Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

Abstract

<p style='text-indent:20px;'>We study the regularity properties of the second order linear operator in <inline-formula><tex-math id="M1">\begin{document}$ {{\mathbb {R}}}^{N+1} $\end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \mathscr{L} u : = \sum\limits_{j,k = 1}^{m} a_{jk}\partial_{x_j x_k}^2 u + \sum\limits_{j,k = 1}^{N} b_{jk}x_k \partial_{x_j} u - \partial_t u, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ A = \left( a_{jk} \right)_{j,k = 1, \dots, m}, B = \left( b_{jk} \right)_{j,k = 1, \dots, N} $\end{document}</tex-math></inline-formula> are real valued matrices with constant coefficients, with <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula> symmetric and strictly positive. We prove that, if the operator <inline-formula><tex-math id="M4">\begin{document}$ {\mathscr{L}} $\end{document}</tex-math></inline-formula> satisfies Hörmander's hypoellipticity condition, and <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> is a Dini continuous function, then the second order derivatives of the solution <inline-formula><tex-math id="M6">\begin{document}$ u $\end{document}</tex-math></inline-formula> to the equation <inline-formula><tex-math id="M7">\begin{document}$ {\mathscr{L}} u = f $\end{document}</tex-math></inline-formula> are Dini continuous functions as well. We also consider the case of Dini continuous coefficients <inline-formula><tex-math id="M8">\begin{document}$ a_{jk} $\end{document}</tex-math></inline-formula>'s. A key step in our proof is a Taylor formula for classical solutions to <inline-formula><tex-math id="M9">\begin{document}$ {\mathscr{L}} u = f $\end{document}</tex-math></inline-formula> that we establish under minimal regularity assumptions on <inline-formula><tex-math id="M10">\begin{document}$ u $\end{document}</tex-math></inline-formula>.</p>