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, 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 term was coined by L. Edholm, but no paper has been published yet). The starting point is the singular spectrum of the Leray transform, which depends on the boundary measure. While these computations have been for convex Reinhardt domains, we will extract information about the norm (for l_p balls), 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 from its singular spectrum?

In the rigid Hartogs setting, the computation of the Leray spectrum will be carried out (using a result of L. Edholm) for a family of measures. Then we will be able to explore similar topics to the above.

Yonatan's advisor is David Barrett. Speaker(s): Yonatan Shelah (UM)