Quasi-extremals for convolution with surface measure on the sphere

TL;DR

This paper characterizes quasi-extremal pairs for convolution with surface measure on the sphere, showing they are essentially comparable to a specific family, extending previous work on paraboloids.

Contribution

The paper explicitly defines a family of quasi-extremal pairs for the spherical convolution operator and proves their fundamental role in approximating all such pairs, extending Christ's paraboloid results.

Findings

Every quasi-extremal pair is comparable to a pair in the family $\\mathcal{F}$

The family $\\mathcal{F}$ captures all quasi-extremal behavior for the operator

Extension of previous paraboloid results to spherical surface measure

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 $\langle T\chi_E, \chi_F \rangle \gtrsim |E|^{d/(d+1)}|F|^{d/(d+1)}$. 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.