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Vector valued estimates for matrix weighted maximal operators and product $\mathrm{BMO}$

Spyridon Kakaroumpas, Odí Soler i Gibert

TL;DR

The paper addresses matrix-weighted vector-valued extensions of Fefferman–Stein inequalities for the Hardy–Littlewood maximal function and the associated product BMO framework. It introduces an extrapolation principle for convex body valued functions to transfer matrix-weight bounds to the convex-body setting, yielding vector-valued estimates for maximal operators and H^1–BMO duality in the biparameter context. The authors then derive two-matrix weighted bounds for biparameter paraproducts and bicommutators, showing that two-matrix weighted product BMO controls the corresponding biparameter operators. This work broadens the scalar theory to the matrix-weight setting, enabling sharp, dimension-dependent bounds with applications to paraproducts and bicommutators in multi-parameter harmonic analysis. The results advance the understanding of matrix weights in product spaces and provide tools for further study of operator theory under matrix weights, with potential implications for related multi-parameter PDEs and analysis on vector-valued function spaces.

Abstract

We consider maximal operators acting on vector valued functions, that is functions taking values on $\mathbb{C}^d,$ that incorporate matrix weights in their definitions. We show vector valued estimates, in the sense of Fefferman--Stein inequalities, for such operators. These are proven using an extrapolation result for convex body valued functions due to Bownik and Cruz-Uribe. Finally, we show an $\mathrm{H}^1$-$\mathrm{BMO}$ duality for matrix valued functions and we apply the previous vector valued estimates to show upper bounds for biparameter paraproducts. For the reader's convenience, we include an appendix explaining how to adapt the extrapolation for real convex body valued functions of Bownik and Cruz-Uribe to the setting of complex convex body valued functions that we treat.

Vector valued estimates for matrix weighted maximal operators and product $\mathrm{BMO}$

TL;DR

The paper addresses matrix-weighted vector-valued extensions of Fefferman–Stein inequalities for the Hardy–Littlewood maximal function and the associated product BMO framework. It introduces an extrapolation principle for convex body valued functions to transfer matrix-weight bounds to the convex-body setting, yielding vector-valued estimates for maximal operators and H^1–BMO duality in the biparameter context. The authors then derive two-matrix weighted bounds for biparameter paraproducts and bicommutators, showing that two-matrix weighted product BMO controls the corresponding biparameter operators. This work broadens the scalar theory to the matrix-weight setting, enabling sharp, dimension-dependent bounds with applications to paraproducts and bicommutators in multi-parameter harmonic analysis. The results advance the understanding of matrix weights in product spaces and provide tools for further study of operator theory under matrix weights, with potential implications for related multi-parameter PDEs and analysis on vector-valued function spaces.

Abstract

We consider maximal operators acting on vector valued functions, that is functions taking values on that incorporate matrix weights in their definitions. We show vector valued estimates, in the sense of Fefferman--Stein inequalities, for such operators. These are proven using an extrapolation result for convex body valued functions due to Bownik and Cruz-Uribe. Finally, we show an - duality for matrix valued functions and we apply the previous vector valued estimates to show upper bounds for biparameter paraproducts. For the reader's convenience, we include an appendix explaining how to adapt the extrapolation for real convex body valued functions of Bownik and Cruz-Uribe to the setting of complex convex body valued functions that we treat.
Paper Structure (27 sections, 18 theorems, 266 equations)

This paper contains 27 sections, 18 theorems, 266 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. Dyadic grids
  5. Haar systems
  6. Haar system on R
  7. Haar system on Rn
  8. Haar system on the product space RxR
  9. Matrix norms in terms of column norms
  10. Matrix weighted matrix-valued Lebesgue spaces
  11. Reducing operators
  12. Matrix Ap weights
  13. One-parameter matrix Ap weights
  14. Biparameter matrix Ap weights
  15. One-parameter restriction of biparameter matrix Ap weights
  16. ...and 12 more sections

Key Result

Theorem 1

Let $1<p<\infty.$ Consider a $(d \times d)$ matrix $A_p$ weight $W$ and a sequence of vector valued functions $\{f_n\}^{\infty}_{n=1}.$ Then, for each $1 < q < \infty$ it holds that where and $M_{W}$ denotes the Christ--Goldberg maximal function corresponding to the weight $W$ and the exponent $p.$

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • proof
  • Theorem A: Bownik, Cruz-Uribe Cruz_Uribe_Bownik_Extrapolation
  • Theorem 5: Vector valued estimates for families of extrapolation pairs
  • Lemma 6
  • proof
  • proof : Proof of Theorem \ref{['thm:VectorValuedEstimates']}
  • ...and 25 more