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

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.