A Study of the Matrix Carleson Embedding Theorem with Applications to Sparse Operators

TL;DR

This paper advances the understanding of the matrix Carleson Embedding Theorem by providing new proofs, exploring its connections to maximal functions and duality, and applying it to bound sparse operators in matrix weighted spaces.

Contribution

It introduces two new proofs of the matrix Carleson Embedding Theorem, establishes boundedness of related maximal functions, and derives operator norm bounds for sparse operators in matrix $A_2$ weights.

Findings

New proofs of the matrix Carleson Embedding Theorem.

Boundedness results for matrix-associated maximal functions.

Operator norm bound for sparse operators in matrix weighted spaces.

Abstract

In this paper, we study the dyadic Carleson Embedding Theorem in the matrix weighted setting. We provide two new proofs of this theorem, which highlight connections between the matrix Carleson Embedding Theorem and both maximal functions and $H^1$-BMO duality. Along the way, we establish boundedness results about new maximal functions associated to matrix $A_2$ weights and duality results concerning $H^1$ and BMO sequence spaces in the matrix setting. As an application, we then use this Carleson Embedding Theorem to show that if $S$ is a sparse operator, then the operator norm of $S$ on $L^2(W)$ satisfies: \[ \| S\|_{L^2(W) \rightarrow L^2(W)} \lesssim [W]_{A_2}^{\frac{3}{2}},\] for every matrix $A_2$ weight $W$.