Absolute continuity of distributions of polynomials on spaces with log-concave measures

Abstract

In the paper, it is proved that the distribution of a measurable polynomial on an infinite-dimensional space with log-concave measure is absolutely continuous if the polynomial is not equal to a constant almost everywhere. A similar assertion is proved for analytic functions and for some other classes of functions. Properties of distributions of norms of polynomial mappings are also studied. For the space of measurable polynomial mappings of a chosen degree, it is proved that the L¹-norm with respect to a log-concave measure is equivalent to the L¹-norm with respect to the restriction of the measure to an arbitrarily chosen set of positive measure.