KASS seminar

Abstract: Consider spaces A^p_α of analytic functions f in the unit disc D = {z : |z| < 1} such that |f (z)|^p is integrable with respect to the measure (α+1)(1 − |z|²)^αdxdy for some 0 < p < ∞, α > −1. It was essentially known since the work of Hardy and Littlewood which of these spaces are contained in which. In this talk we will concern ourselves with the following question: when are these embeddings contractions? The main result that we will discuss is that if p/(α+2) is held constant then the A^p_α norms are decreasing in p, the proof of which notably uses isoperimetric inequality in the hyperbolic plane. If time permits we will also mention the analogous results in other geometries as well as higher-dimensional versions, some of which are still conjectural.