The Laplace and Leray transforms on some (weakly) convex domains in $\mathbb{C}^2$

TL;DR

This paper extends the characterization of Laplace transforms of Hardy-space functions to certain weakly convex Reinhardt domains in ^2, demonstrating the boundedness of the Leray transform and providing new insights into convex domain analysis.

Contribution

It establishes a Paley--Weiner type theorem for weakly convex Reinhardt domains modeled by egg domains, expanding the class of domains with bounded Leray transform.

Findings

Characterization of Laplace transforms on weakly convex Reinhardt domains

Expansion of convex Reinhardt domains with bounded Leray transform

Counterexample showing limitations of previous convex domain results

Abstract

The space of Laplace transforms of holomorphic Hardy-space functions have been characterized as weighted Bergman spaces of entire functions in two cases: that of planar convex domains (Lutsenko--Yumulmukhametov, 1991), and that of strongly convex domains in higher dimensions (Lindholm, 2002). In this paper, we establish such a Paley--Weiner result for a class of (weakly) convex Reinhardt domains in $\mathbb{C}^2$ that are well-modelled by the so-called egg domains. We consider Hardy spaces on these domains with respect to a canonical choice of boundary Monge--Ampere measure. This class of domains was introduced by Barrett--Lanzani (2009) to study the $L^2$-boundedness of the Leray transform in the absence of either strongly convexity or $\mathcal{C}^2$-regularity. The boundedness of the Leray transform plays a crucial role in understanding the image of the Laplace transform. As a supplementary result, we expand the known class of convex Reinhardt domains for which the Leray transform is $L^2$-bounded (with respect to the aforementioned choice of boundary measure). Finally, we also produce an example to show that the Lutsenko--Yumulmukhametov result cannot be expected to generalize to all convex domains in higher dimensions.