---
_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. <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>
  apa: Maas, J., &#38; 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>
  chicago: 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>.
  ieee: 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.
  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.” <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>.
  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
fulldoi: https://doi.org/10.1016/j.jfa.2009.07.001
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'
...