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Vector-valued extensions of operators through multilinear limited range extrapolation

Emiel Lorist, Zoe Nieraeth

Abstract

We give an extension of Rubio de Francia's extrapolation theorem for functions taking values in UMD Banach function spaces to the multilinear limited range setting. In particular we show how boundedness of an $m$-(sub)linear operator \[T:L^{p_1}(w_1^{p_1})\times\cdots\times L^{p_m}(w_m^{p_m})\to L^p(w^p) \] for a certain class of Muckenhoupt weights yields an extension of the operator to Bochner spaces $L^{p}(w^p;X)$ for a wide class of Banach function spaces $X$, which includes certain Lebesgue, Lorentz and Orlicz spaces. We apply the extrapolation result to various operators, which yields new vector-valued bounds. Our examples include the bilinear Hilbert transform, certain Fourier multipliers and various operators satisfying sparse domination results.

Vector-valued extensions of operators through multilinear limited range extrapolation

Abstract

We give an extension of Rubio de Francia's extrapolation theorem for functions taking values in UMD Banach function spaces to the multilinear limited range setting. In particular we show how boundedness of an -(sub)linear operator for a certain class of Muckenhoupt weights yields an extension of the operator to Bochner spaces for a wide class of Banach function spaces , which includes certain Lebesgue, Lorentz and Orlicz spaces. We apply the extrapolation result to various operators, which yields new vector-valued bounds. Our examples include the bilinear Hilbert transform, certain Fourier multipliers and various operators satisfying sparse domination results.

Paper Structure

This paper contains 11 sections, 14 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Muckenhoupt weights
  4. Banach function spaces
  5. The UMD property
  6. The UMDp+p- property of quasi-Banach function spaces
  7. Proof of the main result
  8. Applications
  9. The bilinear Hilbert transform
  10. Multilinear Calderón-Zygmund operators
  11. Limited range extrapolation

Key Result

Theorem \oldthetheorem

Let $m\in\mathbb{N}$ and fix $0< p_j^-<p_j^+\leq\infty$ for $j\in\{1,\ldots,m\}$. Let $T$ be an operator defined on $m$-tuples of functions and suppose there exist $p_j\in (p_j^-,p_j^+)$ such that for all weights $w_j^{p_j} \in A_{{p_j/p_j^-}}\cap RH_{(p_j^+/p_j)'}$ and $f_j\in L^{p_j}(w_j^{p_j})$ w with $w=\prod_{j=1}^mw_j$, $\frac{1}{p}=\sum_{j=1}^m\frac{1}{p_j}$, and where $C>0$ depends only on

Theorems & Definitions (32)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Example \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem: Rubio de Francia
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • ...and 22 more