Restricted type estimates on the Bergman projection of some singular domains

TL;DR

This paper establishes weighted restricted type estimates for the Bergman projection on monomial polyhedra, generalizing known results and revealing potential failures of weak type bounds at domain-specific endpoints.

Contribution

It introduces new weighted restricted type estimates for the Bergman projection on monomial polyhedra, extending previous $L^p$ boundedness results and identifying cases of failure at certain endpoints.

Findings

Weighted restricted type estimates are obtained.

Recovers $L^p$ boundedness results for the Bergman projection.

Shows potential failure of weak type bounds at specific endpoints.

Abstract

We obtain (weighted) restricted type estimates for the Bergman projection operator on monomial polyhedra, a class of domains generalizing the Hartogs triangle. From these estimates, we recapture $L^p$ boundedness results of the Bergman projection on these domains. On some monomial polyhedra, we also discover that the Bergman projection could fail to be of weak type $(q_*,q_*)$ where $q_*$ is the right endpoint of the interval of $L^p$-regularity of the domain.