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res:
bibo_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 <
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.@eng'
bibo_authorlist:
- foaf_Person:
foaf_givenName: Jan
foaf_name: Jan Maas
foaf_surname: Maas
foaf_workInfoHomepage: http://www.librecat.org/personId=4C5696CE-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-0845-1338
- foaf_Person:
foaf_givenName: Jan
foaf_name: van Neerven, Jan M
foaf_surname: Van Neerven
bibo_doi: 10.1016/j.jfa.2009.07.001
bibo_issue: '8'
bibo_volume: 257
dct_date: 2009^xs_gYear
dct_publisher: Academic Press@
dct_title: Boundedness of Riesz transforms for elliptic operators on abstract Wiener
spaces@
...