Sobolev–Morrey spaces related to an ultraparabolic equation

Abstract:

Let us consider the class of hypoelliptic operators

where z=(x,t) ∈ℝ^{N +1}, 0 > m ₀≤N the coefficients a _i,j belong to the space of vanishing mean oscillation functions (VMO _L ) and B=(b _i,j ) is a constant real matrix. In this paper we prove that a strong solution to the differential equation Lu=f, with the known term f in the Morrey space L ^{p , λ}, belongs to a suitable Sobolev–Morrey space S ^{p , λ}. Then we prove some Morrey-type imbedding results that give a local Hölder continuity of the solution u.