Triebel-Lizorkin spaces with general weights

TL;DR

This paper introduces Triebel-Lizorkin spaces with general weights, providing their characterization, embeddings, and atomic decompositions, extending classical results to a broader weighted setting.

Contribution

It generalizes Triebel-Lizorkin spaces with smoothness to include arbitrary weights and develops their fundamental properties and decompositions.

Findings

Established $oldsymbol{ extit{ ext{φ}}}$-transform characterization.

Proved Sobolev embeddings for these spaces.

Developed atomic and molecular decompositions.

Abstract

In this paper, the author introduce Triebel-Lizorkin spaces with general smoothness. We present the $\varphi $-transform characterization of these spaces in the sense of Frazier and Jawerth and we prove their Sobolev embeddings. Also, we establish the smooth atomic and molecular decomposition of these function spaces. To do these we need a generalization of some maximal inequality to the case of general weights.