Schauder estimates for degenerate stable Kolmogorov equations

TL;DR

This paper establishes global Schauder estimates for a class of degenerate stable Kolmogorov equations, enabling the analysis of their regularity and well-posedness in anisotropic Hölder spaces.

Contribution

It introduces a perturbative approach using forward parametrix expansions and duality in Besov spaces to handle degenerate and super-critical cases.

Findings

Derived Schauder estimates for degenerate stable IPDEs.

Proved well-posedness of the equations in suitable functional spaces.

Applicable to super-critical cases with low regularizing properties.

Abstract

We provide here global Schauder-type estimates for a chain of integro-partial differential equations (IPDE) driven by a degenerate stable Ornstein-Uhlenbeck operator possibly perturbed by a deterministic drift, when the coefficients lie in some suitable anisotropic H{\"o}lder spaces. Our approach mainly relies on a perturbative method based on forward parametrix expansions and, due to the low regularizing properties on the degenerate variables and to some integrability constraints linked to the stability index, it also exploits duality results between appropriate Besov Spaces. In particular, our method also applies in some super-critical cases. Thanks to these estimates, we show in addition the well-posedness of the considered IPDE in a suitable functional space.