Sharp Schauder Estimates for some Degenerate Kolmogorov Equations

TL;DR

This paper establishes sharp Schauder estimates for certain degenerate Kolmogorov PDEs with anisotropic Hölder coefficients and nonlinear unbounded first-order terms, using a perturbative approach and duality in Besov spaces.

Contribution

It introduces a novel perturbative method based on forward parametrix expansions and duality techniques to obtain Schauder estimates for degenerate PDEs, offering an alternative to classical approaches.

Findings

Sharp Schauder estimates are derived for degenerate Kolmogorov equations.

The method applies to PDEs with anisotropic Hölder coefficients and nonlinear unbounded terms.

An alternative constructive approach to Schauder estimates is provided even in the non-degenerate case.

Abstract

We provide here some sharp Schauder estimates for degenerate PDEs of Kolmogorov type when the coefficients lie in some suitable anisotropic H{\"o}lder spaces and the first order term is non-linear and unbounded. We proceed through a perturbative approach based on forward parametrix expansions. Due to the low regularizing properties of the degenerate variables, for the procedure to work, we heavily exploit duality results between appropriate Besov spaces. Our method can be seen as constructive and provides, even in the non-degenerate case, an alternative approach to Schauder estimates.