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 $$\begin{aligned} {\mathscr {L}}:=\sum _{i,j=1}^m \partial ^2_{x_i x_j } +\sum _{i,j=1}^N b_{ij}x_j\partial _{x_i}-\partial _t, \end{aligned}$$where \((x,t) \in {{\mathbb {R}}}^{N+1}\), \(1 \le m \le N\) 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 an 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.