Two weight L^{p} inequalities for fractional vector Riesz transforms and doubling measures
Abstract
If T is a fractional vector Riesz transform, 1<p<infinity, and sigma and omega are doubling measures, then the two weight L^{p} norm inequality holds if and only if the quadratic triple testing conditions of Hyt\"onen and Vuorinen hold. We also show that these quadratic triple testing conditions can be relaxed to quadratic local testing conditions, quadratic offset Muckenhoupt conditions, and a quadratic weak boundedness property.
Keywords
Cite
@article{arxiv.2211.01920,
title = {Two weight L^{p} inequalities for fractional vector Riesz transforms and doubling measures},
author = {Eric T. Sawyer and Brett D. Wick},
journal= {arXiv preprint arXiv:2211.01920},
year = {2024}
}
Comments
51 pages, thanks to a referee for many helpful comments leading to the current version. The main change in the current argument is the use of Haar wavelets instead of Alpert wavelets. This has no effect on the main theorem, but does eliminate one of our remarks regarding where the restriction to vector Riesz transforms is needed. Thanks also to I. Uriarte-Tuero, M. Alexis and J.-L. Luna-Garcia