Geometric maximizers of Schatten norms of some convolution type integral operators

TL;DR

This paper proves that for certain convolution integral operators with non-increasing kernels, the ball maximizes the Schatten p-norm among domains of equal measure, and the equilateral triangle maximizes it among triangles of equal area.

Contribution

It establishes geometric maximizers for Schatten norms of convolution operators, identifying the ball and equilateral triangle as extremal shapes.

Findings

The ball maximizes Schatten p-norms among all domains of fixed measure.

The equilateral triangle maximizes Schatten p-norms among triangles of fixed area.

Results have physical motivations and implications.

Abstract

In this paper we prove that the ball is a maximizer of the Schatten $p$-norm of some convolution type integral operators with non-increasing kernels among all domains of a given measure in $\mathbb R^{d}$. We also show that the equilateral triangle has the largest Schatten $p$-norm among all triangles of a given area. Some physical motivations for our results are also presented.