The spectrum of the Leray transform for convex Reinhardt domains in $\mathbb C^2$

TL;DR

This paper investigates the Leray transform on convex Reinhardt domains in a7^2, establishing $L^2$-regularity, computing spectra, and revealing a duality principle that relates operators on dual domains, including cases with less smooth boundaries.

Contribution

It introduces a self-dual class of convex Reinhardt domains, proves $L^2$-regularity, computes essential spectra, and establishes a duality principle with explicit unitary equivalence, extending results to less smooth boundaries.

Findings

Proved $L^2$-regularity for the Leray transform.

Computed essential spectra for boundary operators.

Established a duality principle linking operators on dual domains.

Abstract

The Leray transform and related boundary operators are studied for a class of convex Reinhardt domains in $\mathbb C^2$. Our class is self-dual; it contains some domains with less than $C^2$-smooth boundary and also some domains with smooth boundary and degenerate Levi form. $L^2$-regularity is proved, and essential spectra are computed with respect to a family of boundary measures which includes surface measure. A duality principle is established providing explicit unitary equivalence between operators on domains in our class and operators on the corresponding polar domains. Many of these results are new even for the classical case of smoothly bounded strongly convex Reinhardt domains.