IMPROVED HARDY AND RELLICH TYPE INEQUALITIES WITH TWO WEIGHT FUNCTIONS

In this work, we obtain several improved versions of two weight Hardy and Rellich type inequalities on the sub-Riemannian manifold R2n+1 defined by the vector fields X-j = partial derivative/partial derivative x(j) + 2ky(j)vertical bar z vertical bar(2k-2)partial derivative/partial derivative l, Y-j = partial derivative/partial derivative y(j) - 2kx(j)vertical bar z vertical bar(2k-2)partial derivative/partial derivative l, j = 1, 2, ..., n where (z, l) = (x, y,l) is an element of R2n+1, vertical bar z vertical bar = (vertical bar x vertical bar(2) + vertical bar y vertical bar(2))(1/2) and k >= 1.