Parabolic Muckenhoupt Weights on Spaces of Homogeneous Type

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.