Mapping Properties of Weighted Bergman Projection Operators on Reinhardt Domains

TL;DR

This paper investigates the boundedness of weighted Bergman projection operators on Reinhardt domains, revealing unboundedness on L^p spaces for p≠2 but boundedness on Sobolev spaces on the unit ball.

Contribution

It demonstrates the unboundedness of exponentially weighted Bergman projections on L^p spaces on Reinhardt domains and their boundedness on Sobolev spaces on the unit ball.

Findings

Weighted Bergman projections are unbounded on L^p for p≠2 on smooth complete Reinhardt domains.

Exponentially weighted projections are bounded on Sobolev spaces on the unit ball.

The results highlight domain and weight-dependent boundedness properties of Bergman projections.

Abstract

We show that on smooth complete Reinhardt domains, weighted Bergman projection operators corresponding to exponentially decaying weights are unbounded on $L^p$ spaces for all $p\not=2$. On the other hand, we also show that the exponentially weighted projection operators are bounded on Sobolev spaces on the unit ball.