A smoothing property of the Bergman projection

TL;DR

This paper investigates the smoothing properties of the Bergman projection on certain domains, showing that derivatives of the projection are controlled by derivatives in a specific direction, and characterizing functions mapped to smooth functions.

Contribution

It establishes a new smoothing property of the Bergman projection related to derivatives in a distinguished direction on domains with compact dbar-Neumann operator.

Findings

Derivatives of Bf are controlled by derivatives of f in a single direction.

Functions mapped to smooth functions by B are explicitly characterized.

Provides insight into boundary regularity of the Bergman projection.

Abstract

Let B be the Bergman projection associated to a domain on which the dbar-Neumann operator is compact. We show that arbitrary L^2 derivatives of Bf are controlled by derivatives of f taken in a single, distinguished direction. As a consequence, functions that are not smooth up to the boundary but are mapped by B to functions which are smooth up to the boundary are explicitly described.