H\"ormander's theorem for parabolic equations with coefficients measurable in the time variable

TL;DR

This paper proves that solutions to certain degenerate parabolic equations with measurable time coefficients are infinitely differentiable in space if the coefficients satisfy H"ormander's condition, extending regularity results under minimal temporal regularity.

Contribution

It establishes spatial regularity of solutions for degenerate parabolic equations with coefficients only measurable in time, under H"ormander's hypoellipticity condition.

Findings

Solutions are infinitely differentiable in space variables.

Regularity holds despite coefficients being only measurable in time.

Extends hypoelliptic regularity to time-measurable coefficient setting.

Abstract

We are dealing with possibly degenerate second-order parabolic operators whose coefficients are infinitely differentiable with respect to space variables and only measurable with respect to the time variable. We impose the H\"ormander condition on the diffusion coefficients and prove that the solutions of the corresponding equations with right-hand sides which are infinitely differentiable in the space variables in a space-time domain have also this property.