article
Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces
published
Jan
Maas
author 4C5696CE-F248-11E8-B48F-1D18A9856A870000-0002-0845-1338
Jan
Van Neerven
author
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.
Academic Press2009
Journal of Functional Analysis10.1016/j.jfa.2009.07.001
25782410 - 2475
yes
Maas, J., & Van Neerven, J. (2009). Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces. <i>Journal of Functional Analysis</i>. Academic Press. <a href="https://doi.org/10.1016/j.jfa.2009.07.001">https://doi.org/10.1016/j.jfa.2009.07.001</a>
Maas, Jan, and Jan Van Neerven. “Boundedness of Riesz Transforms for Elliptic Operators on Abstract Wiener Spaces.” <i>Journal of Functional Analysis</i>. Academic Press, 2009. <a href="https://doi.org/10.1016/j.jfa.2009.07.001">https://doi.org/10.1016/j.jfa.2009.07.001</a>.
J. Maas and J. Van Neerven, “Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces,” <i>Journal of Functional Analysis</i>, 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 J, Van Neerven J. Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces. <i>Journal of Functional Analysis</i>. 2009;257(8):2410-2475. doi:<a href="https://doi.org/10.1016/j.jfa.2009.07.001">10.1016/j.jfa.2009.07.001</a>
J. Maas, J. Van Neerven, Journal of Functional Analysis 257 (2009) 2410–2475.
Maas, Jan, and Jan Van Neerven. “Boundedness of Riesz Transforms for Elliptic Operators on Abstract Wiener Spaces.” <i>Journal of Functional Analysis</i>, vol. 257, no. 8, Academic Press, 2009, pp. 2410–75, doi:<a href="https://doi.org/10.1016/j.jfa.2009.07.001">10.1016/j.jfa.2009.07.001</a>.
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