Higher gradients estimates in Morrey spaces for weak solutions to linear ultraparabolic equations

TL;DR

This paper develops higher gradient estimates in Morrey spaces for weak solutions of linear ultraparabolic equations with VMO coefficients, extending regularity theory using inequalities and fundamental solutions.

Contribution

It introduces new Morrey space estimates for ultraparabolic equations with variable coefficients, leveraging hypoellipticity and advanced inequality techniques.

Findings

Established Caccioppoli, Sobolev, and Poincaré inequalities for ultraparabolic equations.

Derived L^p estimates for weak solutions using reverse Hölder inequality.

Proved higher Morrey estimates for solutions with various boundary conditions.

Abstract

The aim of this paper is to consider the linear ultraparabolic equation with bounded and VMO coefficients $a_{ij} (z)$. Assume that the operator $L_0$ obtained by freezing the coefficients $a_{ij}(z)$ at any point ${z_0} \in {\mathbb{R}^{N + 1}}$ is hypoelliptic. We first establish a Caccioppoli type inequality by choosing a cutoff function, a Sobolev type inequality by prosperities of the fundamental solution to $L_0$, and a Poincar\'{e} type inequality with a new cutoff function. Then $L^p$ estimate for weak solutions is derived by using the reverse H\"{o}lder inequality on homogeneous spaces. Finally, higher Morrey estimates for weak solutions to the above equation are shown by investigating a homogeneous ultraparabolic equation of variable coefficients with a nonhomogeneous boundary value condition, and a nonhomogeneous ultraparabolic equation of variable coefficients with homogeneous boundary value condition.