Harmonic Analysis Seminar: On maximizers of generalized Riesz-Sobolev functionals

Seminar | January 23 | 1:10-2 p.m. | 736 Evans Hall

Michael Christ, UCB

Department of Mathematics

The elementary Riesz-Sobolev inequality, which dates to the 1930s, is concerned with the functional $\iint _{\mathbb R^d\times \mathbb R^d} f(x)g(y)h(x+y)\,dx\,dy$, and states that among indicator functions $f,g,h$ of subsets of $\mathbb R^d$ of specified Lebesgue measures, those sets for which the functional attains its maximum value are balls centered at the origin. Burchard's theorem states that under a natural hypothesis on the specified measures, these are the only maximizing configurations, up to a natural action of the group $Sl(d)$ and of translations in $\mathbb R^d$. The Brascamp-Lieb-Luttinger-Rogers inequality generalizes the inequality and involves tuples of linear mappings from $(\mathbb R^d)^m$ to $\mathbb R^d$ that satisfy a certain symmetry hypothesis. A similar characterization of maximizers is known to hold.

I will discuss some partial results concerning the nature of maximizers of integral functionals of this type when the symmetry hypothesis is dropped, indicating some ingredients in the proofs without full details. Ideas used in the classical theory of Riesz, Sobolev, Brascamp-Lieb-Luttinger, Rogers, and Burchard will also be sketched. The new results are joint work with Dominique Maldague.

After this exceptional initial meeting, this will be a student-oriented seminar devoted to systematic presentation of relatively recent works of Bourgain-Demeter and of Guth.

mchrist@berkeley.edu