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A class of multilinear bounded oscillation operators on measure spaces and applications

Mingming Cao, Gonzalo Ibañez-Firnkorn, Israel P. Rivera-Ríos, Qingying Xue, Kôzô Yabuta

Abstract

In this paper, we develop a comprehensive weighted theory for a class of Banach-valued multilinear bounded oscillation operators on measure spaces, which merges multilinear Calderón-Zygmund operators with a quantity of operators beyond the multilinear Calderón-Zygmund theory. We prove that such multilinear operators and corresponding commutators are locally pointwise dominated by two sparse dyadic operators, respectively. We also establish three kinds of typical estimates: local exponential decay estimates, mixed weak type estimates, and sharp weighted norm inequalities. Beyond that, based on Rubio de Francia extrapolation for abstract multilinear compact operators, we obtain weighted compactness for commutators of specific multilinear operators on spaces of homogeneous type. A compact extrapolation allows us to get full range of exponents, while weighted interpolation for multilinear compact operators is crucial to the compact extrapolation. These are due to a weighted Fréchet-Kolmogorov theorem in the quasi-Banach range, which gives a characterization of relative compactness of subsets in weighted Lebesgue spaces. As applications, we illustrate multilinear bounded oscillation operators with examples including multilinear Hardy-Littlewood maximal operators on measure spaces, multilinear $ω$-Calderón-Zygmund operators on spaces of homogeneous type, multilinear Littlewood-Paley square operators, multilinear Fourier integral operators, higher order Calderón commutators, maximally modulated multilinear singular integrals, and $q$-variation of $ω$-Calderón-Zygmund operators.

A class of multilinear bounded oscillation operators on measure spaces and applications

Abstract

In this paper, we develop a comprehensive weighted theory for a class of Banach-valued multilinear bounded oscillation operators on measure spaces, which merges multilinear Calderón-Zygmund operators with a quantity of operators beyond the multilinear Calderón-Zygmund theory. We prove that such multilinear operators and corresponding commutators are locally pointwise dominated by two sparse dyadic operators, respectively. We also establish three kinds of typical estimates: local exponential decay estimates, mixed weak type estimates, and sharp weighted norm inequalities. Beyond that, based on Rubio de Francia extrapolation for abstract multilinear compact operators, we obtain weighted compactness for commutators of specific multilinear operators on spaces of homogeneous type. A compact extrapolation allows us to get full range of exponents, while weighted interpolation for multilinear compact operators is crucial to the compact extrapolation. These are due to a weighted Fréchet-Kolmogorov theorem in the quasi-Banach range, which gives a characterization of relative compactness of subsets in weighted Lebesgue spaces. As applications, we illustrate multilinear bounded oscillation operators with examples including multilinear Hardy-Littlewood maximal operators on measure spaces, multilinear -Calderón-Zygmund operators on spaces of homogeneous type, multilinear Littlewood-Paley square operators, multilinear Fourier integral operators, higher order Calderón commutators, maximally modulated multilinear singular integrals, and -variation of -Calderón-Zygmund operators.
Paper Structure (29 sections, 66 theorems, 593 equations)

This paper contains 29 sections, 66 theorems, 593 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. The hypotheses
  4. Main results
  5. Historical background
  6. Structure of the paper
  7. Applications
  8. Multilinear Hardy-Littlewood maximal operators
  9. Multilinear $\omega$-Calderón-Zygmund operators
  10. Multilinear Littlewood-Paley square operators
  11. Multilinear Fourier integral operators
  12. Higher order Calderón commutators
  13. Maximally modulated multilinear singular integrals
  14. $q$-variation of $\omega$-Calderón-Zygmund operators
  15. Preliminaries
  16. ...and 14 more sections

Key Result

Theorem 1.5

Let $(\Sigma, \mu)$ be a measure space with a ball-basis $\mathscr{B}$. Let $\mathbf{b}=(b_1, \ldots, b_m)$ of measurable functions and $\alpha \in \{0, 1\}^m$. If $T$ is a $\mathbb{B}$-valued multilinear bounded oscillation operator with respect to $\mathscr{B}$ and $r \in [1, \infty)$ such that $T where $\tau = \tau(\alpha) :=\{i: \alpha_i \neq 0\}$ and $\mathfrak{C}(T) := \mathfrak{C}_1(T) + \m

Theorems & Definitions (127)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 117 more