Sharp mixed $A_p$-$A_\infty$ estimates for sparse operators on filtered and nonhomogeneous measure spaces

Abstract

We prove mixed $A_p$-$A_\infty$ estimates for sparse operators in two non-doubling settings. In the first setting, we consider sparse operators defined using stopping times in continuous time. We obtain both strong- and weak-type bounds with the same powers of the weight characteristics as in the classical setting. Some of our weak-type bounds are even new for the dyadic filtration on $\mathbb R^d$ and, in particular, imply a sharp weak-type $(2,2)$ estimate for Rubio de Francia square functions, solving a problem left open by Garg, Roncal and Shrivastava [J. Geom. Anal., 31:748-771, 2021]. In the second setting, we consider dyadic sparse forms in which distinct cubes may interact, provided their dyadic distance is bounded. We obtain strong-type bounds with the same powers of the weight characteristics as in the classical setting. As a one-dimensional application, we obtain strong-type bounds for Haar shifts over balanced non-doubling measures, answering a quantitative question posed by Conde-Alonso, Pipher, and Wagner [Math. Ann., 391:2209-2253, 2025].