Duality of Triebel-Lizorkin spaces of general weights

TL;DR

This paper characterizes the dual spaces of generalized Triebel-Lizorkin function spaces, including weighted cases, using sequence space analysis, $ ext{ϕ}$-transform, and a weighted vector-valued maximal inequality.

Contribution

It provides a duality characterization for Triebel-Lizorkin spaces with generalized smoothness, extending existing theory to weighted and more general settings.

Findings

Duals of Triebel-Lizorkin spaces identified

Extension to weighted Triebel-Lizorkin spaces achieved

New weighted vector-valued maximal inequality established

Abstract

In this paper, we identify the duals of Triebel-Lizorkin spaces of generalized smoothness. In some particular cases these function spaces are just weighted Triebel-Lizorkin spaces. To do these, we will be working at the level of sequence spaces. The $\varphi $-transform characterization of these function spaces in the sense of Frazier and Jawerth, and new weighted version of vector-valued maximal inequality of Fefferman and Stein are the main tools.