L p 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 $L^p$ 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 $M_{\alpha }^{\#}\left(f\right)\in L^{p,\mu }$ , where $\mu$ is a lower doubling measure, $M_{\alpha }^{\#}\left(f\right)$ stands for the sharp maximal function of $f$ , and $0\le \alpha \le 1$ is the degree of smoothness.