Holomorphic function spaces on the Hartogs triangle

TL;DR

This paper introduces a natural family of holomorphic function spaces on the Hartogs triangle, including weighted Bergman, Hardy, and Dirichlet spaces, and studies their properties and relationships.

Contribution

It defines new canonical holomorphic function spaces on the Hartogs triangle and analyzes their projection properties and isometric relations.

Findings

Weighted Bergman and Hardy projections have specific $L^p$ mapping properties.

The Dirichlet space on the Hartogs triangle is isometric to that on the bidisc.

Abstract

The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the $L^p$ mapping properties of Bergman and Szeg\H{o} projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc.