Intrinsic Hölder spaces for fractional kinetic operators

We introduce anisotropic Hölder spaces that are useful for studying the regularity theory for non-local kinetic operators. The Hölder spaces are defined in terms of an anisotropic distance relevant to the Galilean geometric structure, with respect to which the operator is invariant. We prove an intrinsic Taylor-like formula, whose remainder is bounded in terms of the anisotropic distance of the Galilean structure. Our achievements naturally extend analogous known results for purely differential operators on Lie groups.