  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.
