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