Smoothing estimates for the kinetic transport equation at the critical regularity

TL;DR

This paper establishes smoothing estimates for the kinetic transport equation at critical regularity, characterizing the precise regularity exponents and extending results to cone and paraboloid multipliers.

Contribution

It provides the first complete characterization of regularity exponents for smoothing estimates at the critical level, including new mixed-norm estimates for cone and paraboloid multipliers.

Findings

Complete characterization of regularity exponents for smoothing estimates.

New mixed-norm estimates for cone multiplier operators.

Extension of estimates to paraboloid geometry.

Abstract

We prove smoothing estimates for velocity averages of the kinetic transport equation in hyperbolic Sobolev spaces at the critical regularity, leading to a complete characterisation of the allowable regularity exponents. Such estimates will be deduced from some mixed-norm estimates for the cone multiplier operator at a certain critical index. Our argument is not particular to the geometry of the cone and we illustrate this by establishing analogous estimates for the paraboloid.