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: . More properties of that capacity are explored, and applications to a trace inequality for the weak solution of the equation are considered.

1. Introduction

Let , , and denote the fractional power of the spatial Laplacian which can be defined byfor which is the Fourier transform and is its inverse. Many research studies have been carried out on the following fractional heat equation with initial data:

The weak solution of (2) can be written aswhere is the fractional heat kernel.

It is well known that and are the heat kernel and Poisson kernel, respectively. Although it is hard to obtain an explicit formula for when (cf. [1–5]), we are interested to find the following useful estimates (cf. [6, 7]):

To study the traces of , Chang and Xiao [8] introduced the following capacity for a compact set :where stands for the characteristic function of . Using , they characterized a nonnegative Radon measure on such that the mapping is continuous for . Based on the theory of the Hedberg–Wolff potential, a noncapacitary characterization of the extension 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 on and weight (the Muckenhoupt class, cf. [11, 12]), denoted by , if

Here and after, is the classical Euclidean ball centered at with radius for any , and denotes the average of the function on . In the literature, weights can also be given by orfor which are constants independent of the sets and

Let stand for with . A weight is said to be a doubling weight if there is a constant such that . It is well known that weights are doubling weights. The weights we considered in this paper are either the Muckenhoupt classes or weights satisfyingwhich is a larger class than the 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 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 and with and . For each natural number , stands for the space of all functions being times continuously differentiable in , and denotes the space of all functions in having compact support. is the subspace of given by . denotes the class of all nonnegative Radon measures supported by the set . is the usual Lebesgue space, and is the Lebesgue space with measure . The letter is used to express a positive constant which is independent of the main parameters, but may vary from line to line. The symbol means ; moreover, whenever and .

2. A Weighted Capacity

Let be a weight function. The weighted capacity associated with (2) can be defined as

and , the parabolic ball, with center at and radius , is denoted by

It is obvious that its volume . The following theorem gives the estimates for .

Theorem 1. Let . Then,

The proof of Theorem 1 needs a potential theory and will be given in the beginning of Section 3. If , then by Theorem 1, which reduces to the well-known estimate for the unweighted capacity of (cf. Proposition 3 of [8]).

Firstly, we give some fundamental properties of .

Proposition 1. As a nonnegative set function, the mapping enjoys some essential properties as follows:(a).(b)(c)If is a compact sequence, then .(d)For any two compact sets of , one has(e)

Proof. Write– follow immediately from the special case of Proposition 4.2 of [10] ( and ).
For , setandThen, , , andThis clearly forcesIt follows thatHence,To prove , we first note that, for any , there exists a function satisfyingIf is big enough, then the compact set . We conclude from thatConsequently,Next, we proceed to show the second statement. We first conclude from thatTo prove the reverse inequality, there is no loss of generality in assuming . The definition of yields the existence of an open set withSince , a slight modification of the telescoping inequality (Lemma 23 of [14]) (see also Lemma 2.5 of [15]) yields the following inequality:Let be a compact subset of . Then, there exists a natural number such that . Therefore,This along with giveswhich yields the desired result

Secondly, we prove that the infimum for the norm in the definition of can be achieved in . We call this function the capacitary potential.

Proposition 2. Let . Then, there is a unique function such that

Proof. Assume that such that . Since is a closed, convex, and nonempty subset of the reflexive Banach space , by possibly passing to a subsequence, we may assume thatMoreover, we have .
The uniform convexity of the space implies that is unique.

3. Application to a Trace Inequality

As an application of on a given compact set , this section is devoted to some characterizations for the trace inequality of the weak solution to (2). In principle, we investigate nonnegative Radon measures and weight functions such that the mapping is continuous. That is, the following trace inequalityholds for any . To do so, first we need some preliminaries of the potential theory.

Let satisfy (10) and . Then, the energy of with respect to is given by

By a direct calculation, we have that can be rewritten aswhere is called the weighted nonlinear potential of the measure with respect to the weight . When , these potentials have been studied extensively (cf. [7–9]).

Let and be its fractional parabolic maximal function. Then, the first result of this section is about the boundedness of .

Lemma 1. Assume that . Then,

Proof. Sincewe have Hence,
For the converse inequality, we first note that a slight modification of (1.2) of [16] yields that there exist two constants and such that, for any and , one has the following weighted good- inequality:Next, we proceed the proof following Theorem 3.6.1 of [17]. For any , multiplying the above inequality by and integrating in , one hasBy changing of variables, we getWe thus obtainBy letting small enough such that and , therefore,as desired.

Usually, it is difficult to get the estimate for 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 inspired by the idea due to Hedberg and Wolff [18] for the Riesz potential:

Lemma 2. Let satisfy (10). Then,

Proof. The forthcoming demonstration is a modification of Theorem 3.2 of [13]. Sincewe haveWe conclude from Lemma 1 thatSubstituting the following estimateinto (2), one hasTo prove the reverse, we first note thatThe proof is completed by the fact that

The following capacitary inequality for is a straightforward adaption of Proposition 4.1 of [13].

Lemma 3. If satisfies (10), then for any , one has the following inequality:

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

Lemma 4. Let be a compact set in , , and satisfy (10). Then,

Proof. Due to Proposition 4.1 of [10] and the proof of Lemma 2, there is such that on . 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. Sinceapplying Lemma 4 to on ball in which constant will be determined later, and we can obtain

Choosing such that the right side of the above estimate is equal to 1, then on . It follows from Lemma 4 and the doubling property of weights that

If , then for the above , there is such that

Then, a further application of Lemma 4 gives

3.1. Trace Inequality for

In this section, we characterize (33) under the lower case by . In Theorem 5.3 of [10], the authors obtained the following result.

Theorem 2. Let , , and satisfy (10). Then, the following two statements are equivalent:(a)(33) is valid for any nonnegative (b), for which

In particular, if (i.e., ), we have the following deep understanding of (33).

Theorem 3. Assume that , , and satisfies (10). Then, the following four statements are equivalent:(a)(33) holds for all nonnegative (b) for all compact sets (c) for all (d) for all , where

Proof. follows directly from the proof of Theorem 2.
is trivial.
is a consequence of Theorem 1.
: let be defined as in the proof of Theorem 2 and be -restricted to . Following Lemma 2, we can obtain thatFor suitable to be determined later, can be decomposed asUse to derive thatandChooseThen, a further application of and Lemma 2 giveswhich implies (33).

3.2. Trace Inequality for

This section focuses on a further trace inequality under the upper case . The main result can be formulated as follows.

Theorem 4. Let and , and be as in Theorem 2. Then, the following two statements are equivalent:(a)(33) holds for all nonnegative (b)

Proof. The proof can be seen as a weighted fractional heat potential version of the Riesz potential treatment carried out in [19].
To show , we first denote by the -dyadic cube with side length and corners in the set with , namely,as and for . Next, we introduce the following weighted fractional heat Hedberg–Wolff potential generated by —the family of all the above-defined -dyadic cubes in :and we need to proveSince is equivalent to the inequalitywe notice that Lemma 2 is also true with in place of and in place of , and it may be concluded thatnamely,For the convenience of notation, (70) can be rewritten asfor whichSince the following dyadic Hardy–Littlewood maximal functionis bounded on for andwith the choice of under , we deduce thatIt follows thatBy duality, we can getthat is, (67).
We proceed the proof by definingwhere means the -shift of . Then, (67) impliesThe proof will be completed by showing thatSuppose first that is a doubling measure. Then, (80) is a consequence of (67) and the following observation:where is the cube with the same center as and side length two times as . We are left with the task of determining when is a possibly nondoubling measure. For any , we first writeNext, we claim thatIn fact, for fixed and such that with and will be determined later, we haveFor , let be a cube centered at with . Then, for sufficiently small . Let be the set of all points enjoying with being the -dimensional Lebesgue measure, but also, there exists satisfying and . A geometric consideration produces a dimensional constant such that . Therefore,whence reachingwhich is our claim. By Hölder’s inequality and Fubini’s theorem, we givewhich producesby the monotone convergence theorem.
: by duality and Lemma 2, we only need to show thatLetDenote the centered Hardy–Littlewood maximal function of with respect to . We note thatThen, the Hölder inequality yieldswhere we have used the boundedness of on (cf. [20]).

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

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