Smoothness on Weighted Besov–Triebel–Lizorkin Spaces in terms of Sharp Maximal Functions

Abstract

It is known, in harmonic analysis theory, that maximal operators measure local smoothness of functions. These operators are used to study many important problems of function theory such as the embedding theorems of Sobolev type and description of Sobolev space in terms of the metric and measure. We study the Sobolev-type embedding results on weighted Besov–Triebel–Lizorkin spaces via the sharp maximal functions. The purpose of this paper is to study the extent of smoothness on weighted function spaces under the condition , where is a lower doubling measure, stands for the sharp maximal function of , and is the degree of smoothness.

1. Introduction and Main Result

In this paper, we consider the some continuous embeddings on weighted Besov–Triebel–Lizorkin spaces via a general sharp maximal function introduced by Calderón and Scott [6]. Furthermore, we investigate the spaces introduced by Hajłasz [13] that are defined via pointwise inequalities and their connection with the Triebel–Lizorkin spaces. For more details, see [11, 12].

Now, let us begin by recalling some definitions and classical results in harmonic analysis on the -dimensional Euclidean space needed for later sections.

(1) A cube on will always mean a cube with sides parallel to the axes and has nonempty interior. For and , we denote by the dyadic cube , where is its side length, is its lower “left-corner,” and is its center. We set and for all . When the dyadic cube appears as an index, such as , it is understood that runs over all dyadic cubes in . For a function and dyadic cube , setfor all , where .

(2) Throughout the paper, denotes a weight function, i.e., is an almost every (a.e.) positive locally integrable function on . A function and belongs to the weak- spaces, denoted by

If , we do not write the subscription .

A weight function is said to be in the Muckenhoupt classes if there exists a constant such that for every cube ,When ; for ,for a.e. , or equivalently for a.e. , where is the Hardy–Littlewood maximal operator.

The class was introduced by Muckenhoupt [16] in order to characterize the boundedness of the Hardy–Littlewood maximal operator on the weighted Lebesgue spaces [8, 12]. The pioneering work of Muckenhoupt [16] showed that when and

A weight function is in Muckenhoupt’s class , , of weights if there exists a constant such that for all cubes in ,When ; well, for ,for a.e. , or equivalently for a.e. , where is the Hardy–Littlewood maximal operator.

(3) Note that if , then is a doubling measure, i.e., there exists a constant such that for all and all ,

Another class of functions that plays an important role in harmonic analysis and in partial differential equation theory is the class of functions with bounded mean oscillation denoted by , i.e., , if there is a constant :where is the average of on with respect to . The smallest constant for which (11) is satisfied is taken to be the norm of in the space and is denoted by .

(4) The sharp maximal function of is defined bywhere is taken over all cubes in . Let . The sharp fractional maximal function of is defined by

(5) The space of Schwartz functions: let be the space of all Schwartz functions on with the classical topology generated by the family of seminorms:

The topological dual space of is the set of all continuous linear functional the space is endowed with the weak -topology. We denote by the topological subspace of functions in having all vanishing moments:

denotes the topological dual space of , namely, the set of all continuous linear functional on . The space is also endowed with the weak -topology. It is well known that as topological spaces, where denotes the set of all polynomials on ; see, for example, ([21], Proposition 8.1). Similarly, for any , the space is defined to be the set of all Schwartz functions having vanishing moments of order and is its topological dual space. We write .

The Fourier transform, , of Schwartz function is defined by

The convolution of two functions is defined byand still belongs to .

The convolution operator can be extended to via . It makes sense pointwise and is a function on of at most polynomial growth.

To simplify notation, we write often . In some other situations, to avoid confusion, we keep the notation . As usual, denotes the function defined by .

(6) In the rest of this paper, expresses unspecified positive constant, possibly different at each occurrence; the symbol means that . If and , then we write . The Greek letter denotes the characteristics function of a sphere , where is a measurable subset of and represents its Lebesgue measure; and always denote the conjugate index of any and , that is, and .

Function spaces play a crucial role in the genesis of functional analysis and are widely used in the development of the modern analysis of partial differential equations. For instance, the classical Besov–Triebel–Lizorkin spaces are a class of function spaces containing many well-known classical function spaces and are more suitable in the treatment of a large type of partial differential equations (see for instance [5, 10]). A comprehensive treatment of these function spaces and their history can be found in Triebel’s monographs [18, 19] and in the fundamental paper of Frazier and Jawerth [11].

In recent years, there has been increasing interest in a new family of function spaces, called new class of Besov–Triebel–Lizorkin spaces. These spaces unify and generalize many classical spaces including Besov spaces, Morrey spaces, and Triebel–Lizorkin spaces (see for instance [20]).

In this paper, we study the extent of smoothness on weighted function spaces under the condition , where is a lower doubling measure, stands for the sharp maximal function of , and is the degree of smoothness. When , is the classical sharp maximal function. It is well known that the Hardy–Littlewood maximal function is controlled by the sharp maximal function via the celebrated Stein–Fefferman inequality: and in the case of , it is shown that for some range of . As a result, we extend the above results to the some general weighted spaces. Embedding results on weighted Besov–Triebel–Lizorkin spaces are obtained. Namely, (Theorem 1). As a consequence, we obtain , where stands for the fractional Sobolev space.

Now, we are ready to present the main theorem of this section.

Theorem 1. Let be real numbers satisfying and , and is the lower regular doubling measure. Suppose that for every and for . Then, for each ,

Remark 1. The condition that for every is necessary. On the other hand, under this condition, only if .

Proof. If is not a constant function, then there exists a ball such thatTherefore, for all ,Hence,if .

Corollary 1. Under the same conditions in Theorem 1, we have, for each ,for each .

Proof. By Minkowski’s inequality, we haveThen, applying Theorem 1, we obtain (22). This completes the proof.

2. Preliminaries

In this section, we introduce some necessary and important definitions, notations, lemmas, and results.

Definition 1. Let be in the Schwartz space with supp contained in an annulus about the origin andLet be a doubling measure and and ; the homogeneous Triebel–Lizorkin space is the set of all distributions (modulo polynomials) such thatwith the interpretation that when ,The homogeneous Besov–Lipschitz space is the set of all distributions (modulo polynomials) such thatThe supremum is taken over all dyadic cubes , and denotes the length of sides of the cube .

Moreover, it is well known that the Besov–Lipschitz spaces and the Triebel–Lizorkin spaces are independent of the choices of (see, for example [2–4, 11]). Throughout this paper, will be taken as in Definition 1. It is well known that many classical smoothness spaces are covered by the Besov and Triebel–Lizorkin spaces. We recall some examples in the case when and :(1).(2), where denotes the weighted Hardy spaces of for which and is the local weighted Hardy space of for which where is a fixed function in with . By the fundamental work of Fefferman and Stein [9] adapted to the weighted case, or does not depend on the choices of . In particular,(3), where denotes the weighted Bessel potential space defined by In particular, when the exponent is a natural number, say , then the weighted Bessel potential space can be identified with the classical Sobolev space:(4).

All the above identities have to be understood in the sense of equivalent quasi-norms.

Definition 2. We say that a doubling measure is lower regular, where , if there is some constant such thatholds for all ball .

Remark 2. An example of measure lower regular is ,whereIn fact, if and ; hence, is doubling. Moreover, if and , then satisfies for all and all .

Lemma 1. Let and regular. Then, we have

Proof. Let be a cube and . Then,Integrating over the cube with respect to , we getIf , then we have for almost all , . Hence,The last inequality implies thatOn the other hand, if , ; then using (37) and Hölder’s inequality, we obtainTherefore, we conclude thatThis completes the proof.

Lemma 2. We say that is in the fractional Sobolev space , if

Corollary 2. Let and lower regular, , , and . Then,

One can immediately obtain the following corollary.

Corollary 3. Let be real numbers satisfying and . Assume and with . Then, for each ,where is given by .

Recall that for and ; we have (see [17])

In particular,with , and is as before.

Lemma 3. Let and . Then, for every , there is a constant such that

Proof. The proof is an immediate consequence of the well-known Poincaré inequality.
For all ball and all , there is a constant such thatholds.

Corollary 4. Let be a locally integrable function such that and are determined byThen, is in . Moreover, we have

Proof. Let be the Poisson kernel with the constant such that . Then, there exists a constant such that . In fact, if is a cube with diam and , then we haveThus, we haveUsing Lemma 3 with and Proposition 2 (see below), we obtain

Remark 3. If we take , , , and , in Corollary 4, we obtain the classical Sobolev–Gagliardo–Nirenberg inequality:with

3. Some Useful Lemmas

We start this section with some useful lemmas that will be helpful in proving our main result.

Lemma 4 (see [7]). Provided , , and , there exist Schwartz functions and on such that

(1)supp and (2)supp and on (3)

Lemma 5. Assume that for each and each , and let supported on such thatFix a large , and defineThen,

Proof. We adapt here the proof given in [7] in the unweighted case. Use the well-known estimatewhere denotes the characteristic function of the interval , to obtain, for any ,By taking any and using the fact that is supported in the unit ball and has mean equal zero, we obtainwhich holds. Hence,If we choose large enough, we obtainOn the other hand, by (61), we have for any fixed ,Rising (65) to the pth power and integrating over the ball with respect to , one has thatBy using (60), we obtain

Proof. Proof of Theorem 1.

Proof. We consider only the case when . In the case when , estimate (18) follows from the case by the embeddingLet be chosen later and let and be as in Lemma 4. Assume . Then, using (58), we getChoose to deduce thatwhere is given by . Thus, we have

4. Some Extensions

In this section, we will assume that is a nonnegative Borel doubling measure on ; there exists such thatfor all ball . The smallest such is called a doubling constant of .

For each , and , we set

Definition 3. Let , , and . The homogeneous grand Tribel–Lizorkin space is the set of all tempered functions such that when ,and when ,

Proposition 1. Let , , and , and is the doubling measure with a constant equal to . Set . If with , , and , then

Proof. Arguing as in the proof in ([15], Theorem 1.2) and using the almost-diagonality theorem (see [1], Theorem 4.2), we obtain the desired result.

Proposition 2. Let and be real numbers satisfying and and be a lower regular doubling measure. Assume is a smooth function and with . Then, for each ,(1) where is given by (2)For all and ,(3)For all , ,

Proof. We have from (58) that if is a lower -regular measure, thenArguing as in the proof of Proposition 1, we obtain the desired result easily.

Definition 4. Let be a doubling measure and . The homogeneous fractional Hajłasz–Sobolev space is the set of all measurable functions for which there exists a nonnegative function such thatfor .

is equipped with the seminormwhere denotes the class of all nonnegative Borel functions satisfying (80). Thus, Lemma 4.1 in [15] implies the following Sobolev embedding.

Lemma 6. Let , , and be given by . Then, for all , , , and ,

Remark 4. Lemma 6 is due to Hajłasz ([13], Theorem 8.7) when .

Corollary 5. Let , and be real numbers satisfying , , and . Assume . Then, for each ,where .

Proof. Fix a ball . Then, using Lemma 6 and by taking and Hölder’s inequality, we obtainHence,holds, where is the maximal function with respect to the measure . The -boundedness of when and Proposition 2 lead to estimate (83).

Also, recall that for in [15] and for in [14]. Here, denotes, for , the homogeneous Hardy–Sobolev space, i.e., the space of tempered distributions on , such that for each and

Consequently, if with , then

In particular, we have, for , the following well-known result:

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.