$L^p$-parabolic regularity and non-degenerate Ornstein-Uhlenbeck type operators

TL;DR

This paper establishes $L^p$-parabolic regularity estimates for a class of parabolic equations with time-dependent coefficients, including Ornstein-Uhlenbeck type operators, and explores how these estimates depend on the parabolicity constant.

Contribution

It generalizes existing $L^p$-estimates to equations with less restrictive parabolicity conditions and extends results to Ornstein-Uhlenbeck type operators.

Findings

Proved $L^p$-estimates for second spatial derivatives under relaxed conditions.

Analyzed the dependence of estimates on the parabolicity constant.

Extended estimates to Ornstein-Uhlenbeck type operators.

Abstract

We prove $L^p$-parabolic a-priori estimates for $\partial_t u + \sum_{i,j=1}^d c_{ij}(t)\partial_{x_i x_j}^2 u = f $ on $R^{d+1}$ when the coefficients $c_{ij}$ are locally bounded functions on $R$. We slightly generalize the usual parabolicity assumption and show that still $L^p$-estimates hold for the second spatial derivatives of $u$. We also investigate the dependence of the constant appearing in such estimates from the parabolicity constant. Finally we extend our estimates to parabolic equations involving non-degenerate Ornstein-Uhlenbeck type operators.