Elsevier

Journal of Functional Analysis

Volume 257, Issue 8, 15 October 2009, Pages 2410-2475
Journal of Functional Analysis

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Dedicated to Professor Alan McIntosh on the occasion of his 65th birthday
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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 H̲. Given a bounded operator B on H̲, coercive on the range R(V)¯, we consider the operators A:=VBV in H and A̲:=VVB in H̲, as well as the realisations of the operators L:=DVBDV and L̲:=DVDVB in Lp(E,μ) and Lp(E,μ;H̲) respectively, where 1<p<. Our main result asserts that the following four assertions are equivalent:

  • (1)

    D(L)=D(DV) with LfpDVfp for fD(L);

  • (2)

    L̲ admits a bounded H-functional calculus on R(DV)¯;

  • (3)

    D(A)=D(V) with AhVh for hD(A);

  • (4)

    A̲ admits a bounded H-functional calculus on R(V)¯.

Moreover, if these conditions are satisfied, then D(L)=D(DV2)D(DA). The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where H̲=H, V=I, B=12I). A one-sided version of (1)–(4), giving Lp-boundedness of the Riesz transform DV/L 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

Cited by (0)

The authors are supported by VIDI subsidy 639.032.201 (JM and JvN) and VICI subsidy 639.033.604 (JvN) of the Netherlands Organisation for Scientific Research (NWO). The first named author acknowledges partial support by the ARC Discovery Grant DP0558539. The second named author was partially supported by the ARC Discovery Grant DP0559465.