Pointwise and Weighted Hessian Estimates for Kolmogorov-Fokker-Planck type operators

TL;DR

This paper establishes sharp Hessian estimates for Kolmogorov-Fokker-Planck operators in various Banach spaces using advanced harmonic analysis techniques, providing new quantitative bounds.

Contribution

It introduces a novel approach employing sparse domination techniques to derive Hessian estimates for these operators in non-divergence form.

Findings

Sharp Hessian estimates in weighted Lebesgue spaces

Representation formula and sparse domination methods

Quantitative bounds for Kolmogorov-Fokker-Planck operators

Abstract

In this article, we obtain hessian estimates for Kolmogorov-Fokker-Planck operators in non-divergence form in several Banach function spaces. Our approach relies on a representation formula and newly developed sparse domination techniques in Harmonic Analysis. Our result when restricted to weighted Lebesgue spaces yields sharp quantitative hessian estimates for the Kolmogorov-Fokker-Planck operators.