Martingale transforms, the dyadic shift and the Hilbert transform: a sufficient condition for boundedness between matrix weighted spaces

TL;DR

This paper establishes sufficient matrix weight conditions ensuring the boundedness of martingale transforms, dyadic shifts, and the Hilbert transform between weighted L^2 spaces, advancing understanding of operator boundedness in matrix-weighted analysis.

Contribution

It introduces new sufficient conditions on matrix weights that guarantee the boundedness of key operators like martingale transforms, dyadic shifts, and the Hilbert transform between matrix-weighted L^2 spaces.

Findings

Martingale transforms are bounded under the new matrix weight conditions.

Dyadic shifts are uniformly bounded given the same conditions.

The Hilbert transform is bounded between these matrix-weighted spaces.

Abstract

We show sufficient conditions on matrix weights $U$ and $V$ for the martingale transforms to be uniformly bounded from $L^2(V)$ to $L^2(U)$. We also show that these conditions imply the uniform boundedness of the dyadic shifts as well as the boundedness of the Hilbert transform between these spaces.