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The Commutators of $n$-dimensional Rough Fractional Hardy Operators on Two Weighted Grand Herz-Morrey Spaces with Variable Exponents

Shengrong Wang, Pengfei Guo, Jingshi Xu

TL;DR

The paper tackles the problem of establishing boundedness for higher-order commutators of the $n$-dimensional rough fractional Hardy operator $H_{\Omega,\beta}$ and its adjoint on two weighted grand Herz-Morrey spaces with variable exponents. It develops a framework that combines rough kernel estimates, BMO and Lipschitz symbol classes, and variable-exponent two-weight norm inequalities, yielding finite bounds of the form \|...f...\| \lesssim \|b\|^m \|f\| on the target spaces under precise conditions on $q(\cdot)$, $w$, $v$, $\alpha(\cdot)$, $s$, and $\Omega$. The results extend previous Lebesgue and Herz-type bounds to the grand Herz-Morrey setting with variable exponents and two weights, including both the BMO and Lipschitz symbol cases. This advances operator theory for fractional Hardy-type operators in nonstandard function spaces and has potential applications to PDEs in nonhomogeneous media where variable growth and weighted norms are natural. All results are stated under explicit assumptions on the kernel, symbol class, and the exponents, ensuring sharp control over the nonlocal, rough nature of the operators.

Abstract

In this paper, we obtain the boundedness of $m$th order commutators generated by the $n$-dimensional fractional Hardy operator with rough kernel and its adjoint operator with BMO functions on two weighted grand Herz-Morrey spaces with variable exponents. Replacing Lipschitz functions with BMO functions the corresponding result is also given.

The Commutators of $n$-dimensional Rough Fractional Hardy Operators on Two Weighted Grand Herz-Morrey Spaces with Variable Exponents

TL;DR

The paper tackles the problem of establishing boundedness for higher-order commutators of the -dimensional rough fractional Hardy operator and its adjoint on two weighted grand Herz-Morrey spaces with variable exponents. It develops a framework that combines rough kernel estimates, BMO and Lipschitz symbol classes, and variable-exponent two-weight norm inequalities, yielding finite bounds of the form \|...f...\| \lesssim \|b\|^m \|f\| on the target spaces under precise conditions on , , , , , and . The results extend previous Lebesgue and Herz-type bounds to the grand Herz-Morrey setting with variable exponents and two weights, including both the BMO and Lipschitz symbol cases. This advances operator theory for fractional Hardy-type operators in nonstandard function spaces and has potential applications to PDEs in nonhomogeneous media where variable growth and weighted norms are natural. All results are stated under explicit assumptions on the kernel, symbol class, and the exponents, ensuring sharp control over the nonlocal, rough nature of the operators.

Abstract

In this paper, we obtain the boundedness of th order commutators generated by the -dimensional fractional Hardy operator with rough kernel and its adjoint operator with BMO functions on two weighted grand Herz-Morrey spaces with variable exponents. Replacing Lipschitz functions with BMO functions the corresponding result is also given.

Paper Structure

This paper contains 4 sections, 12 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Commutators of the Hardy operator and BMO functions
  4. Commutators by the Hardy operator and Lipschitz functions

Key Result

Lemma 2.7

Let $w$ be a weight on $\mathbb{R}^n$. If $q \in [1,\infty)$ and $w \in A_q$, then there exist constants $\epsilon\in (0,1)$ and $C>0$ such that for all balls $B$ in $\mathbb{R}^n$ and all measurable subsets $S\subset B,$

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Remark 2.9
  • Lemma 2.10: see dhhr-0
  • ...and 13 more