Parabolic Muckenhoupt Weights on Spaces of Homogeneous Type

TL;DR

This paper develops a theory of parabolic Muckenhoupt weights on spaces of homogeneous type, establishing key inequalities, factorizations, and connections to BMO spaces, extending Euclidean parabolic analysis to metric measure spaces.

Contribution

It introduces a characterization of parabolic Muckenhoupt weights via weighted inequalities, reverse H"older inequalities, and factorization results on spaces of homogeneous type.

Findings

Characterization of weights via weighted norm inequalities

A reverse H"older inequality for parabolic weights

A Jones-type factorization for parabolic weights

Abstract

This work discusses parabolic Muckenhoupt weights on spaces of homogeneous type, i.e.\ quasi-metric spaces with both a doubling measure and an additional monotone geodesic property. The main results include a characterization in terms of weighted norm inequalities for parabolic maximal operators, a reverse H\"older inequality, and a Jones-type factorization result for this class of weights. The connection between the space of parabolic bounded mean oscillation and parabolic Muckenhoupt weights is studied by applying a parabolic John--Nirenberg lemma. A Coifman--Rochberg-type characterization of the space of parabolic bounded mean oscillation in terms of parabolic maximal functions is also given. The main challenges in the parabolic theory are related to the time lag in the estimates. The results are motivated by the corresponding Euclidean theory and the regularity theory for parabolic variational problems on metric measure spaces.