Sobolev regularity for the Monge-Ampere equation in the Wiener space

Given the standard Gaussian measure gamma on the countable product of lines R-infinity and a probability measure g . gamma absolutely continuous with respect to gamma, we consider the optimal transportation T(x) = x + del phi(x) of g . gamma to gamma. Assume that the function vertical bar del g vertical bar(2)/g is gamma-integrable. We prove that the function phi is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula g = det(2) (I + D-2 phi) exp(L phi - 1/2 vertical bar del phi vertical bar(2)). We also establish sufficient conditions for the existence of third-order derivatives of phi.