Abstract: | Let (E,H,μ) be an abstract Wiener space and let DV:=VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:=V*BV in H and in , as well as the realisations of the operators and in Lp(E,μ) and respectively, where 1<p<∞. Our main result asserts that the following four assertions are equivalent:- (1) with for ;
- (2) admits a bounded H∞-functional calculus on ;
- (3) with for ;
- (4) admits a bounded H∞-functional calculus on .
Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V=I, ). A one-sided version of (1)–(4), giving Lp-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C0-contraction semigroup on a Hilbert space H and let −L be the Lp-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an Lp-domain characterisation for the operator L. Keywords: Divergence form elliptic operators; Abstract Wiener spaces; Riesz transforms; Domain characterisation in Lp; Kato square root problem; Ornstein–Uhlenbeck operator; Meyer inequalities; Second quantised operators; Square function estimates; H∞-functional calculus; R-boundedness; Hodge–Dirac operators; Hodge decomposition |