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Sparse domination implies vector-valued sparse domination

Emiel Lorist, Zoe Nieraeth

Abstract

We prove that scalar-valued sparse domination of a multilinear operator implies vector-valued sparse domination for tuples of quasi-Banach function spaces, for which we introduce a multilinear analogue of the UMD condition. This condition is characterized by the boundedness of the multisublinear Hardy-Littlewood maximal operator and goes beyond examples in which a UMD condition is assumed on each individual space and includes e.g. iterated Lebesgue, Lorentz, and Orlicz spaces. Our method allows us to obtain sharp vector-valued weighted bounds directly from scalar-valued sparse domination, without the use of a Rubio de Francia type extrapolation result. We apply our result to obtain new vector-valued bounds for multilinear Calderón-Zygmund operators as well as recover the old ones with a new sharp weighted bound. Moreover, in the Banach function space setting we improve upon recent vector-valued bounds for the bilinear Hilbert transform.

Sparse domination implies vector-valued sparse domination

Abstract

We prove that scalar-valued sparse domination of a multilinear operator implies vector-valued sparse domination for tuples of quasi-Banach function spaces, for which we introduce a multilinear analogue of the UMD condition. This condition is characterized by the boundedness of the multisublinear Hardy-Littlewood maximal operator and goes beyond examples in which a UMD condition is assumed on each individual space and includes e.g. iterated Lebesgue, Lorentz, and Orlicz spaces. Our method allows us to obtain sharp vector-valued weighted bounds directly from scalar-valued sparse domination, without the use of a Rubio de Francia type extrapolation result. We apply our result to obtain new vector-valued bounds for multilinear Calderón-Zygmund operators as well as recover the old ones with a new sharp weighted bound. Moreover, in the Banach function space setting we improve upon recent vector-valued bounds for the bilinear Hilbert transform.

Paper Structure

This paper contains 11 sections, 27 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Product quasi-Banach function spaces
  4. Sparse forms and multilinear weight classes
  5. The multisublinear lattice maximal operator
  6. A limited range multilinear UMD property for quasi-Banach function spaces
  7. Main results
  8. Applications
  9. Vector-valued estimates in the quasi-Banach range
  10. Multilinear Calderón-Zygmund operators
  11. The bilinear Hilbert transform

Key Result

Theorem \oldthetheorem

Let $T$ be an operator such that for any $f,g \in L^\infty_c(\mathbf{R}^d)$ there exists a sparse collection of cubes $\mathcal{S}$ such that Let $X$ be a $\mathop{\mathrm{UMD}}\nolimits$ Banach function space over a measure space $(\Omega,\mu)$ and suppose that for any simple function $f \in L^\infty_c(\mathbf{R}^d;X)$ the function $\widetilde{T}f:\mathbf{R}^d \to X$ given by is well-defined an

Theorems & Definitions (59)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • ...and 49 more