  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.
