Title: On the extremizers of a certain L^p norm inequality
Speaker: Michael Christ
Speaker Info: University of California, Berkeley
Brief Description:
Special Note:
Abstract:
Over the last few decades, a vast literature has developed concerning the mapping properties, in the scale of L^p spaces, of linear integral operators which involve singularities and/or curvature. Some of these inequalities have been established by exploiting a combinatorial perspective. There are natural inverse problems: Characterize those functions which extremize the inequalities up to bounded factors; determine whether extremizers exist; identify these if possible; determine their quantitative and qualitative properties in cases when identification is not possible. I will outline a series of works on these problems, for one of the most canonical of the Radon-like transforms. The most recent result is that extremizers for this particular inequality are infinitely differentiable. This is established via analysis of regularity of solutions of a certain nonlinear Euler-Lagrange equation. The main tool is a new family of weighted L^p norm inequalities. (joint work with Qingying Xue)Date: Friday, February 25, 2011