Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Abstract

Let ( E , H , μ ) be an abstract Wiener space and let D V : = V D , where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space H ̲ . Given a bounded operator B on H ̲ , coercive on the range R ( V ) ¯ , we consider the operators A : = V ∗ B V in H and A ̲ : = V V ∗ B in H ̲ , as well as the realisations of the operators L : = D V ∗ B D V and L ̲ : = D V D V ∗ B in L p ( E , μ ) and L p ( E , μ ; H ̲ ) respectively, where 1 < p < ∞ . Our main result asserts that the following four assertions are equivalent: (1) D ( L ) = D ( D V ) with ‖ L f ‖ p ≂ ‖ D V f ‖ p for f ∈ D ( L ) ; (2) L ̲ admits a bounded H ∞ -functional calculus on R ( D V ) ¯ ; (3) D ( A ) = D ( V ) with ‖ A h ‖ ≂ ‖ V h ‖ for h ∈ D ( A ) ; (4) A ̲ admits a bounded H ∞ -functional calculus on R ( V ) ¯ . Moreover, if these conditions are satisfied, then D ( L ) = D ( D V 2 ) ∩ D ( D A ) . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where H ̲ = H , V = I , B = 1 2 I ). A one-sided version of (1)–(4), giving L p -boundedness of the Riesz transform D V / L in terms of a square function estimate, is also obtained. As an application let − A generate an analytic C 0 -contraction semigroup on a Hilbert space H and let − L be the L p -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 L p -domain characterisation for the operator L.