Around $L^1$ (un)boundedness of Bergman and Szeg\"o projections

TL;DR

This paper investigates the conditions under which Bergman and Szeg"o projections are bounded or unbounded on $L^1$ spaces across various complex domains and CR manifolds, expanding understanding of their functional analysis properties.

Contribution

It provides new insights into the $L^1$ boundedness and unboundedness of these projections on broad classes of complex manifolds and CR structures.

Findings

Identifies conditions for $L^1$ boundedness of Bergman projections.

Establishes criteria for $L^1$ unboundedness of Szeg"o projections.

Extends previous results to more general complex and CR manifolds.

Abstract

We consider the problem of $L^1$ (un)boundedness for a wide class of orthogonal projections, including Bergman projections on domains in complex manifolds and Szeg\"o projections on abstract CR manifolds.