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–Wiener 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–Ampère 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.
