A Spectral Exploration of the Leray Transform in Two Different Settings in C2

Abstract

We are concerned with the Leray transform (referred to as Cauchy-Leray or Leray-Aizenberg by some authors), a skew projection acting on the L^2 space of functions defined on the boundary of a suitable domain D in C^n, mapping onto the subspace of boundary values of holomorphic functions in D. We will focus on the case n=2 in two settings, namely convex Reinhardt domains (based on the results of D. Barrett and L. Lanzani) and the so-called rigid Hartogs domains. The starting point is the spectrum of the Leray transform, which depends on the boundary measure. We will extract information about the norm for l_p balls, and more generally about the essential norm and a particular variant. Then we will explore our ability to "hear" a convex Reinhardt domain based on its Leray spectrum. To what extent can we recover the domain?

In the rigid Hartogs setting, the computation of the Leray spectrum will be done for a family of measures. Then we will be able to explore similar topics to the above.