Spatial regularity for a class of degenerate Kolmogorov equations

TL;DR

This paper proves that solutions to certain degenerate Kolmogorov equations gain regularity in all spatial directions, using commutator identities to establish Sobolev estimates without relying on integrability assumptions.

Contribution

It introduces a novel approach based on commutator identities to obtain spatial regularity for degenerate Kolmogorov equations, independent of right-hand side integrability.

Findings

Established spatial a priori estimates for solutions.

Proved solutions are regular in all spatial directions.

Provided an alternative proof for optimal spatial regularity.

Abstract

We establish spatial a priori estimates for the solution u to a class of dilation invariant Kolmogorov equation, where u is assumed to only have a certain amount of regularity in the diffusion's directions. The result is that u is also regular with respect to the remaining directions. The approach we propose is based on the commutators identities and allows us to obtain a Sobolev exponent that does not depend on the integrability assumption of the right-hand side. Lastly, we provide an alternative proof to that of Theorem 1.5 of [9] for the optimal spatial regularity.