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Sharp Bounds for the multilinear Hausdorff operators on mixed radial-angular local Morrey-type spaces

Ronghui Liu, Qi Zhang

TL;DR

This work derives sharp bounds for multilinear Hausdorff operators on mixed radial-angular local Morrey-type spaces with power weights, providing explicit constants and necessary-sufficient conditions. By employing polar-coordinate representations and standard harmonic-analysis tools, it covers several operator forms ($R_{\Phi}$, $\widetilde{R_{\Phi}}$, $S_{\Phi}$, $\widetilde{S_{\Phi}}$) and their weighted/complementary variants, establishing both boundedness and optimal operator norms. The results unify and extend prior local/complementary Morrey-type space bounds to a refined mixed radial-angular setting, with clear criteria that are verifiable from the generating function $\Phi$. Applications recover classical operators such as the Hardy, dual Hardy, and Cesàro operators within this generalized framework, highlighting the practical impact for harmonic analysis on Morrey-type spaces. Overall, the paper broadens the analytical toolkit for multilinear operators in weighted Morrey-type spaces and provides sharp, computable bounds that can drive further research and applications.

Abstract

In this paper, we establish sharp bounds for the multilinear Hausdorff operators on mixed radial-angular local Morrey-type spaces, and we also give ralated applications of these operators. Meanwhile, sharp bounds for the multilinear Hausdorff operators on mixed radial-angular complementary local Morrey-type spaces are also derived.

Sharp Bounds for the multilinear Hausdorff operators on mixed radial-angular local Morrey-type spaces

TL;DR

This work derives sharp bounds for multilinear Hausdorff operators on mixed radial-angular local Morrey-type spaces with power weights, providing explicit constants and necessary-sufficient conditions. By employing polar-coordinate representations and standard harmonic-analysis tools, it covers several operator forms (, , , ) and their weighted/complementary variants, establishing both boundedness and optimal operator norms. The results unify and extend prior local/complementary Morrey-type space bounds to a refined mixed radial-angular setting, with clear criteria that are verifiable from the generating function . Applications recover classical operators such as the Hardy, dual Hardy, and Cesàro operators within this generalized framework, highlighting the practical impact for harmonic analysis on Morrey-type spaces. Overall, the paper broadens the analytical toolkit for multilinear operators in weighted Morrey-type spaces and provides sharp, computable bounds that can drive further research and applications.

Abstract

In this paper, we establish sharp bounds for the multilinear Hausdorff operators on mixed radial-angular local Morrey-type spaces, and we also give ralated applications of these operators. Meanwhile, sharp bounds for the multilinear Hausdorff operators on mixed radial-angular complementary local Morrey-type spaces are also derived.

Paper Structure

This paper contains 4 sections, 21 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications
  4. Supplementary results

Key Result

Theorem 2.1

Suppose that $\Phi$ is a nonnegative, locally integrable, radial function. (i) Let $1 \leq p, \tilde{p}, q \leq \infty$ and $\alpha\in\mathbb{R}$. If (1.3) be satisfied. Then $\widetilde{\mathcal{H}_{\Phi}}$ is bounded on the local Morrey-type space $LML_{rad}^{\tilde{p},\lambda,q}L_{ang}^{p}(\mathb if and only if Moreover, (ii) Let $0 < p,\tilde{p} < 1$, $0 < q < 1$ and $\alpha\in\mathbb{R}$. I

Theorems & Definitions (34)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 24 more