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Grand variable Herz-Morrey type Besove spaces and Triebel-Lizorkin spaces

Mehvish Sultan, Babar Sultan

TL;DR

This work extends harmonic analysis in variable-exponent settings by establishing the boundedness of vector-valued sublinear operators on the new grand variable Herz-Morrey spaces and by introducing Besov-type and Triebel-Lizorkin-type spaces in this framework. The authors develop Peetre-type maximal-characterizations, providing multiple equivalent quasi-norms for these spaces via a Fourier-analytic resolution of unity and detailed maximal-function estimates. The main contributions include vector-valued boundedness results on $M_{\lambda, p(\cdot)}^{\eta(\cdot), q, \theta}$, and robust norm equivalences for the associated Besov- and Triebel-Lizorkin-type spaces under log-Hölder and Tauberian-type conditions, enriching the toolkit for analysis with variable growth and nonhomogeneous scaling. The results broaden the applicability of Besov/Triebel-Lizorkin-type spaces to nonuniform media and variable-growth PDE contexts, offering precise characterizations through Peetre maxima and dyadic decompositions. Overall, the paper advances the theory of grand variable function spaces and their operator theory in a variable-exponent setting.

Abstract

In the article, the boundedness of vector-valued sublinear operators in grand variable Herz-Morrey spaces $M \dot{K}_{ λ, p(\cdot)}^{η(\cdot), q), θ}\left(\mathbb{R}^{n}\right)$ are obtained. Then grand variable Herz-Morrey type Besov and Triebel-Lizorkin spaces are defined. We will also prove the equivalent quasi-norms by Peetre's maximal operators in these spaces.

Grand variable Herz-Morrey type Besove spaces and Triebel-Lizorkin spaces

TL;DR

This work extends harmonic analysis in variable-exponent settings by establishing the boundedness of vector-valued sublinear operators on the new grand variable Herz-Morrey spaces and by introducing Besov-type and Triebel-Lizorkin-type spaces in this framework. The authors develop Peetre-type maximal-characterizations, providing multiple equivalent quasi-norms for these spaces via a Fourier-analytic resolution of unity and detailed maximal-function estimates. The main contributions include vector-valued boundedness results on , and robust norm equivalences for the associated Besov- and Triebel-Lizorkin-type spaces under log-Hölder and Tauberian-type conditions, enriching the toolkit for analysis with variable growth and nonhomogeneous scaling. The results broaden the applicability of Besov/Triebel-Lizorkin-type spaces to nonuniform media and variable-growth PDE contexts, offering precise characterizations through Peetre maxima and dyadic decompositions. Overall, the paper advances the theory of grand variable function spaces and their operator theory in a variable-exponent setting.

Abstract

In the article, the boundedness of vector-valued sublinear operators in grand variable Herz-Morrey spaces are obtained. Then grand variable Herz-Morrey type Besov and Triebel-Lizorkin spaces are defined. We will also prove the equivalent quasi-norms by Peetre's maximal operators in these spaces.

Paper Structure

This paper contains 4 sections, 15 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proofs of the main results
  4. Ethics declarations

Key Result

Lemma 2.1

ref20 Let $f$ belong to the Lebesgue space $L^{p(\cdot)}(\mathbb{R}^{n})$ and $g$ belong to $L^{p'(\cdot)}(\mathbb{R}^{n})$, where $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$. Then, the product function $fg$ is integrable over $\mathbb{R}^{n}$, and the following inequality holds: where $r_{p}$ is defined as $r_{p}=1+\frac{1}{p^{-}} -\frac{1}{p^{+}}$.

Theorems & Definitions (21)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7
  • Theorem 2.8
  • Remark 2.9
  • Lemma 2.10
  • ...and 11 more