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

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The aim of this paper is to consider the linear ultraparabolic equation with bounded and VMO coefficients aij (z). Assume that the operator L₀ obtained by freezing the coefficients aij(z) at any point z₀ ∈ Rⁿ ⁺ 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 L₀, and a Poincaré type inequality with a new cutoff function. Then Lᵖ estimate for weak solutions is derived by using the reverse Hölder 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.