Stein's theorem in the upper-half plane and Bergman spaces with weights

TL;DR

This paper extends Stein's theorem to weighted Bergman spaces in the upper-half plane, establishing a converse condition and exploring dual spaces, multipliers, and operators with logarithmic weights.

Contribution

It introduces a converse to Stein's theorem in the complex upper-half plane and studies weighted Bergman spaces with logarithmic weights, including their duals and operator properties.

Findings

Established a converse Stein's theorem for weighted Bergman spaces.

Characterized dual spaces as logarithmic Bloch type spaces.

Analyzed multipliers, pointwise products, and Hankel operators in this setting.

Abstract

This is a companion paper to our previous one, Avatars of Stein's Theorem in the complex setting. In this previous paper, we gave a sufficient condition for an integrable function in the upper-half plane to have an integrable Bergman projection. Here we push forward methods and establish in particular a converse statement. This naturally leads us to study a family of weighted Bergman spaces for logarithmic weights (1 + ln + (1/___m(z)) + ln + (|z|)) k , which have the same kind of behavior respectively at the boundary and at infinity. We introduce their duals, which are logarithmic Bloch type spaces and interest ourselves in multipliers, pointwise products and Hankel operators.