Conditional Carleson measures and related operators on Bergman spaces

TL;DR

This paper introduces and characterizes conditional Carleson measures on Bergman spaces, providing new insights into their properties and establishing criteria for the boundedness of related weighted conditional expectation operators.

Contribution

It defines generalized and conditional Carleson measures on Bergman spaces and characterizes their properties, linking them to operator boundedness.

Findings

Characterization of conditional Carleson measures on Bergman spaces

Equivalent condition for boundedness of weighted conditional expectation operators

New criteria connecting measures and operator boundedness

Abstract

In this paper first we define generalized Carleson mea- sure. Then we consider a special case of it, named conditional Carleson measure on the Bergman spaces. After that we give a characterization of conditional Carleson measures on Bergman spaces. Moreover, by using this characterization we find an equiv- alent condition to boundedness of weighted conditional expectation operators on Bergman spaces.