Irregularity of the Szeg\"o Projection on Bounded Pseudoconvex Domains in $\mathbb{C}^2$

TL;DR

This paper constructs specific bounded pseudoconvex domains in complex two-dimensional space where the Szeg"o projection operator is unbounded on all L^p spaces except when p=2, revealing irregularity in boundary projections.

Contribution

It provides explicit examples of pseudoconvex domains with unbounded Szeg"o projections on L^p spaces for all p not equal to 2, highlighting irregular boundary behavior.

Findings

Szeg"o projection is unbounded on L^p for all p ≠ 2

Constructs explicit pseudoconvex domains with irregular boundary projections

Shows limitations of Szeg"o projection regularity in complex analysis

Abstract

We construct bounded pseudoconvex domains in $\mathbb{C}^2$ for which the Szeg\"o projection operators are unbounded on $L^p$ spaces of the boundary for all $p\not =2$.