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Hardy type spaces estimates for multilinear fractional integral operators

Jian Tan

TL;DR

The paper develops a unified real-variable framework to establish boundedness of multilinear fractional integral operators ${\mathcal{T}}_{\alpha}$ on products of Hardy spaces associated with ball quasi-Banach spaces and into their targets. Building a general theorem (Theorem s1t1) under Hölder-type and duality assumptions, the authors then instantiate it to a wide range of spaces, including weighted, variable, mixed-norm, Hardy-Lorentz, and Hardy-Orlicz Hardy spaces. The results extend the multilinear operator theory to generalized Hardy spaces and provide new boundedness results in several settings, supported by atomic decompositions and maximal-function techniques. This work unifies disparate Hardy-space theories under a common ball-quasi-Banach framework and broadens potential applications in harmonic analysis and related fields.

Abstract

In this paper, we prove the boundedness of multilinear fractional integral operators from products of Hardy spaces associated with ball quasi-Banach function spaces into their corresponding ball quasi-Banach function spaces. As applications, we establish the boundedness of these operators on various function spaces, including weighted Hardy spaces, variable Hardy spaces, mixed-norm Hardy spaces, Hardy--Lorentz spaces, and Hardy--Orlicz spaces. Notably, several of these results are new, even in special cases, and extend the existing theory of multilinear operators in the context of generalized Hardy spaces.

Hardy type spaces estimates for multilinear fractional integral operators

TL;DR

The paper develops a unified real-variable framework to establish boundedness of multilinear fractional integral operators on products of Hardy spaces associated with ball quasi-Banach spaces and into their targets. Building a general theorem (Theorem s1t1) under Hölder-type and duality assumptions, the authors then instantiate it to a wide range of spaces, including weighted, variable, mixed-norm, Hardy-Lorentz, and Hardy-Orlicz Hardy spaces. The results extend the multilinear operator theory to generalized Hardy spaces and provide new boundedness results in several settings, supported by atomic decompositions and maximal-function techniques. This work unifies disparate Hardy-space theories under a common ball-quasi-Banach framework and broadens potential applications in harmonic analysis and related fields.

Abstract

In this paper, we prove the boundedness of multilinear fractional integral operators from products of Hardy spaces associated with ball quasi-Banach function spaces into their corresponding ball quasi-Banach function spaces. As applications, we establish the boundedness of these operators on various function spaces, including weighted Hardy spaces, variable Hardy spaces, mixed-norm Hardy spaces, Hardy--Lorentz spaces, and Hardy--Orlicz spaces. Notably, several of these results are new, even in special cases, and extend the existing theory of multilinear operators in the context of generalized Hardy spaces.

Paper Structure

This paper contains 9 sections, 13 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Examples and applications
  4. Weighted Hardy spaces
  5. Variable Hardy spaces
  6. Mixed-norm Hardy spaces
  7. Hardy--Lorentz spaces
  8. Hardy--Orlicz spaces
  9. Proof of Theorem \ref{['s1t1']}

Key Result

Theorem 1.1

Given an integer $m\ge 1$, let $X_1,$$\ldots,$$X_m$, $Y_1,$$\ldots,$$Y_m$ and $Y$ be ball quasi-Banach function spaces. Suppose that $X_1,$$\ldots,$$X_m$ have the absolutely continuous quasi-norm satisfying Assumptions ass2.7 and ass2.8 and that $Y$ fulfills Assumption ass2.8. Let $0<p, q, p_k, q_k< and where $k=1, 2, \cdots, m.$ Suppose that the following Hölder's inequality holds true: for any

Theorems & Definitions (23)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.3
  • Lemma 2.4
  • proof : Proof of Lemma \ref{['l2.8']}
  • Definition 2.5
  • Theorem 2.5
  • ...and 13 more