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 on R × R^d × R^d, with respect to which the operators considered are 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.