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

It is known, in harmonic analysis theory, that maximal operators measure local smoothness of L functions. +ese 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. +e purpose of this paper is to study the extent of smoothness on weighted function spaces under the condition Mα (f) ∈ Lp,μ, where μ is a lower doubling measure, M# α (f) stands for the sharp maximal function of f, and 0≤ α≤ 1 is the degree of smoothness.


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 n-dimensional Euclidean space R n needed for later sections.
(1) A cube on R n will always mean a cube with sides parallel to the axes and has nonempty interior. For j ∈ Z and k ∈ Z n , we denote by Q jk the dyadic cube 2 − j ([0, 1] n + k), where l(Q jk ) � 2 − j is its side length, x Q jk � 2 − j k is its lower "left-corner," and c Q jk is its center. We set Q � Q jk : j ∈ Z, k ∈ Z n } and j Q � − log 2 l(Q) for all Q ∈ Q. When the dyadic cube Q appears as an index, such as Q∈Q , it is understood that Q runs over all dyadic cubes in R n . For a function ] and dyadic cube Q � Q jk , set ] Q (x) � |Q| − (1/2) ] 2 j x − k � |Q| (1/2) for all x ∈ R n , where ] j (x) � 2 nj ](2 j x).
(2) roughout the paper, w denotes a weight function, i.e., w is an almost every (a.e.) positive locally integrable function on R n . A function f ∈ L p (w), 0 < p < ∞⟺ and f belongs to the weak-L p spaces, denoted by If w � 1, we do not write the subscription w.
A weight function w is said to be in the Muckenhoupt classes A p , where 1 ≤ p < ∞, if there exists a constant C p > 0 such that for every cube Q, for a.e. x ∈ Q, or equivalently Mw(x) ≤ C 1 w(x) for a.e.
x ∈ R n , where M is the Hardy-Littlewood maximal operator. e class A p was introduced by Muckenhoupt [16] in order to characterize the boundedness of the Hardy-Littlewood maximal operator M on the weighted Lebesgue spaces [8,12]. e pioneering work of Muckenhoupt [16] showed that ⟺w ∈ A p when 1 < p < ∞ and A weight function w is in Muckenhoupt's class A p (R n ), 1 ≤ p < ∞, of weights if there exists a constant C p > 0 such that for all cubes Q in R n , for a.e. x ∈ B, or equivalently Mw(x) ≤ C 1 w(x) for a.e.
x ∈ R n , where M is the Hardy-Littlewood maximal operator.
(3) Note that if w ∈ A p , then w is a doubling measure, i.e., there exists a constant C ≥ 1 such that for all x and all r > 0, 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 BMO(w), i.e., φ ∈ BMO(w), if there is a constant C: where φ Q � (1/w(Q)) Q φ(y)w(y)dy is the average of f on Q with respect to dw. e smallest constant C for which (11) is satisfied is taken to be the norm of φ in the space BMO(w) and is denoted by ‖φ‖ BMO(w) .
where Q is taken over all cubes in R n . Let α ≥ 0. e sharp fractional maximal function M # α (f) of f is defined by (5) e space of Schwartz functions: let S(R n ) be the space of all Schwartz functions on R n with the classical topology generated by the family of seminorms: e topological dual space S ′ (R n ) of S(R n ) is the set of all continuous linear functional the space S(R n ) is endowed with the weak * -topology. We denote by S ∞ (R n ) the topological subspace of functions in S(R n ) having all vanishing moments: S ∞ ′ (R n ) denotes the topological dual space of S ∞ (R n ), namely, the set of all continuous linear functional on S ∞ ′ (R n ). e space S ∞ ′ (R n ) is also endowed with the weak * -topology. It is well known that S ∞ ′ (R n ) � (S ′ (R n )/ P(R n )) as topological spaces, where P(R n )denotes the set of all polynomials on R n ; see, for example, ( [21], Proposition 8.1). Similarly, for any R ∈ N, the space S R (R n ) is defined to be the set of all Schwartz functions having vanishing moments of order R and S R ′ (R n ) is its topological dual space. We write S − 1 (R n ) � S(R n ). e Fourier transform, F] � ], of Schwartz function ] is defined by e convolution of two functions ], μ ∈ S(R n ) is defined by (17) and still belongs to S(R n ). e convolution operator can be extended to It makes sense pointwise and is a C ∞ function on R n of at most polynomial growth.
To simplify notation, we write often ]f � ] * f. In some other situations, to avoid confusion, we keep the notation ] * f. As usual, ] t denotes the function defined by ] t (x) � t − n ](x/t).
(6) In the rest of this paper, C expresses unspecified positive constant, possibly different at each occurrence; the symbol A ≤ B means that A ≤ CB. If A ≤ B and B ≤ A, then we write A≃B. e Greek letter χ S denotes the characteristics function of a sphere S, where S is a measurable subset of R n and |S| represents its Lebesgue measure; p ′ and s ′ always denote the conjugate index of any p > 1 and s > 1, that is, 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. ese 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 M # α f ∈ L p,μ , where μ is a lower doubling measure, M # α f stands for the sharp maximal function of f, and 0 ≤ α ≤ 1 is the degree of smoothness. When α � 0, M # 0 f � M # f is the classical sharp maximal function. It is well known that the Hardy-Littlewood maximal function Mf is controlled by the sharp maximal function M # f via the celebrated Stein-Fefferman inequality: ‖Mf‖ p ≤ ‖M # f‖ p and in the case of α � 1, it is shown that ‖M # 1 f‖ p : ‖f‖ H p for some range of p. 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, ‖f‖ _ stands for the fractional Sobolev space. Now, we are ready to present the main theorem of this section. Theorem 1. Let α and c be real numbers satisfying 0 ≤ α ≤ 1 and c < α, and w is the lower d− regular doubling measure.
Proof. If f is not a constant function, then there exists a ball Hence, for each 0 < p * < q ≤ ∞.

Preliminaries
In this section, we introduce some necessary and important definitions, notations, lemmas, and results.
Let μ be a doubling measure and 0 < p, q ≤ ∞ and c ∈ R; the homogeneous Triebel-Lizorkin space _ F c,q p,v is the set of all distributions f (modulo polynomials) such that Journal of Mathematics with the interpretation that when q � ∞, e supremum is taken over all dyadic cubes Q, and l(Q) denotes the length of sides of the cube Q.
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][3][4]11]). roughout 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 dμ � wdx and w ∈ A ∞ : and h p,w is the local weighted Hardy space of f ∈ S ′ for which where μ is a fixed function in S with R n μ(x)dx ≠ 0. By the fundamental work of Fefferman and Stein [9] adapted to the weighted case, H p,w or h p,w does not depend on the choices of μ. In particular, In particular, when the exponent is a natural number, say c � N ∈ N, then the weighted Bessel potential space can be identified with the classical Sobolev space: All the above identities have to be understood in the sense of equivalent quasi-norms.

Definition 2. We say that a doubling measure μ is lower
holds for all ball B(x, t) ⊂ R n .

Lemma 1. Let w ∈ A p and d− regular. en, we have
Proof. Let Q be a cube and x, y ∈ Q. en, Integrating over the cube Q with respect to y, we get If w ∈ A 1 , then we have for almost all x ∈ Q, On the other hand, if w ∈ A p , p > 1; then using (37) and Hölder's inequality, we obtain 4 Journal of Mathematics erefore, we conclude that is completes the proof.

Lemma 2.
We say that f is in the fractional Sobolev space (42) Corollary 2. Let w ∈ A p and lower d− regular, 0 < α < 1, 1 ≤ p < ∞, and f ∈ _ W α,p (w). en, One can immediately obtain the following corollary.
Lemma 3. Let f ∈ W 1,1 loc (R n ) and 0 ≤ α ≤ 1. en, for every Proof. e proof is an immediate consequence of the wellknown Poincaré inequality.
For all ball B(x, R) and all f ∈ W 1,1 loc (R n ), there is a constant C(n) such that holds.

Corollary 4. Let f be a locally integrable function such that
|∇f| ∈ H r (w) and 0 < p * ≤ ∞ are determined by Proof. Let P t (x) � (c n t/(t 2 + |x| 2 ) (n+1)/2 ) be the Poisson kernel with the constant C(n) such that R n P t (x)dx � 1. en, there exists a constant C � C(n) such that Mf(x) ≤ Csup t>0 P t * |f|. In fact, if Q is a cube with diam (Q) � t and x ∈ Q, then we have us, we have Journal of Mathematics 5 Using Lemma 3 with α � 1 and Proposition 2 (see below), we obtain □ Remark 3. If we take c � 0, q � 2, r > 1, and w � 1, in Corollary 4, we obtain the classical Sobolev-Gagliardo-Nirenberg inequality:

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 c < 1, λ > 0, and 0 < q ≤ 1, there exist Schwartz functions v and μ on R n such that Assume that w(B(x, t)) ≥ C w t d for each x ∈ R n and each t > 0, and let v ∈ S supported on B(0, 1) such that Fix a large λ > 0, and define Proof. We adapt here the proof given in [7] in the unweighted case. Use the well-known estimate where χ denotes the characteristic function of the interval [0, 1], to obtain, for any λ > 0, By taking any z ∈ B(x, 2 k s) and using the fact that ] is supported in the unit ball and has mean equal zero, we obtain which holds. Hence, If we choose λ large enough, we obtain On the other hand, by (61), we have for any fixed x ∈ B(z, s), Rising (65) to the pth power and integrating over the ball B(z, s) with respect to w(x)dx, one has that By using (60), we obtain □ Proof. Proof of eorem 1.

□
Proof. We consider only the case when 0 < q ≤ 1. In the case when 1 < q ≤ ∞, estimate (18) follows from the case q � 1 by the embedding Let k > 0 be chosen later and let μ and ] be as in Lemma 4. Assume 0 < ((α − c < d)/p) and 0 < q ≤ 1. en, using (58), we get 6 Journal of Mathematics (68) where p * is given by □

Some Extensions
In this section, we will assume that μ is a nonnegative Borel doubling measure on R n ; there exists β � β(μ) > 0 such that for all ball B r . e smallest such β is called a doubling constant of μ.