Convex Reinhardt domains lend themselves naturally to spectral analysis of the Leray transform due to their rotational symmetries. In two dimensions and under some hypotheses, D. Barrett and L. Lanzani calculated the SVD spectrum (including the essential part) in terms of integrals involving a useful parametrization, giving conditions for the operator to be bounded. I will discuss the following applications:
1. Calculating the norms for l_p balls, the natural objects of the theory, by finding the maximum of each spectrum.
2. Tackling the inverse problem: Can you "hear" the shape of a bounded, sufficiently smooth, convex Reinhardt domain in C^2? If possible, this can only be done up to the natural symmetries: scaling, variable swap and taking the polar/dual domain. I will describe two approaches to the problem which show that the question has a positive answer at least in some natural cases, assuming that our "hearing" is sensitive enough to recognize the Fourier labeling of the point spectrum (a sequence rather than a set).
3. Calculating a family of measures supported on the essential spectrum, which are defined via limits. These play a key role in the aforementioned problem (specifically the second approach), and may be of separate interest.
Despite the underlying topic, all of the tools used are from real analysis. Very little background is assumed. Speaker(s): Yonatan Shelah (University of Michigan)