interpolation identity, which involves a change of measure, is due to Peetre; see [BL76, Theorem 5.5.1]. The second identity is, again, Hunt's interpolation theorem; see [BL76, Theorem 5.3.1]. Hence, the operator $\omega^{\frac{1}{p}} T_{\delta,R}$ is bounded from $L_{\omega}^{p}(\mathbb{R}^n)$ to $L_{dx}^{q,p}(\mathbb{R}^n)$. The conclusion is then obtained by following the same final steps as in the proof of Theorem 1.1. $\square$

Remark 3.18. The second result would be a corollary of the first one if one could prove that the multiplication operator

$$ M_{\omega^{1/p}} : L_{\omega}^{q,p}(\omega^{\frac{1}{p}-\frac{1}{q}}, \mathbb{R}^n) \to L_{dx}^{q,p}(\mathbb{R}^n), \quad f \mapsto \omega^{1/p} f, $$

defines a bounded operator when $1 < p < q < \infty$ and $\omega \in RH_q(\mathbb{R}^n)$. It is trivially true when $p = q$. One can show the desired inequality when $f$ is an indicator function of a measurable set $A \subset \mathbb{R}^n$, but we have been unable to prove it for general functions or to find a counter-example to the statement. In [CMM22], it is explicitly said in the introduction that while considering $\omega$ as part of the measure or as a multiplicative function yield the same Lebesgue space, it does not necessarily lead to the same Lorentz space. They took the example of $p = \infty$. Here we take $1 < p < q$ and we are ready to assume the reverse Hölder condition on $\omega$, but it does not seem to suffice for a conclusion. Hence, we obtain these two different results and neither one is more natural than the other.

Remark 3.19. As in the unweighted case with Hunt's interpolation theorem, one might expect that starting from the weak-type estimates (3.10) and (3.11), one could deduce that

$$ \omega^{\frac{1}{p}} T_{\delta,R} : L_{\omega}^{p,r}(\mathbb{R}^n) \to L_{dx}^{q,r}(\mathbb{R}^n), \quad \text{for all } 1 \le r \le \infty. $$

However, in general, this holds only for $r = p$, as illustrated by a counterexample in [Fer97]. In fact, by applying the Lizorkin–Freitag formula [Fre78, Theorem 2], we actually obtain that

$$ \omega^{\frac{1}{p}} T_{\delta,R} : L_{dx}^{p,r}(\omega^{1/p}, \mathbb{R}^n) \to L_{dx}^{q,r}(\mathbb{R}^n), \quad \text{for all } 1 \le r \le \infty. $$

The case $r = p$ corresponds to what has already been used in the proof of Theorem 3.17. But when $r \ne p$, the two source spaces in the displayed formulas are not the same, in agreement with both the counter-example and the above remark.
**3.4. Extension to the functional calculus.** We propose here a consequence of Theorem 1.1.

We recall that $\varphi \in \mathcal{H}_0^\infty(S_\mu)$, for $\mu \in (0, \pi)$ where $S_\mu := \{z \in \mathbb{C} \setminus \{0\} : |\arg(z)| < \mu\}$, if $\varphi : S_\mu \to \mathbb{C}$ is holomorphic and there exist constants $C, s > 0$ such that

$$|\varphi(z)| \le C \min(|z|^s, |z|^{-s}), \quad \text{for all } z \in S_\mu.$$

Let $p \in (1, \infty)$. We say that $-\Delta_\omega$ admits a bounded holomorphic functional calculus on $L_\omega^p(\mathbb{R}^n)$ if, for every $\mu \in (0, \pi)$ and every $\varphi \in \mathcal{H}_0^\infty(S_\mu)$, and for all $f \in L_\omega^2(\mathbb{R}^n) \cap L_\omega^p(\mathbb{R}^n)$, the following estimate holds:

$$(3.20) \qquad \|\varphi(-\Delta_{\omega})f\|_{L_{\omega}^{p}(\mathbb{R}^{n})} \le C \|\varphi\|_{\infty, S_{\mu}} \|f\|_{L_{\omega}^{p}(\mathbb{R}^{n})},$$

where $C > 0$ is a constant independent of both $\varphi$ and $f$. By a standard density argument, inequality (3.20) implies that $\varphi(-\Delta_\omega)$ extends to a bounded operator on $L_\omega^p(\mathbb{R}^n)$. Moreover, this estimate remains valid for any bounded holomorphic function $\varphi$ on $S_\mu$. This extension follows readily by combining Fatou's lemma with McIntosh's convergence lemma in $L_\omega^2(\mathbb{R}^n)$ (see [McI86]).

By combining the estimates in (2.9) with the fact that $-\Delta_\omega$ clearly has a bounded $H^\infty$-calculus on $L_\omega^2(\mathbb{R}^n)$, one can apply a result due to Duong and Robinson [DR96, Theorem 3.1] concerning the boundedness of the $H^\infty$-calculus of $-\Delta_\omega$ on $L_\omega^p(\mathbb{R}^n)$.

**Lemma 3.21.** The operator $-\Delta_\omega$ admits a bounded $H^\infty$-calculus on $L_\omega^p(\mathbb{R}^n)$ for each $p \in (1, \infty)$.

We made a point of not using this property so far, using only elementary arguments. Incorporating this property, we can enhance our results as follows.

**Corollary 3.22.** Let $n \ge 1$ and $\omega \in A_2(\mathbb{R}^n)$. Let $1 < p < q < \infty$. Assume that $\omega \in RH_q^p(\mathbb{R}^n)$ and set $\alpha = \frac{n}{2}(\frac{1}{p} - \frac{1}{q})$. Assume that $\varphi$ is a holomorphic function on $S_\mu$ and let $\psi(z) = z^{-\alpha}\varphi(z)$.

(1) Assume $\psi$ is bounded. Then there exists a constant $C = C([\omega]_{A_2}, [\omega]_{RH_q^p}, n, p, q) < \infty$ such that

$$\|\omega^{1/p}\psi(-\Delta_{\omega})f\|_{L_{\text{dx}}^q(\mathbb{R}^n)} \le C\|\varphi(-\Delta_{\omega})f\|_{L_{\omega}^p(\mathbb{R}^n)}, \quad \forall f \in D((-\Delta_{\omega})^{\alpha}).$$

(2) Assume $\varphi$ is bounded. Then $\psi(-\Delta_\omega)$ defines a bounded operator from $L_\omega^p(\mathbb{R}^n)$ to $L_{\omega^{q/p}}^q(\mathbb{R}^n)$ or, equivalently, $w^{1/p}\psi(-\Delta_\omega)$ is bounded from $L_\omega^p(\mathbb{R}^n)$ to $L_{\text{dx}}^q(\mathbb{R}^n)$, with bound $C \|\varphi\|_{\infty, \mu}$ with $C$ a constant as above also depending on $\mu$.

*Proof.* For the first point, we employ the standard argument (see [EHMT24, Theorem 5.15]) stating that if $f \in D((-\Delta_\omega)^\alpha)$ and $\psi$ is bounded, then $g = \psi(-\Delta_\omega)f \in D((-\Delta_\omega)^\alpha)$, with $(-\Delta_\omega)^\alpha g = \psi(-\Delta_\omega)(-\Delta_\omega)^\alpha f = \varphi(-\Delta_\omega)f$. Applying inequality (1.2) to $g$ and substituting completes the argument.

For the second point, let $f \in L_\omega^p(\mathbb{R}^n)$ and set $g = \varphi(-\Delta_\omega)f \in L_\omega^p(\mathbb{R}^n)$ by Lemma 3.21. Then using (1.2) and (3.20),

$$\|\omega^{1/p}\psi(-\Delta_{\omega})f\|_{L_{\text{dx}}^q(\mathbb{R}^n)} = \|\omega^{1/p}(-\Delta_{\omega})^{-\alpha}g\|_{L_{\text{dx}}^q(\mathbb{R}^n)} \le C\|g\|_{L_{\omega}^p(\mathbb{R}^n)} \le C'\|\varphi\|_{\infty, S_{\mu}}\|f\|_{L_{\omega}^p(\mathbb{R}^n)}.$$

□

*Remark 3.23.* By the same interpolation argument as in the proof of Theorem 3.15, when $\varphi$ is bounded, the operator $\psi(-\Delta_\omega)$ is bounded from $L_\omega^p(\mathbb{R}^n)$ to $L_\omega^{q,p}(\omega^{\frac{1}{p}-\frac{1}{q}}, \mathbb{R}^n)$, with a bound of the form $C\|\varphi\|_{\infty, \mu}$, where $C$ is a positive constant depending only on $[\omega]_{A_2}, [\omega]_{RH_q^p}, n, p, q$, and $\mu$.

**3.5. Sharpness of the condition on the weight.** In this subsection, we check that the Hölder inverse condition assumed to $\omega$ in Theorem 1.1 and Lemma 3.1 is the optimal condition.

**Proposition 3.24.** Let $1 \le p < q \le \infty$ and set $\alpha = \frac{n}{2}(\frac{1}{p} - \frac{1}{q})$. Consider the following two assertions:

(1) There is a constant $C < \infty$ such that for all $t > 0$ and $f \in L_\omega^2(\mathbb{R}^n) \cap L_\omega^p(\mathbb{R}^n)$, the following inequality holds:

$$\|e^{t\Delta_{\omega}}f\|_{L_{\omega^{q/p}}^q(\mathbb{R}^n)} \le C t^{-\alpha} \|f\|_{L_{\omega}^p(\mathbb{R}^n)}.$$
(2) Assume $1 < p$ and $q < \infty$. There is a constant $C < \infty$ such that for all $f \in D((-\Delta_{\omega})^{\alpha})$, the following inequality holds:

$$ \|f\|_{L_{\omega^{q/p}}^{q}(\mathbb{R}^n)} \leq C \|(-\Delta_{\omega})^{\alpha} f\|_{L_{\omega}^{p}(\mathbb{R}^n)}. $$

If either (1) or (2) holds, then $\omega \in RH_{\frac{q}{p}}(\mathbb{R}^n)$.

*Proof.* Assume that (1) holds. Consider the case $q < \infty$. Let $Q \subset \mathbb{R}^n$ a cube with length $\ell > 0$. By assumption, we have

$$ \|e^{\ell^2 \Delta_{\omega}} 1_Q\|_{L_{\omega^{q/p}}^{q}(\mathbb{R}^n)} \leq C \ell^{-n(\frac{1}{p}-\frac{1}{q})} \|1_Q\|_{L_{\omega}^{p}(\mathbb{R}^n)}. $$

In particular,

$$ \|e^{\ell^2 \Delta_{\omega}} 1_Q\|_{L_{\omega^{q/p}}^{q}(Q)} \leq C \ell^{-n(\frac{q}{p}-1)} \omega(Q)^{q/p} = C|Q|^{1-\frac{q}{p}} \omega(Q)^{q/p}. $$

By using the lower bound on the heat kernel in Lemma 2.6, we have

$$ \int_Q \left( \int_Q \frac{e^{-\frac{|c|x-y|^2}{\ell^2}}}{\omega_{\ell^2}(y)} \mathrm{d}\omega(y) \right)^q \omega^{q/p}(x) \mathrm{d}x \leq C|Q|^{1-\frac{q}{p}} \omega(Q)^{q/p}. $$

Using the doubling property (2.5) with the equivalence of norms on $\mathbb{R}^n$, we see easily that there exists a constant $\gamma = \gamma(n, D) > 0$ such that

$$ \gamma \leq \inf_{x \in Q} \int_Q \frac{e^{-\frac{|c|x-y|^2}{\ell^2}}}{\omega_{\ell^2}(y)} \mathrm{d}\omega(y) $$

Thus,

$$ \int_Q \omega^{q/p}(x) \mathrm{d}x \leq C|Q|^{1-\frac{q}{p}} \omega(Q)^{q/p}. $$

Therefore, $\omega \in RH_{\frac{q}{p}}(\mathbb{R}^n)$.

When $q = \infty$ we recall that (1) reads $\|w^{1/p}e^{t\Delta_{\omega}}f\|_{L^{\infty}(\mathbb{R}^n)} \leq Ct^{-\alpha}\|f\|_{L_{\omega}^{p}(\mathbb{R}^n)}$ and the argument is similar using the lower bound.

Now, assume that (2) holds. Let $f \in L_{\omega}^2(\mathbb{R}^n) \cap L_{\omega}^p(\mathbb{R}^n)$. First, $T_{\delta,R}f \in D(-\Delta_{\omega})^{\alpha}$, so we have

$$ \|T_{\delta,R}f\|_{L_{\omega^{q/p}}^{q}(\mathbb{R}^n)} \leq C \|(-\Delta_{\omega})^{\alpha} T_{\delta,R}f\|_{L_{\omega}^{p}(\mathbb{R}^n)}. $$

Now, $(-\Delta_{\omega})^{\alpha}T_{\delta,R}f = \varphi_{\delta,R}(-\Delta_{\omega})f$ for a holomorphic function $\varphi_{\delta,R}$ that is bounded on sectors $S_{\mu}$ uniformly in $\delta, R$. Thus, using Lemma 3.21, we obtain

$$ (3.25) \qquad \left\| \int_{\delta}^{R} t^{\alpha} e^{t\Delta_{\omega}} f \frac{\mathrm{d}t}{t} \right\|_{L_{\omega^{q/p}}^{q}(\mathbb{R}^n)} \leq C \Gamma(\alpha) \|f\|_{L_{\omega}^{p}(\mathbb{R}^n)}. $$

Finally, to prove that $\omega \in RH_{\frac{q}{p}}(\mathbb{R}^n)$, we fix an arbitrary cube $Q$ and denote its side length by $\ell$. In the inequality (3.25), we choose $\delta = \ell^2$, $R = 2\ell^2$ and $f = 1_Q$. From the lower bound on the heat kernel in Lemma 2.6, we obtain as before

$$ \int_Q \omega^{q/p}(x) \mathrm{d}x \leq \tilde{C} |Q|^{1-\frac{q}{p}} \omega(Q)^{q/p}, $$

where $\tilde{C} < \infty$ is a positive constant depending only on $C, \alpha, p, q, n, D$ and the choice of $\mu$. Therefore, $\omega \in RH_{\frac{q}{p}}(\mathbb{R}^n)$. $\square$
### 4. COMPARISON WITH RELATED RESULTS

This brief section aims to convince the reader that our results are of a different nature to existing results in the literature. The ones imply the others partially and under restrictive conditions and the implications involve more complex tools.

As we said before, the Sobolev embeddings for the unweighted Laplacian are equivalent to the classical Hardy-Littlewood-Sobolev inequalities. In [MW74], Muckenhoupt and Wheeden extend this embeddings for the unweighted Laplacian to weighted spaces.

For all $\beta > 0$, the Riesz potential $I_\beta$ is defined for suitable functions as

$$I_\beta(f)(x) = \int_{\mathbb{R}^n} \frac{f(y)}{|x - y|^{n-\beta}} dy.$$

We have the classical equality $(-\Delta)^{-\alpha} f = c_{n,\alpha} I_{2\alpha} f$ with $c_{n,\alpha}$ is a positive constant depending on $n$ and $\alpha$, again for suitable $f$. In what follows, think of $\alpha = \beta/2$.

**Theorem 4.1 ([MW74]).** Let $\tilde{\omega}$ be a weight. Let $\beta \in (0, n)$, $p \in [1, \frac{n}{\beta})$ and $q := \frac{np}{n-\beta p}$, that is, $\frac{1}{p} - \frac{1}{q} = \frac{\beta}{n}$. Then the following is true.

(1) If $p > 1$, then $I_\beta$ is bounded from $L_{\tilde{\omega}^p}^p(\mathbb{R}^n)$ to $L_{\tilde{\omega}q}^q(\mathbb{R}^n)$ if and only if $\tilde{\omega} \in A_{p,q}(\mathbb{R}^n)$.

(2) If $p = 1$, then $I_\beta$ is bounded from $L_{\tilde{\omega}}^{1,q}(\mathbb{R}^n)$ to $L_{\tilde{\omega}q}^{q,\infty}(\mathbb{R}^n)$ if and only if $\tilde{\omega} \in A_{1,q}(\mathbb{R}^n)$.

For $\alpha = 1/2$, then $I_1$ can be used to, roughly speaking, invert of the gradient and by construction $\|\nabla f\|_{L_{\tilde{\omega}}^2(\mathbb{R}^n)} = \|(-\Delta_{\tilde{\omega}})^{1/2} f\|_{L_{\tilde{\omega}}^2(\mathbb{R}^n)}$ on $H_{\tilde{\omega}}^1(\mathbb{R}^n)$. So, we may also make use of the following result concerning the boundedness of the Riesz transform, originally established by Cruz-Uribe, Martell, and Rios [CUMR18].

**Theorem 4.2 ([CUMR18]).** The Riesz transform $\nabla(-\Delta_{\tilde{\omega}})^{-1/2}$ is of weak type $(1, 1)$ and is of strong type $(p, p)$ for all $1 < p < q_+( -\Delta_{\tilde{\omega}})$, where this last exponent is greater than 2.

Using these results, we see that one can obtain similar inequalities, only for $0 < \alpha \le 1/2$, in a smaller range of $p$ and assuming the weight in a smaller Muckenhoupt class and in dimension $n \ge 3$.

**Corollary 4.3.** Let $p \in (1, 2]$ and assume that $n \ge 3$ and $\omega \in A_p(\mathbb{R}^n) \cap RH_{\frac{n}{n-2}}(\mathbb{R}^n)$. Then there exists a constant $C = C([\omega]_{A_p}, [\omega]_{RH_{\frac{n}{n-p}}}, n) > 0$ such that for all $\alpha \in [0, 1]$, setting $q := \frac{np}{n-p\alpha}$, we have

$$(4.4) \qquad \|f\|_{L_{\omega q/p}^q(\mathbb{R}^n)} \le C \|(-\Delta_{\omega})^{\alpha/2} f\|_{L_{\omega}^p(\mathbb{R}^n)}, \quad \text{for all } f \in D((-\Delta_{\omega})^{\alpha/2}).$$

In particular, $(-\Delta_{\omega})^{-\alpha/2}$ extends to a bounded operator from $L_{\omega}^p(\mathbb{R}^n)$ to $L_{\omega q/p}^q(\mathbb{R}^n)$. Furthermore, if $p \in [2, \min(n, q_+( -\Delta_{\omega} ))$ then same result holds, provided that $\omega \in A_2(\mathbb{R}^n) \cap RH_{\frac{n}{n-p}}(\mathbb{R}^n)$ and the constant $C$ depends only on $n$, $[\omega]_{A_2}$, $[\omega]_{RH_{\frac{n}{n-p}}}$ and $n$.

*Proof.* We begin with the case $p \in (1, 2]$. We set $p^* := \frac{np}{n-p}$. By assumption, we have $\omega \in A_p(\mathbb{R}^n) \cap RH_{\frac{2^*}{2}}(\mathbb{R}^n)$. In particular, we have $\omega \in A_p(\mathbb{R}^n) \cap RH_{\frac{p^*}{p}}(\mathbb{R}^n)$ and $\omega \in A_2(\mathbb{R}^n) \cap RH_{\frac{2^*}{2}}(\mathbb{R}^n)$. Using point (8) of Proposition 2.2, this is equivalent of saying that $\omega^{\frac{1}{p}} \in A_{p,p^*}(\mathbb{R}^n)$ and $\omega^{\frac{1}{2}} \in A_{2,2^*}(\mathbb{R}^n)$.

The second condition is used qualitatively as follows. We have the pointwise inequality $|f| \le C(n)|I_1(\nabla f)|$ for all $f \in D(\mathbb{R}^n)$. Using the density of $D(\mathbb{R}^n)$ in $H_{\omega}^1(\mathbb{R}^n)$ together with Theorem 4.1 with $\alpha = 1$, $p=2$ and $\tilde{\omega} = \omega^{\frac{1}{2}}$, we deduce that for all $f \in H_{\omega}^1(\mathbb{R}^n)$, $f$ and $I_1(\nabla f)$ are almost everywhere limits of sequences $f_k$ and $I_1(\nabla f_k)$ with $f_k \in D(\mathbb{R}^n)$. Hence $|f| \le C(n)|I_1(\nabla f)|$ almost everywhere on $\mathbb{R}^n$, for all $f \in D((-\Delta_{\omega})^{1/2}) = H_{\omega}^1(\mathbb{R}^n)$.

Next, this inequality and Theorem 4.1 with $\alpha = 1$ and $\tilde{\omega} = \omega^{\frac{1}{p}}$ imply that

$$\|f\|_{L_{\omega p^*/p}^{p^*}(\mathbb{R}^n)} \le C(n) \|I_1(\nabla f)\|_{L_{\omega p^*/p}^{p^*}(\mathbb{R}^n)} \le C(n, p, [\omega]_{A_p}, [\omega]_{RH_{\frac{n}{n-p}}}) \|\nabla f\|_{L_{\omega}^p(\mathbb{R}^n)}, \quad \forall f \in D((-\Delta_{\omega})^{1/2}).$$

Using Theorem 4.2, we deduce that for all $f \in D((-\Delta_{\omega})^{1/2})$,

$$(4.5) \qquad \|f\|_{L_{\omega p^*/p}^{p^*}(\mathbb{R}^n)} \le C \|(-\Delta_{\omega})^{1/2} f\|_{L_{\omega}^p(\mathbb{R}^n)}.$$
Now, we rewrite (4.5), for all $f \in D((-\Delta_{\omega})^{-1/2}) \cap L_{\omega}^p(\mathbb{R}^n)$, as

$$ (4.6) \qquad \|(-\Delta_{\omega})^{-1/2} f\|_{L_{\omega^{p^*}/p}^p(\mathbb{R}^n)} \le C \|f\|_{L_{\omega}^p(\mathbb{R}^n)}. $$

At this point, we apply an abstract version of Stein's interpolation theorem due to Voigt [Voi92] to the analytic family $\Psi(z) := (-\Delta_{\omega})^{-\frac{z}{2}}$ defined for $0 \le \mathrm{Re}(z) \le 1$, proceeding as follows. By the boundedness of the $H^{\infty}$-calculus on $L_{\omega}^p(\mathbb{R}^n)$ in Lemma 3.21, $-\Delta_{\omega}$ has bounded imaginary powers. This, together with inequality (4.6) give us that for all $f \in D((-\Delta_{\omega})^{-1/2}) \cap L_{\omega}^p(\mathbb{R}^n)$,

$$ \begin{aligned} \sup_{t \in \mathbb{R}} \|\Psi(it)f\|_{L_{\omega}^p(\mathbb{R}^n)} &\le c_0 \|f\|_{L_{\omega}^p(\mathbb{R}^n)}, \\ \sup_{t \in \mathbb{R}} \|\Psi(1 + it)f\|_{L_{\omega^{p^*}/p}^{p^*}(\mathbb{R}^n)} &\le c_1 \|f\|_{L_{\omega}^p(\mathbb{R}^n)}, \end{aligned} $$

where $c_0 < \infty$ and $c_1 < \infty$ are finite constants that are independent of $f$ and that depend only on $n, p, [\omega]_{A_p}$ and $[\omega]_{RH_{\frac{n}{n-p}}}$.

$$ \|\Psi(\theta)f\|_{[L_{\omega}^p(\mathbb{R}^n), L_{\omega^{p^*}/p}^{p^*}(\mathbb{R}^n)]_{\theta}} \le c_0^{1-\theta} c_1^{\theta} \|f\|_{L_{\omega}^p(\mathbb{R}^n)}. $$

Choosing $\theta \in (0, 1)$ such that $\frac{1}{q} = \frac{1-\theta}{p} + \frac{\theta}{p^*}$ and using the Stein-Weiss complex interpolation theorem with a change of measure (see [BL76, Theorem 5.5.3]), we have $[L_{\omega}^p(\mathbb{R}^n), L_{\omega^{p^*}/p}^{p^*}(\mathbb{R}^n)]_{\theta} = L_{\omega^{q/p}}^q(\mathbb{R}^n)$, and we obtain

$$ (4.7) \qquad \|(-\Delta_{\omega})^{-\alpha/2} f\|_{L_{\omega^{q/p}}^q(\mathbb{R}^n)} \le C \|f\|_{L_{\omega}^p(\mathbb{R}^n)}, \quad \text{for all } f \in D((-\Delta_{\omega})^{-1/2}) \cap L_{\omega}^p(\mathbb{R}^n). $$

By density, as in step 4 of the proof of Theorem 1.1 this shows that $(-\Delta_{\omega})^{-\alpha/2}$ defines a bounded operator from $L_{\omega}^p(\mathbb{R}^n)$ to $L_{\omega^{q/p}}^q(\mathbb{R}^n)$. Finally, the above inequality also holds for $f \in D((-\Delta_{\omega})^{-\alpha/2}) \cap L_{\omega}^p(\mathbb{R}^n)$ and we use that $D((-\Delta_{\omega})^{-\alpha/2})$ is the range of $(-\Delta_{\omega})^{\alpha/2}$ to obtain (4.4).

The case $p \in [2, \min(n, q_+( -\Delta_{\omega} ))$ is treated similarly and we skip details. $\square$

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

**Remark 4.8.** We could have used [LMPT10, Theorem 2.7] instead of Theorem 4.1. The former provides a weighted Sobolev-Gagliardo-Nirenberg inequality involving the gradient, and it also includes the endpoint case $p = 1$. This inequality is also sharp in the control of the bound with respect to the weight. More precisely, [LMPT10, Theorem 2.7] establishes that for any $p \ge 1$ and any weight $\tilde{\omega} \in A_{p,q}(\mathbb{R}^n)$, where $q < \infty$ is such that $\frac{1}{p} - \frac{1}{q} = \frac{1}{n}$, the following estimate holds for every Lipschitz function $f$ with compact support:

$$ \|f\|_{L_{\tilde{\omega}q}^q(\mathbb{R}^n)} \le c(n, p)[\tilde{\omega}]_{A_{p,q}}^{1/n'} \|\nabla f\|_{L_{\tilde{\omega}p}^p(\mathbb{R}^n)}. $$

Remark that one must assume $n \ge 3$ to use it when $p = 2$. In another direction, one can use [CUMR18, Proposition 6.1] which provides a bound $\|(-\Delta_{\omega})^{1/2} f\|_{L_{\omega}^p(\mathbb{R}^n)} \lesssim \|\nabla f\|_{L_{\omega}^p(\mathbb{R}^n)}$ when $p \in (r_{\omega}, \infty)$ where $r_{\omega} < 2$ is the infimum of exponents $r \ge 1$ for which $w \in A_r(\mathbb{R}^n)$. Our result, together with this inequality, implies the above inequality in this range of $p$ and no information on the control of the constant.

**Remark 4.9.** We have not tried to write down the kernels of $(-\Delta_{\omega})^{-\alpha}$ as we do not know how to exploit them in general. In some cases, they can be estimated in order to use other results (still necessarily with non optimal conditions). Here is one instance. Define the reverse doubling exponent of $\omega$, denoted by RD, as the largest positive constant such that

$$ \omega(B(x, r)) \le 2^{-\mathrm{RD}} \omega(B(x, 2r)), \quad \text{for all } x \in \mathbb{R}^n \text{ and } r > 0. $$

The existence of such a constant is ensured by (2.4). Using the Gaussian upper and lower bounds for the heat kernel given in Lemma 2.6, together with the representation formula

$$ (-\Delta_{\omega})^{-\alpha} f = \frac{1}{\Gamma(\alpha)} \int_{0}^{\infty} t^{\alpha} e^{t\Delta_{\omega}} f \frac{dt}{t}, $$
we can show that, whenever $\alpha < \frac{\mathrm{RD}}{2}$, the following pointwise estimate for nonnegative functions holds

$$(-\Delta_{\omega})^{-\alpha} f(x) \approx \int_{\mathbb{R}^n} \frac{|x-y|^{2\alpha}}{\omega(B(x, |x-y|))} f(y) \, d\omega(y).$$

These weighted Riesz potentials are discussed in [AM08, section 5]. However, further restrictive conditions on $\omega$ must be assumed in order to apply the results obtained therein.

## 5. CASE OF DEGENERATE ELLIPTIC OPERATORS WITH COEFFICIENTS

In this section, we fix a matrix-valued function $A : \mathbb{R}^n \to M_n(\mathbb{C})$ with complex measurable coefficients and such that

$$|A(x)\xi \cdot \zeta| \le M\omega(x) |\xi| |\zeta|, \quad \nu |\xi|^2 \omega(x) \le \mathrm{Re}(A(x)\xi \cdot \bar{\xi})$$

for some $\nu > 0$, $M < \infty$ and for all $\zeta, \xi \in \mathbb{C}^n$ and all $x \in \mathbb{R}^n$.

We define $L_\omega$ as the unbounded operator on $L_\omega^2(\mathbb{R}^n)$ associated with the following bounded, elliptic, and sectorial sesquilinear form on $H_\omega^1(\mathbb{R}^n) \times H_\omega^1(\mathbb{R}^n)$ defined by

$$(u, v) \mapsto \int_{\mathbb{R}^n} A(x) \nabla_x u \cdot \overline{\nabla_x v} \, d\omega.$$

The operator $L_\omega$ is sectorial with sectoriality angle $\varphi \in [\arctan(\frac{M}{\nu}), \frac{\pi}{2})$, and it is injective since the measure $d\omega$ has infinite mass by (2.4). For all $\beta \in \mathbb{R}$, we define $L_\omega^\beta$ as the closed operator $z_\omega^\beta(L_\omega)$, constructed via the functional calculus for sectorial operators (see [Haa06]). We denote the domain of $L_\omega^\beta$ by $D(L_\omega^\beta)$.

When the coefficients of $A$ are real-valued, all the previously stated results concerning the degenerate Laplacian $-\Delta_\omega$, such as the Gaussian upper (and lower) bounds for the heat kernel and the boundedness of the $H^\infty$-calculus on $L_\omega^p(\mathbb{R}^n)$ for every $p \in (1, \infty)$ with the same references (see [AN25, Baa25] and [DR96, Theorem 3.1], respectively) remain valid for the degenerate second-order operator $L_\omega$, with the only modification being the sectoriality angle, which changes from 0 to $\varphi$. We may now adapts our results by replacing $-\Delta_\omega$ with $L_\omega$. As an illustration, the analogs of our results in Theorem 3.15 and Theorem 3.17 that just need the Gaussian upper bounds hold.

**Theorem 5.1.** Let $1 < p < q < \infty$. Assume that $\omega \in RH_{\frac{q}{p}}(\mathbb{R}^n)$ and set $\alpha = \frac{n}{2}(\frac{1}{p} - \frac{1}{q})$. Then, there exists a constant $C = C([\omega]_{A_2}, [\omega]_{RH_{\frac{q}{p}}}, n, p, q) < \infty$ such that

$$\|w^{1/p}f\|_{L_{dx}^{q,p}(\mathbb{R}^n)} + \|f\|_{L_{\omega}^{q,p}(\omega^{\frac{1}{p}-\frac{1}{q}}, \mathbb{R}^n)} \le C \|L_{\omega}^{\alpha}f\|_{L_{\omega}^{p}(\mathbb{R}^n)}, \quad \forall f \in D(L_{\omega}^{\alpha}).$$

In particular, $L_\omega^{-\alpha}$ defines a bounded operator from $L_\omega^p(\mathbb{R}^n)$ to $L_\omega^{q,p}(\omega^{\frac{1}{p}-\frac{1}{q}}, \mathbb{R}^n)$ and $w^{1/p}L_\omega^{-\alpha}$ is bounded from $L_\omega^p(\mathbb{R}^n)$ to $L_{dx}^{q,p}(\mathbb{R}^n)$. Moreover, when $p=1$ and $\omega \in RH_q(\mathbb{R}^n)$, then

$$\|\omega f\|_{L_{dx}^{q,\infty}(\mathbb{R}^n)} \le C \|L_{\omega}^{\alpha}f\|_{L_{\omega}^{1}(\mathbb{R}^n)}, \quad \forall f \in D(L_{\omega}^{\alpha}).$$

In particular, $\omega L_\omega^{-\alpha}$ defines a bounded operator from $L_\omega^1(\mathbb{R}^n)$ to $L_{dx}^{q,\infty}(\mathbb{R}^n)$.

**Remark 5.2.** When the coefficients are complex, the reference [CUMR18] contains a thorough description of the properties of $L_\omega$. The only thing that changes for us is the existence of an interval $(p_-(L_\omega), p_+(L_\omega)) \subset (1, \infty)$ as the largest open interval containing 2 on which the semigroup $e^{-tL_\omega}$ is uniformly bounded on $L_\omega^p(\mathbb{R}^n)$ for $p$ in this interval. One has $p_-( -\Delta_\omega ) = 1$, $p_+( -\Delta_\omega ) = \infty$. The Gaussian upper bounds are replaced by a notion called $L_\omega^p - L_\omega^q$ off-diagonal estimates on balls (and balls could be equivalently replaced by cubes) denoted by $O(L_\omega^p - L_\omega^q)$. See Definition 2.33 and Proposition 3.1 in this reference, showing that they hold for $p < q$ in this interval. We claim that the proof of our key Lemma 2.6 can be adapted to obtain the same result as the first part of Theorem 5.1 but limited to $p_-(L_\omega) < p < q < p_+(L_\omega)$ (no weak type estimates at the endpoint). We shall leave details to the reader.

**Remark 5.3.** Again, the results stated in Remark 5.2 above are partially obtainable from available ones from [MW74, LMPT10] at the expense of complications and restrictions on the hypotheses. For example, one can reproduce the argument in Corollary 4.3 to obtain (4.5) with $-\Delta_\omega$ replaced by
a general degenerate elliptic operator $L_\omega$. However, one must first rely on the solution to the Kato problem [CUR15] for $L_\omega$ and generalisation [CUMR18] before applying the interpolation argument that follows.

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

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UNIVERSITÉ PARIS-SACLAY, CNRS, LABORATOIRE DE MATHÉMATIQUES D'ORSAY, 91405 ORSAY, FRANCE
*Email address: pascal.auscher@universite-paris-saclay.fr*

UNIVERSITÉ PARIS-SACLAY, CNRS, LABORATOIRE DE MATHÉMATIQUES D'ORSAY, 91405 ORSAY, FRANCE
*Email address: khalid.baadi@universite-paris-saclay.fr*