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New fractional type weights and the boundedness of some operators

Xi Cen, Qianjun He, Zichen Song, Zihan Wang

TL;DR

The paper develops a weighted analysis framework for multilinear fractional-type operators on variable exponent Lebesgue spaces. It introduces the new variable-weight class $A_{\vec{p}(\cdot),q(\cdot)}$ and proves that boundedness of multilinear fractional operators (and their commutators) between products of variable-exponent spaces and target variable-exponent spaces is equivalent to membership in this weight class, extending classical scalar results. It also extends the theory to matrix weights by defining $\mathbb{A}_{p(\cdot),q(\cdot)}$ via reducing operators and showing that the uniform boundedness of fractional averaging operators characterizes this matrix class. The proofs combine dyadic domination, Fatou arguments, multilinear Calderón–Zygmund techniques, and extrapolation in the variable-exponent setting. Collectively, these results generalize prior work (Moen, Bernardis–Dalmasso–Pradolini, Cruz-Uribe–Guzmán, etc.) and broaden the reach of weighted multilinear harmonic analysis to variable and matrix-valued contexts with fractional-type operators.

Abstract

Two classes of fractional type variable weights are established in this paper. The first kind of weights ${A_{\vec p( \cdot ),q( \cdot )}}$ are variable multiple weights, which are characterized by the weighted variable boundedness of multilinear fractional type operators, called multilinear Hardy--Littlewood--Sobolev theorem on weighted variable Lebesgue spaces. Meanwhile, the weighted variable boundedness for the commutators of multilinear fractional type operators are also obtained. This generalizes some known work, such as Moen (2009), Bernardis--Dalmasso--Pradolini (2014), and Cruz-Uribe--Guzmán (2020). Another class of weights ${\mathbb{A}_{p( \cdot ),q(\cdot)}}$ are variable matrix weights that also characterized by certain fractional type operators. This generalize some previous results on matrix weights ${\mathbb{A}_{p( \cdot )}}$.

New fractional type weights and the boundedness of some operators

TL;DR

The paper develops a weighted analysis framework for multilinear fractional-type operators on variable exponent Lebesgue spaces. It introduces the new variable-weight class and proves that boundedness of multilinear fractional operators (and their commutators) between products of variable-exponent spaces and target variable-exponent spaces is equivalent to membership in this weight class, extending classical scalar results. It also extends the theory to matrix weights by defining via reducing operators and showing that the uniform boundedness of fractional averaging operators characterizes this matrix class. The proofs combine dyadic domination, Fatou arguments, multilinear Calderón–Zygmund techniques, and extrapolation in the variable-exponent setting. Collectively, these results generalize prior work (Moen, Bernardis–Dalmasso–Pradolini, Cruz-Uribe–Guzmán, etc.) and broaden the reach of weighted multilinear harmonic analysis to variable and matrix-valued contexts with fractional-type operators.

Abstract

Two classes of fractional type variable weights are established in this paper. The first kind of weights are variable multiple weights, which are characterized by the weighted variable boundedness of multilinear fractional type operators, called multilinear Hardy--Littlewood--Sobolev theorem on weighted variable Lebesgue spaces. Meanwhile, the weighted variable boundedness for the commutators of multilinear fractional type operators are also obtained. This generalizes some known work, such as Moen (2009), Bernardis--Dalmasso--Pradolini (2014), and Cruz-Uribe--Guzmán (2020). Another class of weights are variable matrix weights that also characterized by certain fractional type operators. This generalize some previous results on matrix weights .
Paper Structure (6 sections, 31 theorems, 168 equations, 1 figure)

This paper contains 6 sections, 31 theorems, 168 equations, 1 figure.

Table of Contents

  1. Introduction
  2. New variable multiple weights $A_{\vec{p}(\cdot),q(\cdot)}$
  3. Some classical results
  4. Main results
  5. New variable matrix weights ${{\mathbb{A}}_{p( \cdot ),q( \cdot )}}$
  6. The proof of Theorem \ref{['FMVLP']}

Key Result

Lemma 1.1

Let $\omega \in A_{\infty}$. Then, for each $0 < \alpha < 1$, there exists $\beta \in (0,1)$ such that for any cube $Q$ and any subset $E \subseteq Q$ with $\alpha|Q| \leq |E|$, it holds that $\beta \omega(Q) \leq \omega(E)$. Similarly, for each $0 < \gamma < 1$, there exists $\delta \in (0,1)$ such

Figures (1)

  • Figure 1: The relationships between weights

Theorems & Definitions (51)

  • Definition 1.1: Gra1
  • Definition 1.2: Gra1
  • Lemma 1.1: Cruz2020
  • Definition 1.3
  • Remark 1.1
  • Remark 1.2
  • Lemma 1.2
  • proof : Proof:
  • Lemma 1.3
  • Remark 1.3
  • ...and 41 more