Matrix weights, Littlewood Paley inequalities and the Riesz transforms

TL;DR

This paper investigates weighted estimates for Riesz transforms with matrix weights, showing that near-identity weights lead to operator norms close to the unweighted case, using Bellman functions.

Contribution

It introduces a new approach to estimate Riesz transforms with matrix weights, especially when weights are close to the identity, via Bellman function techniques.

Findings

Operator norm of R^{2} is close to 1 when W is near the identity.

Weighted estimates depend continuously on the matrix weight W.

Bellman function method effectively handles matrix weight inequalities.

Abstract

We discuss weighted estimates for the squares of the Riesz transforms, R^{2}, on L^{2}(W) where W is a matrix A2 weight. We prove that if W is close to the Identity matrix Id, then the operator norm of R^{2} is close to its unweighted norm on L^{2} which is one. This is done by the use of the Bellman function technique.