Pointwise estimates for degenerate Kolmogorov equations with $L^p$-source term

Abstract

The aim of this paper is to establish new pointwise regularity results for solutions to degenerate second order partial differential equations with a Kolmogorov-type operator of the form $$\mathcal{L}:=\sum _{i,j=1}^m{\partial }_{x_i{x}_j}^2+\sum _{i,j=1}^N{b}_{ij}{x}_j{\partial }_{x_i}-{\partial }_t,$$ where $(x,t)\in {\mathbb{R}}^{N+1}$, $1\le m\le N$ and the matrix $B:=(b_{ij}{)}_{i,j=1,\dots ,N}$ has real constant entries. In particular, we show that if the modulus of $L^p$-mean oscillation of $\mathcal{L}u$ at the origin is Dini, then the origin is a Lebesgue point of continuity in $L^p$ average for the second order derivatives ${\partial }_{x_i{x}_j}^{2}u$, $i,j=1,\dots ,m$, and the Lie derivative $\left({\sum }_{i,j=1}^N{b}_{ij}{x}_j{\partial }_{x_i}-{\partial }_t\right)u$. Moreover, we are able to provide a Taylor-type expansion up to second order with estimate of the rest in $L^p$ norm. The proof is based on decay estimates, which we achieve by contradiction, blow-up and compactness results.