On weak solutions of ultraparabolic equations

Let us consider the following second order differential operators Lu = Sigma (m0)(i,j=1) partial derivative (xi)(a(i,j)(x,y)partial derivative (xj)u) + Sigma (N)(i,j=1) b(i,j)x(i)partial derivativex(j)u - partial derivative (t)u where z = (x, t) is an element of RN+1, 0 < m(0) < N. This kind of linear operatos of Fokker-Plank type are used in probability and in mathematical phisics, as for istance in the study of brownian motions of a particle in a fluid. We point out that the natural geometry for the above operator is not euclidean but is given by a suitable groups structure. In this note lot us suppose that the matrix A(z) is the N x N matrix A(z) = ((A0(z)) (0) (0) (0)) where A(0)(z) = (a(i,j)(z))(i,j=1),...,(m0) is symmetric and there exists Lambda > 0 such that Lambda (-1)/xi/(2) less than or equal to < A0(z)xi,xi less than or equal to Lambda/xi/(2) For All xi is an element of R-m0, For Allz is an element of RN+1. and B = (b(i,j)) is a suitable N x N constant real matrix. This paper deals with the study interior regularity for weak solutions to the following equation: Lu = Sigma (m0)(j=1) partial derivative F-xj(j)(z) where F-j belong to a function space of Morrey type. It's also proved the local Holder continuity of the solution a. We observe that the above equation is a linearized version of the equation studied by Landau and the last one is a simplified model for the Boltzmann equation.