Caltech/UCLA Joint Analysis Seminar
UCLA, MS 6627
It has long been understood that Strichartz estimates for the homogeneous Schrödinger equation correspond to adjoint Fourier restriction estimates on the paraboloid. The study of extremizers and sharp constants for the corresponding inequalities has a short but rich history. In this talk, I will summarize it briefly, and then specialize to the case of certain planar power curves. A geometric comparison principle for convolution measures can be used to establish the corresponding sharp Strichartz inequality, and to decide whether extremizers exist. The mechanism underlying the possible lack of compactness is explained by the behaviour of extremizing sequences and will be described via concentration-compactness. Time permitting, I will show how this resolves a dichotomy from the recent literature concerning the existence of extremizers for the fourth order Schrödinger equation in one spatial dimension. This talk is based on joint work with Gianmarco Brocchi and René Quilodrán.