Sharp inequalities and regularity of heat semigroup on infinite dimensional spaces (Q1087248)

The author investigates the regularity properties of the heat semigroup \(P_ t\) on an abstract Wiener space (H,B) defined by \[ P_ tf(x)=\int_{B}f(x+y)p_ t(dy) \] where \(p_ t\) denotes the abstract Wiener measure with covariance t. It is shown that if \(q>1\) then for every \(f\in L^ q(p_ t(x,dy))\), \(p_ tf\) is infinitely often differentiable in direction H \((p_ t(x,E):=p_ t(E-x)).\) The n-fold H-derivative is a symmetric n-linear Hilbert-Schmidt operator and is computed explicitly as \[ D^ nP_ tf(x)h_ 1...h_ n=t^{- n}\int_{B}f(x+y)\sigma_ t(\prod^{n}_{j=1}\hat h_ j)(y)p_ t(dy) \] where \(h_ 1,...,h_ n\in H\). \(\sigma_ t(\hat h)\) is the Gauss transform of the \(L^ 2(p_ t)\)-function \(y\to <y,h>\). A sharp estimate for the Hilbert-Schmidt norm of \(D^ nP_ tf(x)\) is given. As a corollary, the same results are obtained for the Ornstein-Uhlenbeck semigroup - which is of importance in Malliavin's calculus - for x in a set of full \(P_ 1\)-measure.