Sharp inequalities and regularity of heat semigroup on infinite dimensional spaces

Abstract

This paper discusses the regularity of the heat semigroup P t generated from the abstract Wiener measure p t on an abstract Wiener space ( H, B). It is shown that if f ϵ L x ( p t ( x, dy)) for some α > 1, then P t f( x) is infinitely H-differentiable and D n P t f( x) is a symmetric n-linear Hilbert-Schmidt operator in L n ( H). A compact formula of D n P t f( x) and the estimation of ∥ D n P t f( x)∥ HS are also given. In the course of proving the above results, we obtain some sharp integral inequalities.