Sharp inequalities and regularity of heat semigroup on infinite dimensional spaces
type: Item
instance of: scholarly article
title: Sharp inequalities and regularity of heat semigroup on infinite dimensional spaces
zbMATH Open document ID: 0611.60078
DOI: 10.1016/0022-1236(87)90016-4
author: Yuh-Jia Lee
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://purl.umn.edu/4373
review text: 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.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 3988416
zbMATH Keywords: regularity properties of the heat semigroup
zbMATH Keywords: abstract Wiener space
zbMATH Keywords: Hilbert-Schmidt norm
zbMATH Keywords: Ornstein-Uhlenbeck semigroup
zbMATH Keywords: Malliavin's calculus
MaRDI profile type: Publication
OpenAlex ID: W2064533477
cites work: An Extension of Weyl's Lemma to Infinite Dimensions
cites work: Quasi-Differentiable Functions on Banach Spaces
cites work: Potential theory on Hilbert space
cites work: Gaussian measures in Banach spaces
cites work: On Gross differentiation on Banach spaces
cites work: Q4148193
cites work: Q3882391
cites work: Applications of the Fourier-Wiener Transform to Differential Equations on Infinite Dimensional Spaces. I
cites work: Q3916897
cites work: Integral transforms of analytic functions on abstract Wiener spaces
cites work: Smooth functions on Banach spaces
Recommended article: Non-differentiability of heat semigroups in infinite dimensional Hilbert spaces
Recommended article: Q4224940
Recommended article: Q4284339
Recommended article: Regularity of.the heat semigroup generated by certain operators on a hilbert space
Recommended article: Sharp inequalities for heat kernels of Schrödinger operators and applications to spectral gaps
Recommended article: Q2934419
Recommended article: Comparison inequalities for heat semigroups and heat kernels on metric measure spaces
Recommended article: Heat semigroup bounds on \(BV(\Omega)\) and related estimates
Recommended article: Q3058137
description: scientific article; zbMATH DE number 3988416