Nonlinear Transformations of Convex Measures

摘要

Given a uniformly convex measure mu on {f R}^infty that is equivalent to its translation to the vector (1,0,0,ldots) and a probability measure u that is absolutely continuous with respect to mu, we show that there is a Borel mapping T=(T_k)_{k=1}^infty of {f R}^infty transforming mu into u and having the form T(x)=x+F(x), where F has values in l^2. Moreover, if mu is a product-measure, then T can be chosen triangular in the sense that each component T_k is a function of x_1,ldots,x_k. In addition, for any uniformly convex measure mu on {f R}^infty and any probability measure u with finite entropy Ent_mu(u) with respect to mu, the canonical triangular mapping T=I+F transforming mu into u satisfies the inequality |F|_{L^2(mu,l^2)}^2le C(mu)Ent_mu (u). Several inverse assertions are proved. Our results apply, in particular, to the standard Gaussian product-measure. As an application we obtain a new sufficient condition for the absolute continuity of a nonlinear image of a convex measure and the membership of the corresponding Radon--Nikodym derivative in the class Llog L.