Intrinsic H\"older spaces for fractional kinetic operators

TL;DR

This paper develops anisotropic H"older spaces tailored for fractional kinetic operators, establishing an intrinsic Taylor formula that respects the Galilean geometric structure, advancing regularity theory for these non-local operators.

Contribution

Introduction of anisotropic H"older spaces for fractional kinetic operators and proof of an intrinsic Taylor formula aligned with the Galilean structure.

Findings

Defined anisotropic H"older spaces for kinetic operators

Proved an intrinsic Taylor-like expansion with anisotropic distance estimates

Extended regularity results to non-local kinetic operators

Abstract

We introduce anisotropic H\"older spaces useful for the study of the regularity theory for non local kinetic operators $\mathcal{L}$ whose prototypal example is \begin{equation} \mathcal{L} u (t,x,v) = \int_{\mathbb{R}^d} \frac{C_{d,s}}{|v - v'|^{d+2s}} (u(t,x,v') - u(t,x,v)) d v' + \langle v , \nabla_x \rangle + \partial_t, \quad (t,x,v)\in\mathbb{R}\times\mathbb{R}^{2d}. \end{equation} The H\"older spaces are defined in terms of an anisotropic distance relevant to the Galilean geometric structure on $\mathbb{R}\times\mathbb{R}^{2d}$ the operator $\mathcal{L}$ is invariant with respect to. We prove an intrinsic Taylor-like formula, whose reminder is estimated in terms of the anisotropic distance of the Galilean structure. Our achievements naturally extend analogous known results for purely differential operators on Lie groups.