Some dual realizations of Bergman and Hardy spaces on convex domains via integral transforms

Abstract

The Fantappi{\`e} and Laplace transforms identify abstract spaces of analytic functionals with concrete function spaces, converting geometric support conditions into analytic growth conditions. Specifically, the space of analytic functionals carried by a compact convex set $K\subset\mathbb C^n$ is isomorphic, via the Fantappi{\`e} transform, to the space of holomorphic functions on the dual complement of $K$, and, via the Laplace transform, to a space of entire functions of exponential type. The latter result can be viewed as a type of Paley--Wiener theorem.

In this thesis, we consider the restrictions of the Fantappi{\`e} and Laplace transforms to special subspaces of the space of analytic functionals carried by the closure of a bounded convex domain in $\mathbb C^n$. These are the Bergman and (holomorphic) Hardy spaces of the convex domain. We establish normed space isomorphisms between these spaces and certain (weighted) Bergman spaces.

In the first half of the thesis, we discuss our results for Bergman spaces. For a bounded convex domain $D$ in the complex plane, Napalkov Jr. and Yulmukhametov (1995, 2004) have completely characterized the range of the Fantappi{\`e} and Laplace transforms restricted to the Bergman space of $D$. The former as the Bergman space of the dual complement of $D$, and the latter as a weighted Bergman space of entire functions. We extend these results to higher dimensions under certain strong convexity assumptions on $D$. We also provide counterexamples to show that the planar results do not extend to higher dimensions in full generality.

In the latter half, we turn our attention to Hardy spaces. For holomorphic Hardy spaces of bounded convex domains in the complex plane, Lutsenko and Yulmukhametov (1991) have established results analogous to those in the Bergman-space setting. Lindholm (2002) has extended these results to strongly convex domains in higher dimensions. We characterize the Laplace transforms of Hardy-space functions on a class of bounded $\mathcal C^1$-smooth (weakly) convex Reinhardt domains in $\mathbb {C}^2$, which are modeled by egg-type domains away from the coordinate axes. These domains were previously considered by Barrett and Lanzani (2009) to study the $L^2$-boundedness of the so-called Leray transform, a higher-dimensional analogue of the Cauchy transform, which plays a key role in our analysis.