Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

TL;DR

This paper establishes Schauder-type estimates for degenerate Kolmogorov equations with Dini continuous coefficients, showing that solutions' second derivatives inherit Dini continuity under hypoellipticity conditions.

Contribution

It proves Dini continuity of second derivatives for solutions to degenerate Kolmogorov equations with Dini continuous data and coefficients, extending regularity results under minimal assumptions.

Findings

Second derivatives of solutions are Dini continuous when data is Dini continuous.

Established a Taylor formula for solutions under minimal regularity.

Extended regularity results to equations with Dini continuous coefficients.

Abstract

We study the regularity properties of the second order linear operator in $\mathbb{R}^{N+1}$: \begin{equation*} \mathscr{L} u := \sum_{j,k= 1}^{m} a_{jk}\partial_{x_j x_k}^2 u + \sum_{j,k= 1}^{N} b_{jk}x_k \partial_{x_j} u - \partial_t u, \end{equation*} where $A = \left( a_{jk} \right)_{j,k= 1, \dots, m}, B= \left( b_{jk} \right)_{j,k= 1, \dots, N}$ are real valued matrices with constant coefficients, with $A$ symmetric and strictly positive. We prove that, if the operator $\mathscr{L}$ satisfies H\"ormander's hypoellipticity condition, and $f$ is a Dini continuous function, then the second order derivatives of the solution $u$ to the equation $\mathscr{L} u = f$ are Dini continuous functions as well. We also consider the case of Dini continuous coefficients $a_{jk}$'s. A key step in our proof is a Taylor formula for classical solutions to $\mathscr{L} u = f$ that we establish under minimal regularity assumptions on $u$.