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Duality of Triebel-Lizorkin spaces of general weights

Douadi Drihem

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.

Duality of Triebel-Lizorkin spaces of general weights

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 -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.
Paper Structure (7 sections, 20 theorems, 154 equations)

This paper contains 7 sections, 20 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Background tools
  3. Notation and conventions
  4. Muckenhoupt weights
  5. The weight class $\dot{X}_{\alpha ,\sigma ,p}$
  6. Function spaces
  7. Duality

Key Result

Lemma 2.5

Let $1\leqslant p<\infty$. $\mathrm{(i)}$ Let $1<p<\infty$. $\gamma \in A_{p}(\mathbb{R}^{n})$ if and only if $\gamma ^{1-p^{\prime }}\in A_{p^{\prime }}(\mathbb{R}^{n})$. $\mathrm{(ii)}$ Let $\gamma \in A_{p}(\mathbb{R}^{n})$. There is $C>0$ such that for any cube $Q$ and a measurable subset $E\sub $\mathrm{(iii)}$ Let $1\leqslant p<\infty$ and $\gamma \in A_{p}(\mathbb{R}^{n})$. Then there exist

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.3
  • Lemma 2.5
  • Definition 2.6
  • Remark 2.9
  • Example 2.10
  • Remark 2.11
  • Lemma 2.12
  • Remark 2.14
  • Lemma 2.15
  • ...and 34 more