Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundedness of a class of multilinear operators and their iterated commutators on Morrey-Banach function spaces

Jiawei Tan, Jiahui Wang, Qingying Xue

TL;DR

This work develops a unified framework to study the boundedness of a broad class of multilinear operators and their iterated commutators on Morrey-Banach spaces, unifying Morrey-Lorentz and variable-exponent Morrey settings. The authors introduce two central hypotheses—pointwise sparse domination and the $W_r$ property—that enable quantitative Coifman–Fefferman type bounds and control of iterated commutators in Morrey-Banach spaces. They prove robust results for multilinear Calderón–Zygmund-type operators, rough operators, and related square and maximal operators, with explicit dependence on Hardy–Littlewood maximal operator norms and sparse-dominated forms. Applications include Morrey-Lorentz spaces and Morrey spaces with variable exponents, as well as specific operators such as multilinear maximal singular integrals and multilinear Bochner–Riesz square functions. The findings significantly extend Morrey-type harmonic analysis to a broad operator class, providing tools for PDE-related contexts where local control and nonuniform integrability are essential.

Abstract

This paper investigates the boundedness of a broad class of operators within the framework of generalized Morrey-Banach function spaces. This class includes multilinear operators such as multilinear $ω$-Calderón-Zygmund operators, multilinear maximal singular integral operators, multilinear pseudo-differential operators, and multilinear square functions, as well as linear operators such as rough singular integral operators, nonintegral operators, and Stein's square functions. The boundedness of this class of operators and their commutators on Morrey-Banach spaces is established provided they satisfy either a pointwise sparse domination assumption or the $W_{r}$ property (Lerner, Lorist and Ombrosi, Math. Z. 2024), which significantly generalize the classical theory of Morrey spaces. As an application, the boundedness of iterated commutators of this class of multilinear operators is extended to Morrey-Lorentz spaces and Morrey spaces with variable exponents.

Boundedness of a class of multilinear operators and their iterated commutators on Morrey-Banach function spaces

TL;DR

This work develops a unified framework to study the boundedness of a broad class of multilinear operators and their iterated commutators on Morrey-Banach spaces, unifying Morrey-Lorentz and variable-exponent Morrey settings. The authors introduce two central hypotheses—pointwise sparse domination and the property—that enable quantitative Coifman–Fefferman type bounds and control of iterated commutators in Morrey-Banach spaces. They prove robust results for multilinear Calderón–Zygmund-type operators, rough operators, and related square and maximal operators, with explicit dependence on Hardy–Littlewood maximal operator norms and sparse-dominated forms. Applications include Morrey-Lorentz spaces and Morrey spaces with variable exponents, as well as specific operators such as multilinear maximal singular integrals and multilinear Bochner–Riesz square functions. The findings significantly extend Morrey-type harmonic analysis to a broad operator class, providing tools for PDE-related contexts where local control and nonuniform integrability are essential.

Abstract

This paper investigates the boundedness of a broad class of operators within the framework of generalized Morrey-Banach function spaces. This class includes multilinear operators such as multilinear -Calderón-Zygmund operators, multilinear maximal singular integral operators, multilinear pseudo-differential operators, and multilinear square functions, as well as linear operators such as rough singular integral operators, nonintegral operators, and Stein's square functions. The boundedness of this class of operators and their commutators on Morrey-Banach spaces is established provided they satisfy either a pointwise sparse domination assumption or the property (Lerner, Lorist and Ombrosi, Math. Z. 2024), which significantly generalize the classical theory of Morrey spaces. As an application, the boundedness of iterated commutators of this class of multilinear operators is extended to Morrey-Lorentz spaces and Morrey spaces with variable exponents.

Paper Structure

This paper contains 22 sections, 26 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Morrey type spaces
  3. Multilinear operators
  4. Motivation
  5. Basic assumptions
  6. Main results
  7. Results under the Hypothesis \ref{['hyp']}
  8. Results under the Hypothesis \ref{['hyp-new']}
  9. Preliminary
  10. Weights
  11. BMO spaces
  12. Morrey-Banach function space
  13. Lorentz space
  14. Young function and Orlicz space
  15. Sparse family
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $I \coloneqq \{i_1,\dots,i_l\} = \{1,\dots,l\} \subseteq \{1,\dots,m\}$, $X$ be a Banach function space and $u: \mathbb{R}^n \times (0,+\infty) \to (0,+\infty)$ be Lebesgue measurable. Suppose that $\vec{b} \in \mathrm{BMO}^l$ and $\mathcal{T}_{\vec{b}}$ meets Hypothesis hyp. If $0<p<\infty$, $X where $X' \in \mathbb{M}$ represents that the Hardy-Littlewood maximal operator $M$ is bounded on $

Theorems & Definitions (63)

  • Definition 1.1: Iterated commutators of multilinear operators
  • Remark 1
  • Definition 1.2: Morrey-Banach space
  • Definition 1.3: $\mathbb{W}_X^\alpha$ class
  • Remark 2
  • Theorem 1.1
  • Remark 3
  • Remark 4
  • Theorem 1.2
  • Remark 5
  • ...and 53 more