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The Weighted Grand Herz-Morrey-Lizorkin-Triebel Spaces with Variable Exponents

Shengrong Wang, Pengfei Guo, Jingshi Xu

TL;DR

This work extends harmonic analysis in variable-exponent and weighted settings by establishing a boundedness theory for vector-valued sublinear operators on weighted grand Herz–Morrey spaces with variable exponents, under a size condition and a vector-valued bound on $L^{q(\cdot)}(w)$. It proves that $\| (\sum_j |T f_j|^r)^{1/r} \|_{M\dot{K}_{q(\cdot),\lambda}^{\alpha(\cdot),p,\theta}(w)} \le C \| (\sum_j |f_j|^r)^{1/r} \|_{M\dot{K}_{q(\cdot),\lambda}^{\alpha(\cdot),p,\theta}(w)}$, and extends these ideas to derive maximal-function based characterizations via corollaries for the maximal operator. Furthermore, it introduces weighted grand Herz–Morrey–Triebel–Lizorkin spaces with variable exponents and proves equivalent quasi-norms using Peetre maximal operators and Calderón reproducing formulas, providing robust norm equivalences across resolutions of unity. These results broaden the toolbox for analysis in nonstandard growth and weighted settings, with potential applications to PDEs and interpolation theory in variable exponent frameworks.

Abstract

Let a vector-valued sublinear operator satisfy the size condition and be bounded on weighted Lebesgue spaces with variable exponent. Then we obtain its boundedness on weighted grand Herz-Morrey spaces with variable exponents. Next we introduce weighted grand Herz-Morrey-Triebel-Lizorkin spaces with variable exponents and provide their equivalent quasi-norms via maximal functions.

The Weighted Grand Herz-Morrey-Lizorkin-Triebel Spaces with Variable Exponents

TL;DR

This work extends harmonic analysis in variable-exponent and weighted settings by establishing a boundedness theory for vector-valued sublinear operators on weighted grand Herz–Morrey spaces with variable exponents, under a size condition and a vector-valued bound on . It proves that , and extends these ideas to derive maximal-function based characterizations via corollaries for the maximal operator. Furthermore, it introduces weighted grand Herz–Morrey–Triebel–Lizorkin spaces with variable exponents and proves equivalent quasi-norms using Peetre maximal operators and Calderón reproducing formulas, providing robust norm equivalences across resolutions of unity. These results broaden the toolbox for analysis in nonstandard growth and weighted settings, with potential applications to PDEs and interpolation theory in variable exponent frameworks.

Abstract

Let a vector-valued sublinear operator satisfy the size condition and be bounded on weighted Lebesgue spaces with variable exponent. Then we obtain its boundedness on weighted grand Herz-Morrey spaces with variable exponents. Next we introduce weighted grand Herz-Morrey-Triebel-Lizorkin spaces with variable exponents and provide their equivalent quasi-norms via maximal functions.

Paper Structure

This paper contains 4 sections, 13 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Boundedness of the vector-valued sublinear operator on weighted grand Herz-Morrey spaces
  4. Characterizations of Herz-Morrey-Lizorkin-Triebel Spaces

Key Result

Lemma 2.5

Let $w$ be a weight on $\mathbb{R}^n$, $q(\cdot)\in\mathcal{P}(\mathbb{R}^n),$$p\in(1,\infty)$, $\lambda \in [0, \infty)$, $\alpha(\cdot) \in L^\infty(\mathbb{R}^n)$ and $\theta>0$. If $\alpha(\cdot) \in C^{\log}_0(\mathbb{R}^n)\cap C^{\log}_{\infty}(\mathbb{R}^n),$ then for all $f\in L^{q(\cdot)}_{

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: see zhz1
  • Lemma 2.6
  • Remark 2.7
  • Theorem 3.1
  • Lemma 3.2: see cw-1
  • Corollary 3.3
  • ...and 17 more