A Sharpened Rearrangement Inequality for Convolution on the Sphere

TL;DR

This paper proves a sharpened rearrangement inequality for a convolution form on the sphere, showing that symmetric rearrangements uniquely maximize the form for indicator functions under certain conditions.

Contribution

It establishes a new sharpened inequality that characterizes the unique maximizers as symmetric rearrangements for indicator functions on the sphere.

Findings

Symmetric rearrangements are the unique maximizers under natural hypotheses.

The inequality sharpens the classical Baernstein-Taylor symmetrization result.

The result applies specifically to indicator functions on the sphere.

Abstract

One may define a trilinear convolution form on the sphere involving two functions on the sphere and a monotonic function on the interval $[-1,1]$. A symmetrization inequality of Baernstein and Taylor states that this form is maximized when the two functions on the sphere are replaced with their nondecreasing symmetric rearrangements. In the case of indicator functions, we show that under natural hypotheses, the symmetric rearrangements are the only maximizers up to symmetry by establishing a sharpened inequality.