Hypoelliptic estimates for linear transport operators

TL;DR

This paper investigates how non-commutation between linear transport operators and fractional diffusion leads to hypoelliptic estimates, enhancing understanding of regularity in kinetic equations using classical multiplier methods.

Contribution

It introduces a broad class of linear transport operators for which hypoelliptic estimates are established via multiplier techniques, advancing the analysis of kinetic equations.

Findings

Hypoelliptic estimates are proven for various linear transport operators.

The multiplier method effectively demonstrates regularity properties.

Results contribute to the understanding of kinetic equation regularity.

Abstract

We aim at understanding how the non-commutation phenomena between a linear transport operator and a fractional diffusion allow the transport operator to satisfy hypoelliptic estimates on the whole space. Such hypoelliptic estimates are obtained for a large class of linear transport operators by using the classical multiplier method. The main motivation of this work arises from the study of the hypoelliptic regularity of the solutions of kinetic equations associated with a free transport operator.