We introduce anisotropic H & ouml;lder spaces that are useful for studying the regularity theory for non-local kinetic operators L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}$$\end{document}, whose prototypical example is Lu(t,x,v)=integral RdCd,s|v-v '|d+2s(u(t,x,v ')-u(t,x,v))dv '+< v,del x >+partial derivative t,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathscr {L}u (t,x,v) = \int _{{{\mathbb {R}}}<^>d} \frac{C_{d,s}}{|v - v'|<^>{d+2s}} (u(t,x,v') - u(t,x,v)) \textrm{d}v' + \langle v, \nabla _x \rangle + \partial _t, \end{aligned}$$\end{document}with (t,x,v)is an element of RxR2d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t,x,v)\in {{\mathbb {R}}}\times {{\mathbb {R}}}<^>{2d}$$\end{document}. The H & ouml;lder spaces are defined in terms of an anisotropic distance relevant to the Galilean geometric structure on RxR2d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}\times {{\mathbb {R}}}<^>{2d}$$\end{document}, with respect to which the operator L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}$$\end{document} 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.
