# Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Maas J, Van Neerven J. 2009. Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces. Journal of Functional Analysis. 257(8), 2410–2475.

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Abstract
Let (E, H, μ) be an abstract Wiener space and let DV : = V D, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space under(H, {combining low line}). Given a bounded operator B on under(H, {combining low line}), coercive on the range over(R (V), -), we consider the operators A : = V* B V in H and under(A, {combining low line}) : = V V* B in under(H, {combining low line}), as well as the realisations of the operators L : = DV* B DV and under(L, {combining low line}) : = DV DV* B in Lp (E, μ) and Lp (E, μ ; under(H, {combining low line})) respectively, where 1 &lt; p &lt; ∞. Our main result asserts that the following four assertions are equivalent: (1)D (sqrt(L)) = D (DV) with {norm of matrix} sqrt(L) f {norm of matrix}p {minus tilde} {norm of matrix} DV f {norm of matrix}p for f ∈ D (sqrt(L));(2)under(L, {combining low line}) admits a bounded H∞-functional calculus on over(R (DV), -);(3)D (sqrt(A)) = D (V) with {norm of matrix} sqrt(A) h {norm of matrix} {minus tilde} {norm of matrix} V h {norm of matrix} for h ∈ D (sqrt(A));(4)under(A, {combining low line}) admits a bounded H∞-functional calculus on over(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 under(H, {combining low line}) = H, V = I, B = frac(1, 2) I). A one-sided version of (1)-(4), giving Lp-boundedness of the Riesz transform DV / sqrt(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.
Publishing Year
Date Published
2009-10-15
Journal Title
Journal of Functional Analysis
Volume
257
Issue
8
Page
2410 - 2475
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### Cite this

Maas J, Van Neerven J. Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces. Journal of Functional Analysis. 2009;257(8):2410-2475. doi:10.1016/j.jfa.2009.07.001
Maas, J., & Van Neerven, J. (2009). Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces. Journal of Functional Analysis. Academic Press. https://doi.org/10.1016/j.jfa.2009.07.001
Maas, Jan, and Jan Van Neerven. “Boundedness of Riesz Transforms for Elliptic Operators on Abstract Wiener Spaces.” Journal of Functional Analysis. Academic Press, 2009. https://doi.org/10.1016/j.jfa.2009.07.001.
J. Maas and J. Van Neerven, “Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces,” Journal of Functional Analysis, vol. 257, no. 8. Academic Press, pp. 2410–2475, 2009.
Maas J, Van Neerven J. 2009. Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces. Journal of Functional Analysis. 257(8), 2410–2475.
Maas, Jan, and Jan Van Neerven. “Boundedness of Riesz Transforms for Elliptic Operators on Abstract Wiener Spaces.” Journal of Functional Analysis, vol. 257, no. 8, Academic Press, 2009, pp. 2410–75, doi:10.1016/j.jfa.2009.07.001.
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