Estimates for the Boltzmann collision operator via radial symmetry and Fourier transform

TL;DR

This paper advances the mathematical understanding of the Boltzmann collision operator by leveraging radial symmetry and Fourier analysis to simplify proofs, extend theoretical results, and derive explicit sharp constants in related inequalities.

Contribution

It introduces new techniques exploiting radial symmetry to extend the $L^p$-theory and obtain explicit sharp constants, simplifying existing proofs and broadening the operator's analytical framework.

Findings

Extended $L^p$-theory for the Boltzmann collision operator

Simplified proofs using radial symmetry and Fourier analysis

Derived explicit sharp constants in convolution-like inequalities

Abstract

We extend the $L^p$-theory of the Boltzmann collision operator by using classical techniques based in the Carleman representation and Fourier analysis, allied to new ideas that exploit the radial symmetry of this operator. We are then able to greatly simplify existent technical proofs in this theory, extend the range, and obtain explicit sharp constants in some convolution-like inequalities for the gain part of the Boltzmann collision operator.