Nonlinear transformations of convex measures

Given a uniformly convex measure $\mu$ on $R^\infty$ that is equivalent to its translation to the vector $(1,0,0,\ldots)$ and a probability measure $\nu$ that is absolutely continuous with respect to $\mu$, we show that there is a Borel mapping $T=(T_k)_{k=1}^\infty$ of $R^\infty$ transforming $\mu$ into $\nu$ 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 $x_1,\dots,x_k$. In addition, for any uniformly convex measure $\mu$ on $R^\infty$ and any probability measure $\nu$ with finite entropy $\textrm{ent}_\mu(\nu)$ with respect to $\mu$, the canonical triangular mapping $T=I+F$ transforming $\mu$ into $\nu$ satisfies the inequality $\|F\|_{L^2(\mu,l^2)}^2\le C(\mu)\textrm{ent}_\mu (\nu)$. 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 $L\log L$.