Endpoint estimates and sparse domination in nonhomogeneous trees

TL;DR

This paper establishes endpoint and sparse bounds for Bergman projectors on nonhomogeneous radial trees, introducing a new Calderón-Zygmund theory for non-doubling metric spaces with applications to weighted inequalities.

Contribution

It develops a novel Calderón-Zygmund framework for non-locally doubling spaces and applies it to Bergman projections on nonhomogeneous trees, extending classical results.

Findings

Proved endpoint bounds for Bergman projectors on nonhomogeneous trees.

Established sparse domination results in non-doubling metric spaces.

Derived weighted inequalities consistent with classical disk results.

Abstract

We prove endpoint and sparse-like bounds for Bergman projectors on nonhomogeneous, radial trees $X$ that model manifolds with possibly unbounded geometry. The natural Bergman measures on $X$ may fail to be doubling, and even locally doubling, with respect to the right metric in our setting. Weighted consequences of our sparse domination results are also considered, and are in line with the known results in the disk. Our endpoint results are partly a consequence of a new Calder\'on-Zygmund theory for discrete, non-locally doubling metric spaces.