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arXiv · 2309.13943

Balanced measures, sparse domination and complexity-dependent weight classes

José M. Conde-Alonso·Jill Pipher·Nathan A. Wagner
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Abstract

We study sparse domination for operators defined with respect to an atomic filtration on a space equipped with a general measure $μ$. In the case of Haar shifts, $L^p$-boundedness is known to require a weak regularity condition, which we prove to be sufficient to have a sparse domination-like theorem. Our result allows us to characterize the class of weights where Haar shifts are bounded. A surprising novelty is that said class depends on the complexity of the Haar shift operator under consideration. Our results are qualitatively sharp.

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José M. Conde-Alonso, Jill Pipher, Nathan A. Wagner. 2023-09-25. Balanced measures, sparse domination and complexity-dependent weight classes. https://arxiv.org/abs/2309.13943

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