Harmonic Analysis Seminar: Maximizers of Rogers-Brascamp-Lieb-Luttinger functionals in higher dimensions

Seminar | January 30 | 1:10-2 p.m. | 736 Evans Hall

 Kevin O'Neill, UC Berkeley

 Department of Mathematics

The Riesz-Sobolev inequality states that for indicator functions $f,g,h$ of subsets of $ℝ^d$ of specified measure, the functional $\iint_{ℝ^d\times ℝ^d} f(x)g(y)h(x+y)dxdy$ is maximized when the subsets are balls centered at the origin. Burchard '96 showed that under suitable hypotheses, the functional is maximized precisely when the subsets are balls centered at the origin and their orbit under its symmetries (translation and the diagonal action of the special linear group).

In this talk, I will discuss a similar characterization of maximizers for a generalization of the Riesz-Sobolev inequality known as the Rogers-Bracsamp-Lieb-Luttinger inequality. A byproduct of the method of proof is a sharpened version of the Rogers-Brascamp-Lieb-Luttinger inequality, controlling the underlying functional by its value when the sets are balls centered at the origin minus a quadratic term involving the distance of the given sets from maximizers.

Results are joint work with Michael Christ.