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

Ferit Gürbüz,Ahmed Loulit

doi:10.1155/2021/8104815

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

Ferit Gürbüz, Ahmed Loulit

Open Access

https://doi.org/10.1155/2021/8104815

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Abstract

It is known, in harmonic analysis theory, that maximal operators measure local smoothness of L p 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 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.

Highlights

Introduction and Main ResultIn this paper, we consider the some continuous embeddings on weighted Besov–Triebel–Lizorkin spaces via a general sharp maximal function introduced by Calderon and Scott [6]
We consider the some continuous embeddings on weighted Besov–Triebel–Lizorkin spaces via a general sharp maximal function introduced by Calderon and Scott [6]
We investigate the spaces introduced by Hajłasz [13] that are defined via pointwise inequalities and their connection with the Triebel–Lizorkin spaces

Summary

Introduction and Main Result

We consider the some continuous embeddings on weighted Besov–Triebel–Lizorkin spaces via a general sharp maximal function introduced by Calderon and Scott [6]. A weight function w is said to be in the Muckenhoupt classes Ap, where 1 ≤ p < ∞, if there exists a constant Cp > 0 such that for every cube Q,. E class Ap 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]. We study the extent of smoothness on weighted function spaces under the condition M#α 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. En, applying eorem 1, we obtain (22). pletes the proof

Preliminaries

Some Useful Lemmas

Some Extensions