We introduce anisotropic Hölder spaces that are useful for studying the regularity theory for non-local kinetic operators \(\mathscr {L}\) , whose prototypical example is \(\begin{aligned} \mathscr {L}u (t,x,v) = \int _{{{\mathbb {R}}}^d} \frac{C_{d,s}}{|v - v'|^{d+2s}} (u(t,x,v') - u(t,x,v)) \textrm{d}v' + \langle v, \nabla _x \rangle + \partial _t, \end{aligned}\) with \((t,x,v)\in {{\mathbb {R}}}\times {{\mathbb {R}}}^{2d}\) . The Hölder spaces are defined in terms of an anisotropic distance relevant to the Galilean geometric structure on \({{\mathbb {R}}}\times {{\mathbb {R}}}^{2d}\) , with respect to which the operator \(\mathscr {L}\) 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.
