Quasi-extremals for convolution with surface measure on the sphere

Abstract

If $T$ is the operator given by convolution with surface measure on the sphere, $(E,F)$ is a quasi-extremal pair of sets for $T$ if $⟨T{\chi }_E,{\chi }_F⟩\gtrsim |E{|}^{d/(d+1)}|F{|}^{d/(d+1)}$. In this article, we explicitly define a family $\mathcal{F}$ of quasi-extremal pairs of sets for $T$. We prove that $\mathcal{F}$ is fundamental in the sense that every quasi-extremal pair $(E,F)$ is comparable (in a rather strong sense) to a pair from $\mathcal{F}$. This extends work carried out by M. Christ for convolution with surface measure on the paraboloid.

This is a preprint version of an article published as Illinois J. Math. 53 (2009), no. 2, 391-412