A matrix weighted bilinear Carleson Lemma and Maximal Function

TL;DR

This paper establishes a bilinear Carleson embedding theorem with matrix weights and scalar measures, extending recent scalar results to the matrix setting and introducing a maximal function with dimension-independent bounds.

Contribution

It provides the first bilinear Carleson embedding theorem with matrix weights and measures, extending scalar results and defining a maximal function with bounded dimension growth.

Findings

Proves a bilinear Carleson embedding theorem with matrix weights.

Introduces a maximal function with dimension-independent bounds.

Extends scalar Carleson theorems to the matrix setting.

Abstract

We prove a bilinear Carleson embedding theorem with matrix weight and scalar measure. In the scalar case, this becomes exactly the well known weighted bilinear Carleson embedding theorem. Although only allowing scalar Carleson measures, it is to date the only extension to the bilinear setting of the recent Carleson embedding theorem by Culiuc and Treil that features a matrix Carleson measure and a matrix weight. It is well known that a Carleson embedding theorem implies a Doob's maximal inequality and this holds true in the matrix weighted setting with an appropriately defined maximal operator. It is also known that a dimensional growth must occur in the Carleson embedding theorem with matrix Carleson measure, even with trivial weight. We give a definition of a maximal type function whose norm in the matrix weighted setting does not grow with dimension.