Intrinsic Hölder spaces for fractional kinetic operators

We introduce anisotropic Hölder 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 Hölder 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.