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 $$\mathscr{L} :=\sum_{i,j=1}^m \partial^2_{x_i x_j } +\sum_{i,j=1}^N b_{ij}x_j\partial_{x_i}-\partial_t, $$ where $(x,t) \in \mathbb{R}^{N+1}$, $1 \leq m \le N$ and the matrix $B:=(b_{ij})_{i,j=1,\ldots,N}$ has real constant entries. In particular, we show that if the modulus of $L^p$-mean oscillation of $\mathscr{L} u$ at the origin is Dini, then the origin is a Lebesgue point of continuity in $L^p$ average for the second order derivatives $\partial^2_{x_i x_j} u$, $i,j=1,\ldots,m$, 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.