Dominating Sets in Bergman Spaces on Strongly Pseudoconvex Domains

TL;DR

This paper establishes local estimates and three sphere inequalities for analytic functions in several variables, characterizing dominating sets in Bergman spaces on strongly pseudoconvex domains through density and kernel testing conditions.

Contribution

It introduces new propagation of smallness estimates and characterizes dominating sets in Bergman spaces using boundary separation and kernel conditions.

Findings

Derived local estimates for analytic functions in multiple variables.

Established a three sphere-type inequality with flexible boundary sets.

Characterized dominating sets in Bergman spaces via density and kernel testing conditions.

Abstract

We obtain local estimates, also called propagation of smallness or Remez-type inequalities, for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be any sets satisfying a boundary separation property, and the inner sphere can be any set of positive Lebesgue measure. We apply this local result to characterize the dominating sets for Bergman spaces on strongly pseudoconvex domains in terms of a density condition or a testing condition on the reproducing kernels. Our methods also yield a sufficient condition for arbitrary domains and lower-dimensional sets.