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

TL;DR

This paper establishes new pointwise regularity results for solutions to degenerate Kolmogorov equations with $L^p$-source terms, showing conditions under which derivatives are well-behaved and providing Taylor expansions with $L^p$ estimates.

Contribution

It introduces novel pointwise regularity criteria for degenerate Kolmogorov equations with $L^p$ sources, including Taylor expansions and $L^p$ derivative estimates.

Findings

Regularity of second derivatives under Dini mean oscillation conditions

Existence of Taylor-type expansions with $L^p$ estimates

Decay estimates achieved through contradiction and compactness

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.