A Capacity Associated with the Weighted Lebesgue Space and Its Applications

Abstract

In this paper, we focus on a further study of the weighted Lebesgue capacity associated with the following fractional heat equation: $\begin{array}{c}{\partial }_t+{-{\mathrm{\Delta }}_x}^{\alpha }ut,x=0,\hspace{1em}\forall \alpha ,t,x\in \mathrm{0,1}\times 0,\infty \times {ℝ}^n,\hfill \\ u0,x=fx,\hspace{1em}\forall x\in {ℝ}^n.\hfill \end{array}$. More properties of that capacity are explored, and applications to a trace inequality for the weak solution of the equation are considered.

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1. Introduction

Let $\alpha \in \mathrm{0,1}$, ${ℝ}_{+}^{1+n}=0,\infty \times {ℝ}^n$, and ${-{\mathrm{\Delta }}_x}^{\alpha }$ denote the fractional power of the spatial Laplacian which can be defined by$$\begin{array}{}\text{(1)}& {-{\Delta }_x}^{\alpha }u\cdot ,x={\mathrm{ℱ}}^{-1}{\xi }^{2\alpha }\mathrm{ℱ}u\cdot ,\xi x,\hspace{1em}\forall x\in {ℝ}^n,\end{array}$$for which $\mathrm{ℱ}$ is the Fourier transform and ${\mathrm{ℱ}}^{-1}$ is its inverse. Many research studies have been carried out on the following fractional heat equation with initial data:$$\begin{array}{}\text{(2)}& \begin{array}{c}{\partial }_t+{-{\Delta }_x}^{\alpha }ut,x=0,\hspace{1em}\forall t,x\in {ℝ}_{+}^{1+n};\hfill \\ u0,x=fx,\hspace{1em}\forall x\in {ℝ}^n.\hfill \end{array}\end{array}$$

The weak solution of (2) can be written as$$\begin{array}{}\text{(3)}& ut,x=R_{\alpha }ft,x=e^{-t{-{\Delta }_x}^{\alpha }}fx={\displaystyle {\int }_{{ℝ}^n}{K}_t^{\alpha }}x-yfy\text{d}y,\end{array}$$where $K_t^{\alpha }x$ is the fractional heat kernel.$$\begin{array}{}\text{(4)}& K_t^{\alpha }x≔{2\pi }^{-n/2}{\displaystyle {\int }_{{ℝ}^n}{e}^{ix\cdot y-t{y}^{2\alpha }}}dy,\hspace{1em}\forall t,x\in {ℝ}_{+}^{1+n}.\end{array}$$

It is well known that $K_t^{1}x$ and $K_t^{1/2}x$ are the heat kernel and Poisson kernel, respectively. Although it is hard to obtain an explicit formula for $K_t^{\alpha }x$ when $\alpha \in \mathrm{0,1}\mathrm{\setminus }1/2$ (cf. [1–5]), we are interested to find the following useful estimates (cf. [6, 7]):$$\begin{array}{}\text{(5)}& \begin{array}{c}{K}_t^{\alpha }x\approx \min t^{-n/2\alpha },t{x}^{-n+2\alpha }\approx \frac{t}{{t^{1/2\alpha }+x}^{n+2\alpha }},\hspace{1em}\forall t,x\in {ℝ}_{+}^{1+n};\hfill \\ {\displaystyle {\int }_{{ℝ}^n}{K}_t^{\alpha }}x\text{d}x=1,\hspace{1em}\forall t\in 0,\infty .\hfill \end{array}\end{array}$$

To study the traces of $u$, Chang and Xiao [8] introduced the following $L^p$ capacity for a compact set $K\subset {ℝ}_{+}^{1+n}$:$$\begin{array}{}\text{(6)}& C_{\alpha p}K=\inf f_{L^p{ℝ}^n}^p:f\ge 0\&R_{\alpha }f\ge 1_K,\end{array}$$where $1_K$ stands for the characteristic function of $K$. Using $C_{\alpha p}\cdot$, they characterized a nonnegative Radon measure $\mu$ on $R_{+}^{1+n}$ such that the mapping $R_{\alpha }:L^p{ℝ}^n⟶L_{\mu }^q{ℝ}_{+}^{1+n}$ is continuous for $1<p,q<\infty$. Based on the theory of the Hedberg–Wolff potential, a noncapacitary characterization of the extension $R_{\alpha }{L}^p{ℝ}^n\subseteq L_{\mu }^q{ℝ}_{+}^{1+n}$ was given in [9]. In [10], the authors considered the fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations. As a complement of [10], when the measure coincided with the Muckenhoupt class, this article addresses a weighted version of that extension.

We first recall a nonnegative measurable function $w$ on ${ℝ}^n$ and $A_p$ weight (the Muckenhoupt class, cf. [11, 12]), denoted by $w\in A_p$, if$$\begin{array}{}\text{(7)}& \begin{array}{c}\underset{B\subset {ℝ}^n}{\sup }{w}_B{{w^{1-p^{\prime }}}_B}^{p-1}<\infty \text{as\hspace{0.17em}}1<p<\infty ,\\ \underset{B\subset {ℝ}^n}{\sup }{w}_B{\underset{x\in B}{\inf }wx}^{-1}<\infty \text{as\hspace{0.17em}}p=1,\\ \underset{B\subset {ℝ}^n}{\sup }{w}_B\exp {\log w^{-1}}_B<\infty \text{as\hspace{0.17em}}p=\infty .\end{array}\end{array}$$

Here and after, $B≔B_{r}x=y\in {ℝ}^n:x-y<r$ is the classical Euclidean ball centered at $x$ with radius $r$ for any $r,x\in {ℝ}_{+}^{1+n}$, and $g_B≔1/B{\displaystyle {\int }_{B}g}x\text{d}x$ denotes the average of the function $g$ on $B$. In the literature, $A_{\infty }$ weights can also be given by $A_{\infty }={\cup }_{1<p<\infty }{A}_p$ or$$\begin{array}{}\text{(8)}& w\in A_{\infty }\iff C_1{\frac{E}{\Omega }}^{\xi }\le \frac{wE}{w\Omega }\le C_2{\frac{E}{\Omega }}^{\eta }\text{as\hspace{0.17em}}\xi \ge 1,\eta \le 1,\text{\hspace{0.17em}and\hspace{0.17em}}E\subset \Omega ,\end{array}$$for which $C_1,C_2$ are constants independent of the sets $E,\mathrm{\Omega }$ and$$\begin{array}{}\text{(9)}& wE={\displaystyle {\int }_{E}w}\text{x}\text{d}x\&E={\displaystyle {\int }_E\text{d}x}.\end{array}$$

Let $\lambda B$ stand for $B_{\lambda r}x$ with $\lambda >0$. A weight $w$ is said to be a doubling weight if there is a constant $C$ such that $w2B\le CwB$. It is well known that $A_p$ weights are doubling weights. The weights we considered in this paper are either the Muckenhoupt classes $A_p$ or weights satisfying$$\begin{array}{}\text{(10)}& w^{1-p^{\prime }}\in A_{\infty },\end{array}$$which is a larger class than the $A_p$ class. Condition (10) was first introduced to study weighted nonlinear potential theory by Adams in [13].

The plan of this paper is as follows. Section 2 presents the definition of a weighted $L^p$ capacity and its properties. Section 3 addresses the application of the weighted capacity to characterize the trace inequalities of weak solutions to (2). Some of the results can be seen as not only the weighted extension of [8, 9] but also the parabolic case of [13].

Until further statement, we always assume throughout the paper that $0<\alpha <1$ and $1<p,q<\infty$ with $1/p+1/p^{\prime }=1$ and $1/q+1/q^{\prime }=1$. For each natural number $k$, $C^k\mathbb{X}$ stands for the space of all functions being $k$ times continuously differentiable in $\mathbb{X}$, and $C_c^k\mathbb{X}$ denotes the space of all functions in $C^k\mathbb{X}$ having compact support. $C_c^{\infty }\mathbb{X}$ is the subspace of $C_c^k\mathbb{X}$ given by $C_c^{\infty }\mathbb{X}≔{\cap }_k{C}_c^k\mathbb{X}$. ${\mathcal{U}}^{+}\mathbb{X}$ denotes the class of all nonnegative Radon measures supported by the set $\mathbb{X}$. $L^p\mathbb{X}$ is the usual Lebesgue $p$ space, and $L_{\mu }^p\mathbb{X}$ is the Lebesgue space with measure $\mu$. The letter $C$ is used to express a positive constant which is independent of the main parameters, but may vary from line to line. The symbol $a\mathrm{\lesssim }b$ means $a\le Cb$; moreover, $a\approx b$ whenever $a\mathrm{\lesssim }b$ and $b\mathrm{\lesssim }a$.

2. A Weighted $L^p$ Capacity

Let $w$ be a weight function. The weighted $L^p$ capacity associated with (2) can be defined as$$\begin{array}{}\text{(11)}& \begin{array}{c}{C}_{\alpha p}^{w}K=\inf f_{L_w^p{ℝ}^n}^p:f\ge 0\&R_{\alpha }f\ge 1_K\text{for\hspace{0.17em}compact\hspace{0.17em}set\hspace{0.17em}}K\subset {ℝ}_{+}^{1+n},\\ C_{\alpha p}^{w}O=\text{sup}{C}_{\alpha p}^{w}K:\text{compact\hspace{0.17em}}K\subset O\text{for\hspace{0.17em}open\hspace{0.17em}set\hspace{0.17em}}O\subset {ℝ}_{+}^{1+n},\\ C_{\alpha p}^{w}E=\inf C_{\alpha p}^{w}O:\text{open\hspace{0.17em}}OE\text{for\hspace{0.17em}any\hspace{0.17em}set\hspace{0.17em}}E\subset {ℝ}_{+}^{1+n}.\end{array}\end{array}$$

$\forall t_0,x_0\in {ℝ}_{+}^{1+n}$ and $r>0$, the parabolic ball, with center at $t_0,x_0$ and radius $r$, is denoted by$$\begin{array}{}\text{(12)}& B_r^{\alpha }{t}_0,x_0≔t,x\in {ℝ}_{+}^{1+n}:r^{2\alpha }<t-t_0<2r^{2\alpha }\&x-x_0<r.\end{array}$$

It is obvious that its volume $B_r^{\alpha }{t}_0,x_0\approx r^{n+2\alpha }$. The following theorem gives the estimates for $C_{\alpha p}^w{B}_r^{\alpha }{t}_0,x_0$.

Theorem 1.

Let $r,t_0,x_0\in {ℝ}_{+}\times {ℝ}_{+}^{1+n}$. Then,$$\begin{array}{}\text{(13)}& C_{\alpha p}^w{B}_r^{\alpha }{t}_0,x_0\begin{array}{c}\mathrm{\lesssim }{{\displaystyle {\int }_r^{\infty }{s}^{-n/p-1}}{w}_{B_s{x}_0}^{1-p^{\prime }}\frac{\text{d}s}{s}}^{1-p},\hspace{1em}\text{as\hspace{0.17em}}w\text{satisfies}10,\hfill \\ \approx {{\displaystyle {\int }_r^{\infty }{s}^{-n/p-1}}{w}_{B_s{x}_0}^{1-p^{\prime }}\frac{\text{d}s}{s}}^{1-p},\hspace{1em}\text{as\hspace{0.17em}}w\in A_p.\hfill \end{array}\end{array}$$

The proof of Theorem 1 needs a potential theory and will be given in the beginning of Section 3. If $w\equiv 1$, then $C_{\alpha p}^1{B}_r^{\alpha }{t}_0,x_0\approx r^n$ by Theorem 1, which reduces to the well-known estimate for the unweighted $L^p$ capacity of $B_r^{\alpha }{t}_0,x_0$ (cf. Proposition 3 of [8]).

Firstly, we give some fundamental properties of $C_{\alpha p}^w.$.

Proposition 1.

As a nonnegative set function, the mapping $K⟶C_{\alpha p}^{w}K$ enjoys some essential properties as follows:

(a) $C_{\alpha p}^w\varnothing =0$.

(b) $\begin{array}{cc}{C}_{\alpha p}^w{K}_1\le C_{\alpha p}^w{K}_2\text{as\hspace{0.17em}}{K}_1,K_2\text{are\hspace{0.17em}compact\hspace{0.17em}subsets\hspace{0.17em}of\hspace{0.17em}}{ℝ}_{+}^{1+n}\hfill & K_1\subseteq K_2,\hfill \\ C_{\alpha p}^{w_1}K\le C_{\alpha p}^{w_2}K\text{as\hspace{0.17em}}{w}_1,w_2\text{are\hspace{0.17em}nonnegative\hspace{0.17em}weight\hspace{0.17em}functions\hspace{0.17em}on\hspace{0.17em}}{ℝ}^n,\hfill & w_1\le w_2.\hfill \end{array}$

(c) If $K_i\subset {ℝ}_{+}^{1+n}$ is a compact sequence, then $C_{\alpha p}^w{\cup }_{i=1}^{\infty }{K}_i\le {\displaystyle {\sum }_{i=1}^{\infty }{C}_{\alpha p}^w}{K}_i$.

(d) For any two compact sets $K_1,K_2$ of ${ℝ}_{+}^{1+n}$, one has$$\begin{array}{}\text{(14)}& C_{\alpha p}^w{K}_1\cup K_2+C_{\alpha p}^w{K}_1\cap K_2\le C_{\alpha p}^w{K}_1+C_{\alpha p}^w{K}_2.\end{array}$$

(e) $\begin{array}{c}\underset{i⟶\infty }{\text{lim}}{C}_{\alpha p}^w{K}_i=C_{\alpha p}^{w}K\text{as\hspace{0.17em}compact\hspace{0.17em}}{K}_i\subset {ℝ}_{+}^{1+n}\text{is\hspace{0.17em}decreasing\hspace{0.17em}and\hspace{0.17em}}K={\cap }_{i=1}^{\infty }{K}_i,\\ \underset{i⟶\infty }{\text{lim}}{C}_{\alpha p}^w{E}_i=C_{\alpha p}^{w}E\text{as\hspace{0.17em}any}{E}_i\subset {ℝ}_{+}^{1+n}\text{is\hspace{0.17em}increasing\hspace{0.17em}and\hspace{0.17em}}E={\cup }_{i=1}^{\infty }{E}_i.\end{array}$

Proof.

Write$$\begin{array}{}\text{(15)}& X_{\alpha p}^{w}K\text{:}=f:0\le f\in L_w^p{ℝ}^n\&R_{\alpha }f\ge 1_K.\end{array}$$

$\text{a}$–$\text{c}$ follow immediately from the special case of Proposition 4.2 of [10] ($\mathbb{M}={ℝ}^n$ and $\text{d}\mu =wx\text{d}x$).

For $\text{d}$, set$$\begin{array}{}\text{(16)}& f=\max f_1,f_{2}g=\min f_1,f_2\text{with\hspace{0.17em}}{f}_i\in X_{\alpha p}^w{K}_i,i=\mathrm{1,2},\end{array}$$and$$\begin{array}{}\text{(17)}& E_1=x\in {ℝ}^n:f_{1}x\le f_{2}x{E}_2={ℝ}^n\backslash E_1.\end{array}$$

Then, $f\in X_{\alpha p}^w{K}_1\cup K_2$, $g\in X_{\alpha p}^w{K}_1\cap K_2$, and$$\begin{array}{}\text{(18)}& \begin{array}{c}{\displaystyle {\int }_{{ℝ}^n}{fx}^p}wx\text{d}x={\displaystyle {\int }_{E_1}{f_{2}x}^p}wx\text{d}x+{\displaystyle {\int }_{E_2}{f_{1}x}^p}wx\text{d}x,\hfill \\ {\displaystyle {\int }_{{ℝ}^n}{gx}^p}wx\text{d}x={\displaystyle {\int }_{E_1}{f_{1}x}^p}wx\text{d}x+{\displaystyle {\int }_{E_2}{f_{2}x}^p}wx\text{d}x.\hfill \end{array}\end{array}$$

This clearly forces$$\begin{array}{}\text{(19)}& f_{L_w^p{ℝ}^n}^p+g_{L_w^p{ℝ}^n}^p={f_1}_{L_w^p{ℝ}^n}^p+{f_2}_{L_w^p{ℝ}^n}^p.\end{array}$$

It follows that$$\begin{array}{}\text{(20)}& C_{\alpha p}^w{K}_1\cup K_2+C_{\alpha p}^w{K}_1\cap K_2\le {f_1}_{L_w^p{ℝ}^n}^p+{f_2}_{L_w^p{ℝ}^n}^p.\end{array}$$

Hence,$$\begin{array}{}\text{(21)}& C_{\alpha p}^w{K}_1\cup K_2+C_{\alpha p}^w{K}_1\cap K_2\le C_{\alpha p}^w{K}_1+C_{\alpha p}^w{K}_2.\end{array}$$

To prove $\text{e}$, we first note that, for any $0<\epsilon <1$, there exists a function $f\in X_{\alpha p}^{w}K$ satisfying$$\begin{array}{}\text{(22)}& f_{L_w^p{ℝ}^n}^p<C_{\alpha p}^{w}K+\epsilon .\end{array}$$

If $i$ is big enough, then the compact set $K_i\subseteq K_{\epsilon }=x\in {ℝ}^n:R_{\alpha }fx\ge 1-\epsilon$. We conclude from $\text{b}$ that$$\begin{array}{}\text{(23)}& \underset{i⟶\infty }{\text{lim}}{C}_{\alpha p}^w{K}_i\le C_{\alpha p}^w{K}_{\epsilon }\le {1-\epsilon }^{-p}{f}_{L_w^p{ℝ}^n}^p.\end{array}$$

Consequently,$$\begin{array}{}\text{(24)}& C_{\alpha p}^{w}K\le \underset{i⟶\infty }{\text{lim}}{C}_{\alpha p}^w{K}_i\le C_{\alpha p}^{w}K\text{as\hspace{0.17em}}\epsilon ⟶0.\end{array}$$

Next, we proceed to show the second statement. We first conclude from $\text{b}$ that$$\begin{array}{}\text{(25)}& C_{\alpha p}^{w}E\ge \underset{i⟶\infty }{\text{lim}}{C}_{\alpha p}^w{E}_i.\end{array}$$

To prove the reverse inequality, there is no loss of generality in assuming $C_{\alpha p}^w{E}_i<\infty$. The definition of $C_{\alpha p}^w{E}_i$ yields the existence of an open set $O_i\supseteq E_i$ with$$\begin{array}{}\text{(26)}& C_{\alpha p}^w{O}_i\le C_{\alpha p}^w{E}_i+2^{-i}\epsilon ,\hspace{1em}\forall \epsilon >0.\end{array}$$

Since $C_{\alpha p}^w{\cup }_{i=1}^k{E}_i=C_{\alpha p}^w{E}_k<\infty$, a slight modification of the telescoping inequality (Lemma 23 of [14]) (see also Lemma 2.5 of [15]) yields the following inequality:$$\begin{array}{}\text{(27)}& C_{\alpha p}^w{\cup }_{i=1}^k{O}_i-C_{\alpha p}^w{\cup }_i^k{E}_i<\epsilon .\end{array}$$

Let $K$ be a compact subset of ${\cup }_{i=1}^{\infty }{O}_i$. Then, there exists a natural number $k$ such that $K\subseteq {\cup }_{i=1}^k{O}_i$. Therefore,$$\begin{array}{}\text{(28)}& C_{\alpha p}^{w}K\le C_{\alpha p}^w{\cup }_{i=1}^k{O}_i\le C_{\alpha p}^w{\cup }_i^k{E}_i+\epsilon \le \underset{k⟶\infty }{\text{lim}}{C}_{\alpha p}^w{E}_k+\epsilon .\end{array}$$

This along with $\text{b}$ gives$$\begin{array}{}\text{(29)}& C_{\alpha p}^{w}E\le C_{\alpha p}^w{\cup }_{i=1}^{\infty }{O}_i=\underset{K\subset {\cup }_{i=1}^{\infty }}{\text{sup}}{O}_i{C}_{\alpha p}^{w}K\le \underset{k⟶\infty }{\text{lim}}{C}_{\alpha p}^w{E}_k+\epsilon ,\end{array}$$which yields the desired result$$\begin{array}{}\text{(30)}& C_{\alpha p}^{w}E\le \underset{i⟶\infty }{\text{lim}}{C}_{\alpha p}^w{E}_i\text{as\hspace{0.17em}}\epsilon ⟶0.\end{array}$$

Secondly, we prove that the infimum for the norm ${\cdot }_{L_w^p{ℝ}^n}$ in the definition of $C_{\alpha p}^w\cdot$ can be achieved in $X_{\alpha p}^w\cdot$. We call this function the capacitary potential.

Proposition 2.

Let $C_{\alpha p}^{w}K<\infty$. Then, there is a unique function $f_0\in X_{\alpha p}^{w}K$ such that$$\begin{array}{}\text{(31)}& {f_0}_{L_w^p{ℝ}^n}^p=C_{\alpha p}^{w}K.\end{array}$$

Proof.

Assume that $f_n\subset X_{\alpha p}^{w}K$ such that ${\text{lim}}_{n⟶\infty }{f_n}_{L_w^p{ℝ}^n}^p=C_{\alpha p}^{w}K$. Since $X_{\alpha p}^{w}K$ is a closed, convex, and nonempty subset of the reflexive Banach space $L_w^p{ℝ}^n$, by possibly passing to a subsequence, we may assume that$$\begin{array}{}\text{(32)}& f_n⟶f_0\text{weakly\hspace{0.17em}in\hspace{0.17em}}{L}_w^p{ℝ}^n{f}_0\in X_{\alpha p}^{w}K.\end{array}$$

Moreover, we have ${f_0}_{L_w^p{ℝ}^n}^p=C_{\alpha p}^{w}K$.

The uniform convexity of the space $L_w^p{ℝ}^n$ implies that $f_0$ is unique.

3. Application to a Trace Inequality

As an application of $C_{\alpha p}^{w}K$ on a given compact set $K\subset {ℝ}_{+}^{1+n}$, this section is devoted to some characterizations for the trace inequality of the weak solution to (2). In principle, we investigate nonnegative Radon measures $\mu \in {\mathcal{U}}^{+}K$ and weight functions $w$ such that the mapping $R_{\alpha }:L_w^p{ℝ}^n⟶L_{\mu }^q{ℝ}_{+}^{1+n}$ is continuous. That is, the following trace inequality$$\begin{array}{}\text{(33)}& {R_{\alpha }f}_{L_{\mu }^q{ℝ}_{+}^{1+n}}\mathrm{\lesssim }{f}_{L_w^p{ℝ}^n}\end{array}$$holds for any $f\in L_w^p{ℝ}^n$. To do so, first we need some preliminaries of the potential theory.

Let $w$ satisfy (10) and $\mu \in {\mathcal{U}}^{+}{ℝ}_{+}^{1+n}$. Then, the energy of $\mu$ with respect to $w$ is given by$$\begin{array}{}\text{(34)}& E_{\alpha p}^w\mu ={\displaystyle {\int }_{{ℝ}^n}{R}_{\alpha }^{\ast }\mu x^{p^{\prime }}}w{x}^{1-p^{\prime }}\text{d}x.\end{array}$$

By a direct calculation, we have that $E_{\alpha p}^w\mu$ can be rewritten as$$\begin{array}{}\text{(35)}& E_{\alpha p}^w\mu ={\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{R}_{\alpha }}{w}^{-1}{R}_{\alpha }^{\ast }\mu {t,x}^{p^{\prime }-1}\text{d}\mu t,x≔{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{W}_{\alpha p}^w}\mu t,x\text{d}\mu t,x,\end{array}$$where $W_{\alpha p}^w\mu$ is called the weighted nonlinear potential of the measure $\mu$ with respect to the weight $w$. When $w\equiv 1$, these potentials have been studied extensively (cf. [7–9]).

Let $\mu \in {\mathcal{U}}^{+}{ℝ}_{+}^{1+n}$ and $M_{\alpha }\mu x={\text{sup}}_{r>0}{r}^{-n}\mu B_r^{\alpha }{r}^{2\alpha },x$ be its fractional parabolic maximal function. Then, the first result of this section is about the $L^p-$boundedness of $M_{\alpha }\mu$.

Lemma 1.

Assume that $w\in A_{\infty }$. Then,$$\begin{array}{}\text{(36)}& {R_{\alpha }^{\ast }\mu }_{L_w^p{ℝ}^n}\approx {M_{\alpha }\mu }_{L_w^p{ℝ}^n}.\end{array}$$

Proof.

Since$$\begin{array}{}\text{(37)}& R_{\alpha }^{\ast }\mu x\underset{˜}{>}{\displaystyle {\int }_{B_r^{\alpha }{r}^{2\alpha },x}\frac{t}{{t^{1/2\alpha }+x-y}^{n+2\alpha }}}\text{d}\mu t,y\underset{˜}{>}\frac{\mu B_r^{\alpha }{r}^{2\alpha },x}{r^n},\hspace{1em}\forall r>0,\end{array}$$we have $R_{\alpha }^{\ast }\mu x\underset{˜}{>}{M}_{\alpha }\mu x.$ Hence, ${M_{\alpha }\mu }_{L_w^p{ℝ}^n}\mathrm{\lesssim }{R_{\alpha }^{\ast }\mu }_{L_w^p{ℝ}^n}.$

For the converse inequality, we first note that a slight modification of (1.2) of [16] yields that there exist two constants $a>1$ and $b>0$ such that, for any $\lambda >0$ and $0<\epsilon \le 1$, one has the following weighted good-$\lambda$ inequality:$$\begin{array}{c}\text{(38)}& wx\in {ℝ}^n:R_{\alpha }^{\ast }\mu x>a\lambda \le b{\epsilon }^{n+2\alpha /n}wx\in {ℝ}^n:R_{\alpha }^{\ast }\mu x>\lambda +wx\in {ℝ}^n:M_{\alpha }\mu x>\epsilon \lambda .\hfill \end{array}$$

Next, we proceed the proof following Theorem 3.6.1 of [17]. For any $\gamma >0$, multiplying the above inequality by ${\lambda }^{p-1}$ and integrating in $\lambda$, one has$$\begin{array}{c}\text{(39)}& {\displaystyle {\int }_0^{\gamma }w}x\in {ℝ}^n:R_{\alpha }^{\ast }\mu x>a\lambda {\lambda }^{p-1}\text{d}\lambda \le b{\epsilon }^{n+2\alpha /n}{\displaystyle {\int }_0^{\gamma }w}x\in {ℝ}^n:R_{\alpha }^{\ast }\mu x>\lambda {\lambda }^{p-1}\text{d}\lambda +{\displaystyle {\int }_0^{\gamma }w}x\in {ℝ}^n:M_{\alpha }\mu x>\epsilon \lambda {\lambda }^{p-1}\text{d}\lambda .\hfill \end{array}$$

By changing of variables, we get$$\begin{array}{c}\text{(40)}& a^{-p}{\displaystyle {\int }_0^{a\gamma }w}x\in {ℝ}^n:R_{\alpha }^{\ast }\mu x>\lambda {\lambda }^{p-1}\text{d}\lambda \le b{\epsilon }^{n+2\alpha /n}{\displaystyle {\int }_0^{\gamma }w}x\in {ℝ}^n:R_{\alpha }^{\ast }\mu x>\lambda {\lambda }^{p-1}\text{d}\lambda +{\epsilon }^{-p}{\displaystyle {\int }_0^{\epsilon \gamma }w}x\in {ℝ}^n:M_{\alpha }\mu x>\lambda {\lambda }^{p-1}\text{d}\lambda .\hfill \end{array}$$

We thus obtain$$\begin{array}{}\text{(41)}& a^{-p}{\displaystyle {\int }_{{ℝ}^n}{R_{\alpha }^{\ast }\mu x}^p}wx\text{d}x\le 2{\epsilon }^{-p}{\displaystyle {\int }_{{ℝ}^n}{M_{\alpha }\mu x}^p}wx\text{d}x.\end{array}$$

By letting $\epsilon$ small enough such that $b{\epsilon }^{n+2\alpha /n}\le 1/2a^{-p}$ and $\gamma ⟶\infty$, therefore,$$\begin{array}{}\text{(42)}& {R_{\alpha }^{\ast }\mu }_{L_w^p{ℝ}^n}\mathrm{\lesssim }{M_{\alpha }\mu }_{L_w^p{ℝ}^n}\end{array}$$as desired.

Usually, it is difficult to get the estimate for $E_{\alpha p}^w\mu$ directly. One way around this difficulty is to try to give some simpler equivalent expression first. To do so, we introduce the following weighted parabolic Hedberg–Wolff potential for $R_{\alpha }$ inspired by the idea due to Hedberg and Wolff [18] for the Riesz potential:$$\begin{array}{}\text{(43)}& P_{\alpha p}^w\mu t,x={\displaystyle {\int }_0^{\infty }{\frac{\mu B_r^{\alpha }t,x}{r^n}}^{p^{\prime }-1}}\frac{{w^{1-p^{\prime }}}_{B_{r}x}\text{d}r}{r},\hspace{1em}\forall t,x\in {ℝ}_{+}^{1+n}.\end{array}$$

Lemma 2.

Let $w$ satisfy (10). Then,$$\begin{array}{}\text{(44)}& E_{\alpha p}^w\mu \approx {\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{P}_{\alpha p}^w}\mu \text{d}\mu .\end{array}$$

Proof.

The forthcoming demonstration is a modification of Theorem 3.2 of [13]. Since$$\begin{array}{}\text{(45)}& {\frac{\mu B_r^{\alpha }{r}^{2\alpha },x}{r^n}}^{p^{\prime }}\approx {\displaystyle {\int }_r^{2r}{\frac{\mu B_s^{\alpha }{s}^{2\alpha },x}{s^n}}^{p^{\prime }}}\frac{\text{d}s}{s}\mathrm{\lesssim }{\displaystyle {\int }_0^{\infty }{\frac{\mu B_s^{\alpha }{s}^{2\alpha },x}{s^n}}^{p^{\prime }}}\frac{\text{d}s}{s},\end{array}$$we have$$\begin{array}{}\text{(46)}& M_{\alpha }\mu x\mathrm{\lesssim }{{\displaystyle {\int }_0^{\infty }{\frac{\mu B_r^{\alpha }{r}^{2\alpha },x}{r^n}}^{p^{\prime }}}\frac{\text{d}r}{r}}^{1/p^{\prime }}.\end{array}$$

We conclude from Lemma 1 that$$\begin{array}{}\text{(47)}& E_{\alpha p}^w\mu \mathrm{\lesssim }{\displaystyle {\int }_{{ℝ}^n}{M_{\alpha }\mu x}^{p^{\prime }}}w{x}^{1-p^{\prime }}\text{d}x\mathrm{\lesssim }{\displaystyle {\int }_{{ℝ}^n}{\displaystyle {\int }_0^{\infty }{\frac{\mu B_r^{\alpha }{r}^{2\alpha },x}{r^n}}^{p^{\prime }}}}\frac{w{x}^{1-p^{\prime }}\text{d}r}{r}\text{d}x.\end{array}$$

Substituting the following estimate$$\begin{array}{c}\text{(48)}& {\displaystyle {\int }_{{ℝ}^n}\mu }{B_r^{\alpha }{r}^{2\alpha },x}^{p^{\prime }}w{x}^{1-p^{\prime }}\text{d}x={\displaystyle {\int }_{{ℝ}^n}\mu }{B_r^{\alpha }{r}^{2\alpha },x}^{p^{\prime }-1}{\displaystyle {\int }_{B_r^{\alpha }{r}^{2\alpha },x}\text{d}\mu }t,yw{x}^{1-p^{\prime }}\text{d}x\mathrm{\lesssim }{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}\mu }{B_r^{\alpha }2r^{2\alpha },y}^{p^{\prime }-1}{\displaystyle {\int }_{B_{r}y}w}{x}^{1-p^{\prime }}\text{d}x\text{d}\mu t,y\hfill \end{array}$$into (2), one has$$\begin{array}{c}\text{(49)}& E_{\alpha p}^w\mu \mathrm{\lesssim }{\displaystyle {\int }_0^{\infty }{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}\mu }}{B_r^{\alpha }2r^{2\alpha },y}^{p^{\prime }-1}{\displaystyle {\int }_{B_{r}y}w}{x}^{1-p^{\prime }}\text{d}x\text{d}\mu t,y\frac{\text{d}r}{r^{n{p}^{\prime }+1}}\mathrm{\lesssim }{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{\displaystyle {\int }_0^{\infty }{\frac{\mu B_r^{\alpha }t,y}{r^n}}^{p^{\prime }-1}}}\frac{{w^{1-p^{\prime }}}_{B_{r}y}\text{d}r}{r}\text{d}\mu t,y\hfill \\ ={\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{P}_{\alpha p}^w}\mu \text{d}\mu .\hfill \end{array}$$

To prove the reverse, we first note that$$\begin{array}{c}\text{(50)}& W_{\alpha p}^w\mu t,x=K_t^{\alpha }\ast {w^{-1}{R}_{\alpha }^{\ast }\mu }^{p^{\prime }-1}t,x\underset{˜}{>}{\displaystyle \sum _{m\in \mathbb{Z}}{\displaystyle {\int }_{B_{2^{-m}}x}{t}^{-n/2\alpha }}}{w^{-1}y{\displaystyle {\int }_{B_{2^{-m}}^{\alpha }t,x}{s}^{-n/2\alpha }}\text{d}\mu s,z}^{p^{\prime }-1}\text{d}y\hfill \\ \underset{˜}{>}{\displaystyle \sum _{m\in \mathbb{Z}}{\displaystyle {\int }_{B_{2^{-m}}x}{2}^{mn}}}{\frac{\mu B_{2^{-m}}^{\alpha }t,x}{2^{-mn}}}^{p^{\prime }-1}{w}^{1-p^{\prime }}y\text{d}y\hfill \\ \underset{˜}{>}{\displaystyle {\int }_0^{\infty }{\frac{\mu B_r^{\alpha }t,x}{r^n}}^{p^{\prime }-1}}\frac{{w^{1-p^{\prime }}}_{B_{r}x}\text{d}r}{r}\hfill \\ =P_{\alpha p}^w\mu t,x.\hfill \end{array}$$

The proof is completed by the fact that$$\begin{array}{}\text{(51)}& E_{\alpha p}^w\mu ={\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{W}_{\alpha p}^w}\mu t,x\text{d}\mu t,x\underset{˜}{>}{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{P}_{\alpha p}^w}\mu t,x\text{d}\mu t,x.\end{array}$$

The following capacitary inequality for $P_{\alpha p}^w\mu$ is a straightforward adaption of Proposition 4.1 of [13].

Lemma 3.

If $w$ satisfies (10), then for any $\mu \in {\mathcal{U}}^{+}{ℝ}_{+}^{1+n}$, one has the following inequality:$$\begin{array}{}\text{(52)}& C_{\alpha p}^w{P}_{\alpha p}^w\mu >\lambda \mathrm{\lesssim }{\lambda }^{1-p}\mu .\end{array}$$

On account of the above analysis, we have the following.

Lemma 4.

Let $K$ be a compact set in ${ℝ}_{+}^{1+n}$, $\mu \in {\mathcal{U}}^{+}{ℝ}_{+}^{1+n}$, and $w$ satisfy (10). Then,$$\begin{array}{}\text{(53)}& C_{\alpha p}^{w}K\begin{array}{c}\underset{˜}{>}\mu \text{as\hspace{0.17em}}{P}_{\alpha p}^w\mu \le 1\text{on\hspace{0.17em}}K,\hfill \\ \mathrm{\lesssim }\mu \text{as\hspace{0.17em}}{P}_{\alpha p}^w\mu \ge 1\text{on\hspace{0.17em}}K.\hfill \end{array}\end{array}$$

Proof.

Due to Proposition 4.1 of [10] and the proof of Lemma 2, there is ${\mu }_K\in {\mathcal{U}}^{+}K$ such that $P_{\alpha p}^w{\mu }_K\mathrm{\lesssim }1$ on $\mathrm{supp}{\mu }_K$. Then, combining with Lemma 3, we give the proof.

Using Lemma 4, we are now in a position to show the proof of Theorem 1 in Section 2. Since$$\begin{array}{c}\text{(54)}& P_{\alpha p}^w\mu t_0,x_0={\displaystyle {\int }_0^{2r}{\frac{\mu B_s^{\alpha }{t}_0,x_0}{s^n}}^{p^{\prime }-1}}\frac{{w^{1-p^{\prime }}}_{B_s{x}_0}\text{d}s}{s}+{\displaystyle {\int }_{2r}^{\infty }{\frac{\mu B_s^{\alpha }{t}_0,x_0}{s^n}}^{p^{\prime }-1}}\frac{{w^{1-p^{\prime }}}_{B_s{x}_0}\text{d}s}{s}\hfill \\ ≔I+II,\hfill \end{array}$$applying Lemma 4 to $\text{d}\mu =Cw\text{d}x\text{d}t$ on ball $B_s^{\alpha }{t}_0,x_0$ in which constant will be determined later, and we can obtain$$\begin{array}{}\text{(55)}& II\approx C^{p^{\prime }-1}{\displaystyle {\int }_{2r}^{\infty }{\frac{{\displaystyle {\int }_{B_s^{\alpha }{t}_0,x_0}w}x\text{d}t\text{d}x}{s^n}}^{p^{\prime }-1}}\frac{{w^{1-p^{\prime }}}_{B_s{x}_0}\text{d}s}{s}.\end{array}$$

Choosing $C$ such that the right side of the above estimate is equal to 1, then $P_{\alpha p}^w\mu \ge 1$ on $B_r^{\alpha }{t}_0,x_0$. It follows from Lemma 4 and the doubling property of $A_{\infty }$ weights that$$\begin{array}{}\text{(56)}& C_{\alpha p}^w{B}_r^{\alpha }{t}_0,x_0\mathrm{\lesssim \Vert }\mu \mathrm{\Vert }\approx {{\displaystyle {\int }_r^{\infty }{s}^{-\frac{n}{p-1}}}\frac{{w^{1-p^{\prime }}}_{B_s{x}_0}ds}{s}}^{1-p}.\end{array}$$

If $w\in A_p$, then for the above $C$, there is $r$ such that$$\begin{array}{}\text{(57)}& I\mathrm{\lesssim }{\displaystyle {\int }_0^{2r}{s}^{2\alpha p^{\prime }-1}}{w_{B_s{x}_0}}^{p^{\prime }-1}{w^{1-p^{\prime }}}_{B_s{x}_0}\frac{\text{d}s}{s}\mathrm{\lesssim }{r}^{2\alpha p^{\prime }-1}\mathrm{\lesssim }{C}^{p^{\prime }-1}II\approx 1.\end{array}$$

Then, a further application of Lemma 4 gives$$\begin{array}{}\text{(58)}& C_{\alpha p}^w{B}_r^{\alpha }{t}_0,x_0\underset{˜}{>}\mu \approx {{\displaystyle {\int }_r^{\infty }{s}^{-n/p-1}}\frac{{w^{1-p^{\prime }}}_{B_s{x}_0}\text{d}s}{s}}^{1-p}.\end{array}$$

3.1. Trace Inequality for $1<p\le q<\infty$

In this section, we characterize (33) under the lower case $1<p\le q<\infty$ by $C_{\alpha p}^w\cdot$. In Theorem 5.3 of [10], the authors obtained the following result.

Theorem 2.

Let $1<p\le q<\infty$, $\mu \in {\mathcal{U}}^{+}{ℝ}_{+}^{1+n}$, and $w$ satisfy (10). Then, the following two statements are equivalent:

(a) (33) is valid for any nonnegative $f\in L_w^p{ℝ}^n$

(b) ${\sup }_{\lambda >0}{\lambda }^{p/q}/{\mathrm{\Phi }}_{\lambda ,\mu }^w<\infty$, for which ${\mathrm{\Phi }}_{\lambda ,\mu }^w=\inf C_{\alpha p}^{w}K:\text{compact\hspace{0.17em}}K\subset {ℝ}_{+}^{1+n}\&\mu K\ge \lambda$

In particular, if $p\ne q$ (i.e., $1<p<q<\infty$), we have the following deep understanding of (33).

Theorem 3.

Assume that $1<p<q<\infty$, $\mu \in {\mathcal{U}}^{+}{ℝ}_{+}^{1+n}$, and $w$ satisfies (10). Then, the following four statements are equivalent:

(a) (33) holds for all nonnegative $f\in L_w^p{ℝ}^n$

(b) $\mu K\mathrm{\lesssim }{C}_{\alpha p}^w{K}^{p/q}$ for all compact sets $K\subset {ℝ}_{+}^{1+n}$

(c) $\mu B_r^{\alpha }t,x\mathrm{\lesssim }{C}_{\alpha p}^w{B_r^{\alpha }t,x}^{p/q}$ for all $r,t,x\in {ℝ}_{+}\times {ℝ}_{+}\times {ℝ}^n$

(d) $\mu B_r^{\alpha }t,x\mathrm{\lesssim }{\mathrm{\Psi }}_x{r}^{-q/p^{\prime }}$ for all $r,t,x\in {ℝ}_{+}\times {ℝ}_{+}\times {ℝ}^n$, where ${\mathrm{\Psi }}_{x}r={\displaystyle {\int }_r^{\infty }{w^{1-p^{\prime }}}_{B_{s}x}/s^{n{p}^{\prime }-1}\text{d}s/s}$

Proof.

$\text{a}\Rightarrow \text{b}$ follows directly from the proof of Theorem 2.

$\text{b}\Rightarrow \text{c}$ is trivial.

$\text{c}\Rightarrow \text{d}$ is a consequence of Theorem 1.

$\text{d}\Rightarrow \text{a}$: let ${\mathcal{K}}_{\lambda }f$ be defined as in the proof of Theorem 2 and ${\mu }_{{\mathcal{K}}_{\lambda }}$ be $\mu$-restricted to ${\mathcal{K}}_{\lambda }f$. Following Lemma 2, we can obtain that$$\begin{array}{}\text{(59)}& \lambda \mu {\mathcal{K}}_{\lambda }\le {\displaystyle {\int }_{{\mathcal{K}}_{\lambda }}{R}_{\alpha }}f\text{d}\mu ={\displaystyle {\int }_{{ℝ}^n}f}x{R}_{\alpha }^{\ast }{\mu }_{{\mathcal{K}}_{\lambda }}x\text{d}x\mathrm{\lesssim }{f}_{L_w^p{ℝ}^n}{{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{P}_{\alpha p}^w}{\mu }_{{\mathcal{K}}_{\lambda }}\text{d}{\mu }_{{\mathcal{K}}_{\lambda }}}^{1/p^{\prime }}.\end{array}$$

For suitable $\rho >0$ to be determined later, $P_{\alpha p}^w{\mu }_{{\mathcal{K}}_{\lambda }}$ can be decomposed as$$\begin{array}{c}\text{(60)}& P_{\alpha p}^w{\mu }_{{\mathcal{K}}_{\lambda }}t,x={\displaystyle {\int }_0^{\rho }{\frac{{\mu }_{{\mathcal{K}}_{\lambda }}{B}_r^{\alpha }t,x}{r^n}}^{p^{\prime }-1}}\frac{{w^{1-p^{\prime }}}_{B_{r}x}\text{d}r}{r}+{\displaystyle {\int }_{\rho }^{\infty }{\frac{{\mu }_{{\mathcal{K}}_{\lambda }}{B}_r^{\alpha }t,x}{r^n}}^{p^{\prime }-1}}\frac{{w^{1-p^{\prime }}}_{B_{r}x}\text{d}r}{r}\hfill \\ ≔J+JJ.\hfill \end{array}$$

Use $\text{d}$ to derive that$$\begin{array}{}\text{(61)}& J\mathrm{\lesssim }{\displaystyle {\int }_0^{\rho }{{\Psi }_{x}r}^{-q/p}}\frac{{w^{1-p^{\prime }}}_{B_{r}x}}{r^{n{p}^{\prime }-1}}\frac{\text{d}r}{r}={\displaystyle {\int }_0^{\rho }{{\Psi }_{x}r}^{-q/p}}\text{d}{\Psi }_{x}r={\Psi }_x{\rho }^{-q/p}-{\Psi }_x{0}^{-q/p},\end{array}$$and$$\begin{array}{}\text{(62)}& JJ\mathrm{\lesssim }\mu {{\mathcal{K}}_{\lambda }}^{p^{\prime }-1}{\Psi }_x\rho .\end{array}$$

Choose$$\begin{array}{}\text{(63)}& \rho =\begin{array}{c}\rho x,\lambda \text{so\hspace{0.17em}that\hspace{0.17em}}{\Psi }_x\rho ={\mu }_{{\mathcal{K}}_{\lambda }}^{-p^{\prime }/q}\text{as\hspace{0.17em}}{\Psi }_x{0}^{-q/p}<{\mu }_{{\mathcal{K}}_{\lambda }}^{p^{\prime }-1},\hfill \\ 0\text{as\hspace{0.17em}}{\Psi }_x{0}^{-q/p}\ge {\mu }_{{\mathcal{K}}_{\lambda }}^{p^{\prime }-1}.\hfill \end{array}\end{array}$$

Then, a further application of $\text{d}$ and Lemma 2 gives$$\begin{array}{}\text{(64)}& \lambda {\mu }_{{\mathcal{K}}_{\lambda }}\mathrm{\lesssim }\begin{array}{c}{f}_{L_w^p{ℝ}^n}{{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{\mu }_{{\mathcal{K}}_{\lambda }}^{p^{\prime }-1}}{\Psi }_x\rho \text{d}{\mu }_{{\mathcal{K}}_{\lambda }}}^{1/p^{\prime }}\mathrm{\lesssim }{f}_{L_w^p{ℝ}^n}{\mu }_{{\mathcal{K}}_{\lambda }}^{1/q^{\prime }}\text{as\hspace{0.17em}}{\Psi }_x{0}^{-q/p}<{\mu }_{{\mathcal{K}}_{\lambda }}^{p^{\prime }-1},\hfill \\ f_{L_w^p{ℝ}^n}{{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{\mu }_{{\mathcal{K}}_{\lambda }}^{p^{\prime }-1}}{\Psi }_{x}0\text{d}{\mu }_{{\mathcal{K}}_{\lambda }}}^{1/p^{\prime }}\mathrm{\lesssim }{f}_{L_w^p{ℝ}^n}{\mu }_{{\mathcal{K}}_{\lambda }}^{1/q^{\prime }}\text{as\hspace{0.17em}}{\Psi }_x{0}^{-q/p}\ge {\mu }_{{\mathcal{K}}_{\lambda }}^{p^{\prime }-1},\hfill \end{array}\end{array}$$which implies (33).

3.2. Trace Inequality for $1<q<p<\infty$

This section focuses on a further trace inequality under the upper case $1<q<p<\infty$. The main result can be formulated as follows.

Theorem 4.

Let $1<q<p<\infty$ and $\mu ,w$, and ${\mathrm{\Phi }}_{\lambda ,\mu }^w$ be as in Theorem 2. Then, the following two statements are equivalent:

(a) (33) holds for all nonnegative $f\in L_w^p{ℝ}^n$

(b) $P_{\alpha p}^w\mu \in L_{\mu }^{qp-1/p-q}{ℝ}_{+}^{1+n}$

Proof.

The proof can be seen as a weighted fractional heat potential version of the Riesz potential treatment carried out in [19].

To show $\text{a}\Rightarrow \text{b}$, we first denote by $Q_l^{\alpha }$ the $\alpha$-dyadic cube with side length $l\equiv l{Q}_l^{\alpha }$ and corners in the set $l^{2\alpha }{\mathbb{Z}}_{+},l{\mathbb{Z}}^n$ with ${\mathbb{Z}}_{+}=\mathrm{0,1,2},\dots$, namely,$$\begin{array}{}\text{(65)}& Q_l^{\alpha }\equiv k_0{l}^{2\alpha },k_0+1l^{2\alpha }\times Q≔k_0{l}^{2\alpha },k_0+1l^{2\alpha }\times k_{1}l,k_1+1l\times \cdots \times k_{n}l,k_n+1l,\end{array}$$as $k_0\in {\mathbb{Z}}_{+}$ and $k_i\in \mathbb{Z}$ for $i=\mathrm{1,2},\dots ,n$. Next, we introduce the following weighted fractional heat Hedberg–Wolff potential generated by ${\mathcal{D}}^{\alpha }$—the family of all the above-defined $\alpha$-dyadic cubes in ${ℝ}_{+}^{1+n}$:$$\begin{array}{}\text{(66)}& P_{\alpha p}^{d,w}\mu t,x={\displaystyle \sum _{Q_l^{\alpha }\in {\mathcal{D}}^{\alpha }}{\frac{\mu Q_l^{\alpha }}{l^n}}^{p^{\prime }-1}}{w^{1-p^{\prime }}}_Q{1}_{Q_l^{\alpha }}t,x,\end{array}$$and we need to prove$$\begin{array}{}\text{(67)}& \text{a}\Rightarrow P_{\alpha p}^{d,w}\mu \in L_{\mu }^{qp-1/p-q}{ℝ}_{+}^{1+n}.\end{array}$$

Since $\text{a}$ is equivalent to the inequality$$\begin{array}{}\text{(68)}& {R_{\alpha }^{\ast }\mathbf{g}\text{d}\mu }_{L_{w^{1-p^{\prime }}}^{p^{\prime }}{ℝ}^n}^{p^{\prime }}\mathrm{\lesssim }{\mathbf{g}}_{L_{\mu }^{q^{\prime }}{ℝ}_{+}^{1+n}}^{p^{\prime }},\forall \mathbf{g}\in L_{\mu }^{q^{\prime }}{ℝ}_{+}^{1+n},\end{array}$$we notice that Lemma 2 is also true with $P_{\alpha p}^{d,w}\mu$ in place of $P_{\alpha p}^w\mu$ and $\mathbf{g}\text{d}\mu$ in place of $\text{d}\mu$, and it may be concluded that$$\begin{array}{c}\text{(69)}& {R_{\alpha }^{\ast }\mathbf{g}\text{d}\mu }_{L_{w^{1-p^{\prime }}}^{p^{\prime }}{ℝ}^n}^{p^{\prime }}\underset{˜}{>}{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{P}_{\alpha p}^{d,w}}\mathbf{g}\text{d}\mu t,x\mathbf{g}t,x\text{d}\mu t,x\approx {\displaystyle \sum _{Q_l^{\alpha }}{\frac{{\displaystyle {\int }_{Q_l^{\alpha }}\mathbf{g}}t,x\text{d}\mu t,x}{l^n}}^{p^{\prime }}}{l}^n{w^{1-p^{\prime }}}_Q,\hfill \end{array}$$namely,$$\begin{array}{}\text{(70)}& {\displaystyle \sum _{Q_l^{\alpha }}{\frac{{\displaystyle {\int }_{Q_l^{\alpha }}\mathbf{g}}t,x\text{d}\mu t,x}{l^n}}^{p^{\prime }}}{l}^n{w^{1-p^{\prime }}}_Q\mathrm{\lesssim }{\mathbf{g}}_{L_{\mu }^{q^{\prime }}{ℝ}_{+}^{1+n}}^{p^{\prime }}.\end{array}$$

For the convenience of notation, (70) can be rewritten as$$\begin{array}{}\text{(71)}& {\displaystyle \sum _{Q_l^{\alpha }}{{\rm Y}}_{Q_l^{\alpha }}}{\frac{{\displaystyle {\int }_{Q_l^{\alpha }}\mathbf{g}}\text{d}\mu }{\mu Q_l^{\alpha }}}^{p^{\prime }}{w^{1-p^{\prime }}}_Q\mathrm{\lesssim }{\mathbf{g}}_{L_{\mu }^{q^{\prime }}{ℝ}_{+}^{1+n}}^{p^{\prime }},\end{array}$$for which$$\begin{array}{}\text{(72)}& {{\rm Y}}_{Q_l^{\alpha }}={\frac{\mu Q_l^{\alpha }}{l^n}}^{p^{\prime }}{l}^n.\end{array}$$

Since the following dyadic Hardy–Littlewood maximal function$$\begin{array}{}\text{(73)}& M_{\mu }^{d}ht,x=\underset{t,x\in Q^{\alpha }}{\text{sup}}\frac{1}{\mu Q^{\alpha }}{\displaystyle {\int }_{Q^{\alpha }}hs,y}\text{d}\mu s,y,\forall Q^{\alpha }\in {\mathcal{D}}^{\alpha },\end{array}$$is bounded on $L_{\mu }^p{ℝ}_{+}^{1+n}$ for $1<p<\infty$ and$$\begin{array}{}\text{(74)}& {\frac{{\displaystyle {\int }_{Q_l^{\alpha }}\mathbf{g}}t,x\text{d}\mu t,x}{\mu Q_l^{\alpha }}}^{p^{\prime }}\underset{˜}{>}\frac{{\displaystyle {\int }_{Q}h}t,x\text{d}\mu t,x}{\mu Q_l^{\alpha }},\end{array}$$with the choice of $\mathbf{g}t,x={M_{\mu }^{d}h}^{1/p^{\prime }}t,x$ under $0\le h\in L_{\mu }^{q^{\prime }/p^{\prime }}{ℝ}_{+}^{1+n}$, we deduce that$$\begin{array}{}\text{(75)}& {\mathbf{g}}_{L_{\mu }^{q^{\prime }}{ℝ}_{+}^{1+n}}^{p^{\prime }}\mathrm{\lesssim }{h}_{L_{\mu }^{q^{\prime }/p^{\prime }}{ℝ}_{+}^{1+n}}.\end{array}$$

It follows that$$\begin{array}{}\text{(76)}& {\displaystyle \sum _{Q_l^{\alpha }}{{\rm Y}}_{Q_l^{\alpha }}}\frac{{\displaystyle {\int }_{Q_l^{\alpha }}h}t,x\text{d}\mu t,x}{\mu Q_l^{\alpha }}{w^{1-p^{\prime }}}_Q\mathrm{\lesssim }{h}_{L_{\mu }^{q^{\prime }/p^{\prime }}{ℝ}_{+}^{1+n}},\hspace{1em}\forall h\in L_{\mu }^{q^{\prime }/p^{\prime }}{ℝ}_{+}^{1+n}.\end{array}$$

By duality, we can get$$\begin{array}{}\text{(77)}& {\displaystyle \sum _{Q_l^{\alpha }}\frac{{\lambda }_{Q_l^{\alpha }}}{\mu Q_l^{\alpha }}}{1}_{Q_l^{\alpha }}{w^{1-p^{\prime }}}_Q\in L_{\mu }^{q^{\prime }/q^{\prime }-p^{\prime }}{ℝ}_{+}^{1+n},\end{array}$$that is, (67).

We proceed the proof by defining$$\begin{array}{}\text{(78)}& P_{\alpha p}^{d,\tau ,w}\mu t,x={\displaystyle \sum _{Q_l^{\alpha }\in {\mathcal{D}}_{\tau }^{\alpha }}{\frac{\mu Q_l^{\alpha }}{l^n}}^{p^{\prime }-1}}{w^{1-p^{\prime }}}_Q{1}_{Q_l^{\alpha }}t,x\&{\mathcal{D}}_{\tau }^{\alpha }={\mathcal{D}}^{\alpha }+\tau ={Q_l^{\alpha }+\tau }_{Q_l^{{\alpha }^{\prime }}\in {\mathcal{D}}^{\alpha }},\end{array}$$where $Q_l^{\alpha }+\tau =t,x+\tau :t,x\in Q_l^{\alpha }$ means the ${ℝ}_{+}^{1+n}\mathrm{\ni }\tau$-shift of $Q_l^{\alpha }$. Then, (67) implies$$\begin{array}{}\text{(79)}& \underset{\tau \in {ℝ}_{+}^{1+n}}{\text{sup}}{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{P_{\alpha p}^{d,\tau ,w}\mu t,x}^{qp-1/p-q}}\text{d}\mu t,x<\infty .\end{array}$$

The proof will be completed by showing that$$\begin{array}{}\text{(80)}& P_{\alpha p}^w\mu \in L_{\mu }^{qp-1/p-q}{ℝ}_{+}^{1+n}.\end{array}$$

Suppose first that $\mu$ is a doubling measure. Then, (80) is a consequence of (67) and the following observation:$$\begin{array}{}\text{(81)}& P_{\alpha p}^w\mu t,x\mathrm{\lesssim }{\displaystyle \sum _{Q_l^{\alpha }}{\frac{\mu Q_l^{\alpha \ast }}{l^n}}^{p^{\prime }-1}}{w^{1-p^{\prime }}}_Q{1}_{Q_l^{\alpha }}t,x,\end{array}$$where $Q_l^{\alpha \ast }$ is the cube with the same center as $Q_l^{\alpha }$ and side length two times as $Q_l^{\alpha }$. We are left with the task of determining when $\mu$ is a possibly nondoubling measure. For any $\delta >0$, we first write$$\begin{array}{}\text{(82)}& P_{\alpha p,\delta }^w\mu t,x={\displaystyle {\int }_0^{\delta }{\frac{\mu B_r^{\alpha }t,x}{r^n}}^{p^{\prime }-1}}\frac{{w^{1-p^{\prime }}}_{B_{r}x}\text{d}r}{r}.\end{array}$$

Next, we claim that$$\begin{array}{}\text{(83)}& P_{\alpha p,\delta }^w\mu t,x\mathrm{\lesssim }{\delta }^{-n+1}{\displaystyle {\int }_{\tau \mathrm{\lesssim }\delta }{P}_{\alpha p}^{d,\tau ,w}}\mu t,x\text{d}\tau .\end{array}$$

In fact, for fixed $t,x\in {ℝ}_{+}^{1+n}$ and $\delta >0$ such that $2^{i-1}\rho \le \delta <2^i\rho$ with $i\in \mathbb{Z}$ and $\rho >0$ will be determined later, we have$$\begin{array}{}\text{(84)}& P_{\alpha p,\delta }^w\mu t,x\mathrm{\lesssim }{\displaystyle \sum _{j=-\infty }^i{\frac{\mu B_{2^j\rho }^{\alpha }t,x}{{2^j\rho }^n}}^{p^{\prime }-1}}{w^{1-p^{\prime }}}_{B_{2^j\rho }x}.\end{array}$$

For $j\le i$, let $Q_{l,j}^{\alpha }$ be a cube centered at $x$ with $2^{j-1}<l\le 2^j$. Then, $B_{2^j\rho }^{\alpha }t,x\subseteq Q_{l,j}^{\alpha }$ for sufficiently small $\rho$. Let $E$ be the set of all points $\tau \in {ℝ}_{+}^{1+n}$ enjoying $\tau \mathrm{\lesssim }\delta$ with $E$ being the $1+n$-dimensional Lebesgue measure, but also, there exists $Q_l^{\alpha ,\tau }\in {\mathcal{D}}_{\tau }^{\alpha }$ satisfying $l=2^{j+1}$ and $Q_{l,j}^{\alpha }\subseteq Q_l^{\alpha ,\tau }$. A geometric consideration produces a dimensional constant $cn>0$ such that $E\ge cn{\delta }^{n+1}\hspace{1em}\forall j\le i$. Therefore,$$\begin{array}{c}\text{(85)}& \mu {B_{2^j\rho }^{\alpha }t,x}^{p^{\prime }-1}\mathrm{\lesssim }{E}^{-1}{\displaystyle {\int }_E{\displaystyle \sum _{l=2^{j+1}}\mu }}{Q_l^{\alpha ,\tau }}^{p^{\prime }-1}{1}_{Q_l^{\alpha ,\tau }}t,x\text{d}\tau \mathrm{\lesssim }{\delta }^{-n+1}{\displaystyle {\int }_{\tau \mathrm{\lesssim }\delta }{\displaystyle \sum _{l=2^{j+1}}\mu }}{Q_l^{\alpha ,\tau }}^{p^{\prime }-1}{1}_{Q_l^{\alpha ,\tau }}t,x\text{d}\tau ,\hfill \end{array}$$whence reaching$$\begin{array}{c}\text{(86)}& P_{\alpha p,\delta }^w\mu t,x\mathrm{\lesssim }{\delta }^{-n+1}{\displaystyle {\int }_{\tau \mathrm{\lesssim }\delta }{\displaystyle \sum _{j=-\infty }^i{\displaystyle \sum _{l=2^{j+1}}{\frac{\mu Q_l^{\alpha ,\tau }}{{2^j\rho }^n}}^{p^{\prime }-1}}}}{w^{1-p^{\prime }}}_{B_{2^j\rho }x}{1}_{Q_l^{\alpha ,\tau }}t,x\text{d}s\mathrm{\lesssim }{\delta }^{-n+1}{\displaystyle {\int }_{\tau \mathrm{\lesssim }\delta }{P}_{\alpha p}^{d,\tau ,w}}\mu t,x\text{d}\tau ,\hfill \end{array}$$which is our claim. By Hölder’s inequality and Fubini’s theorem, we give$$\begin{array}{c}\text{(87)}& {\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{P_{\alpha p,\delta }^w\mu t,x}^{qp-1/p-q}}\text{d}\mu t,x\mathrm{\lesssim }{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{{\delta }^{-n+1}{{\displaystyle {\int }_{\tau \le C\delta }{P_{\alpha p}^{d,\tau ,w}\mu }^{qp-1/p-q}}\text{d}\tau }^{p-q/qp-1}{{\displaystyle {\int }_{\tau \mathrm{\lesssim }\delta }\text{d}\tau }}^{1-p-q/qp-1}}^{qp-1/p-q}}\text{d}\mu \\ \mathrm{\lesssim }{\delta }^{-n+1}{\displaystyle {\int }_{\tau \mathrm{\lesssim }\delta }{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{P_{\alpha p}^{d,\tau ,w}\mu }^{qp-1/p-q}}\text{d}\mu }\text{d}\tau \\ <\infty ,\end{array}$$which produces$$\begin{array}{}\text{(88)}& P_{\alpha p}^w\mu \in L_{\mu }^{qp-1/p-q}{ℝ}_{+}^{1+n}\text{as\hspace{0.17em}}\delta ⟶\infty ,\end{array}$$by the monotone convergence theorem.

$\text{b}\Rightarrow \text{a}$: by duality and Lemma 2, we only need to show that$$\begin{array}{}\text{(89)}& {\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{P}_{\alpha p}^w}\mathbf{g}\text{d}\mu t,x\mathbf{g}t,x\text{d}\mu \mathrm{\lesssim }{\mathbf{g}}_{L_{\mu }^{q^{\prime }}{ℝ}_{+}^{1+n}}^{p^{\prime }},\hspace{1em}\forall \mathbf{g}\in L_{\mu }^{q^{\prime }}{ℝ}_{+}^{1+n}.\end{array}$$

Let$$\begin{array}{}\text{(90)}& M_{\mu }\mathbf{g}t,x=\underset{r>0}{\text{sup}}\frac{1}{\mu B_r^{\alpha }t,x}{\displaystyle {\int }_{B_r^{\alpha }t,x}\mathbf{g}}s,y\text{d}\mu s,y.\end{array}$$

Denote the centered Hardy–Littlewood maximal function of $\mathbf{g}$ with respect to $\mu$. We note that$$\begin{array}{c}\text{(91)}& P_{\alpha p}^w\mathbf{g}\text{d}\mu t,x\approx {\displaystyle {\int }_0^{\infty }{\frac{\mu B_r^{\alpha }t,x}{r^n}}^{p^{\prime }-1}}{\frac{{\displaystyle {\int }_{B_r^{\alpha }t,x}\mathbf{g}}t,x\text{d}\mu }{\mu B_r^{\alpha }t,x}}^{p^{\prime }-1}\frac{{w^{1-p^{\prime }}}_{B_{r}x}\text{d}r}{r}\mathrm{\lesssim }{M_{\mu }\mathbf{g}t,x}^{p^{\prime }-1}{P}_{\alpha p}^w\mu t,x.\hfill \end{array}$$

Then, the Hölder inequality yields$$\begin{array}{c}\text{(92)}& {\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{P}_{\alpha p}^w}\mathbf{g}\text{d}\mu t,x\text{d}\mu t,x\mathrm{\lesssim }{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{M_{\mu }\mathbf{g}t,x}^{p^{\prime }-1}}{P}_{\alpha p}^w\mu t,x\mathbf{g}t,x\text{d}\mu t,x\\ \mathrm{\lesssim }{{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{M_{\mu }\mathbf{g}t,x}^{q^{\prime }}}\text{d}\mu t,x}^{q^{\prime }/p^{\prime }-1}{{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{\mathbf{g}t,x{P}_{\alpha p}^w\mu t,x}^{q^{\prime }/q^{\prime }-p^{\prime }+1}}\text{d}\mu t,x}^{q^{\prime }-p^{\prime }+1/q^{\prime }}\\ \mathrm{\lesssim }{\mathbf{g}}_{L_{\mu }^{q^{\prime }}{ℝ}_{+}^{1+n}}^{p^{\prime }}{{\displaystyle {\int }_{{ℝ}_{+}^{1+n}}{P_{\alpha p}^w\mu }^{qp-1/p-q}}\text{d}\mu }^{p-q/qp-1}\\ \mathrm{\lesssim }{\mathbf{g}}_{L_{\mu }^{q^{\prime }}{ℝ}_{+}^{1+n}}^{p^{\prime }},\end{array}$$where we have used the boundedness of $M_{\mu }$ on $L_{\mu }^{q^{\prime }}{ℝ}_{+}^{1+n}$ (cf. [20]).

Acknowledgments

This work was supported by the NSF of China (Grant no. 11771195) and NSF of Shandong Province (Grant nos. ZR2019YQ04 and 2020KJI002).

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