Cyclotomic Units and Class Groups in Z_p extensions of real abelian fields

For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the field. We determine the kernel and the cokernel of this map assuming Greenberg's conjecture on the vanishing of the lambda-invariant of the extension.