--- _id: '2119' abstract: - lang: eng text: '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 < p < ∞. 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.' author: - first_name: Jan full_name: Jan Maas id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Jan full_name: van Neerven, Jan M last_name: Van Neerven citation: ama: 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 apa: 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 chicago: 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. ieee: 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. ista: 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. mla: 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. short: J. Maas, J. Van Neerven, Journal of Functional Analysis 257 (2009) 2410–2475. date_created: 2018-12-11T11:55:49Z date_published: 2009-10-15T00:00:00Z date_updated: 2021-01-12T06:55:25Z day: '15' doi: 10.1016/j.jfa.2009.07.001 extern: 1 intvolume: ' 257' issue: '8' main_file_link: - open_access: '1' url: http://arxiv.org/abs/0804.1432 month: '10' oa: 1 page: 2410 - 2475 publication: Journal of Functional Analysis publication_status: published publisher: Academic Press publist_id: '4913' quality_controlled: 0 status: public title: Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces type: journal_article volume: 257 year: '2009' ...