The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, ANU, Canberra, 13-17 July 2009

251 Pages·2010·4.105 MB·English

by  Andrew Hassell, Alan McIntosh, Robert Taggart

Proceedings of the
CENTRE FOR MATHEMATICS
AND ITS APPLICATIONS
AUSTRALIAN NATIONAL UNIVERSITY
Volume 44, 2010
The AMSI–ANU Workshop on
Spectral Theory and Harmonic Analysis
(ANU, Canberra, 13–17 July 2009)
Edited by
Andrew Hassell, Alan McIntosh and Robert Taggart
Centre for Mathematics and its Applications
Mathematical Sciences Institute
The Australian National University
First published in Australia 2010
c Centre for Mathematics and its Applications
￿
Mathematical Sciences Institute
The Australian National University
CANBERRA ACT 0200
AUSTRALIA
This book is copyright. Apart from any fair dealing for the purpose of private
study, research, criticism or review as permitted under the Copyright Act, no part
may be reproduced by any process without permission. Inquiries should be made
to the publisher.
Edited by
Andrew Hassell, Alan McIntosh and Robert Taggart
Centre for Mathematics and its Applications, Mathematical Sciences Institute
The Australian National University
The AMSI–ANU Workshop on
Spectral Theory and Harmonic Analysis
ISBN 0 7315 5208 3
ysis speakers include: pierre albin ivana alexandrova matthew blair jean-marc bouclet kiril datchev ian doust tom ter elst colin guillarmou luc hillairet tuomas hytönen josé maría martell marius mitrea jan van neerven ane nonnenmacher cristian rios david rule adam sikora hart smith chris sogge tatiana toro lixin yan nces Institute of the ANU
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The AMSI – ANU Wor on Spectra 13 - 17 July 2009 ANU, Canberra, Australia The workshop aims to bring together leading internati ogether with top Australian mathematicians in two ra ncreasingly intertwined, fields of analysis: spectral thanalysis. The workshop will include both research se ectures, the latter designed for graduate students an enowned expositors Jan van Neerven and Hart Smit For further information and registration, g http://wwwmaths.anu.edu.au/events/Spec rganising committee: daniel daners (sydney), xuan duong (macquarie), ndrew hassell (anu), alan mcintosh (anu), robert taggart (anu) ontact: [email protected]
t i l r oa c

Preface
ThisvolumecontainstheproceedingsoftheAMSI–ANUWorkshoponSpectral
TheoryandHarmonicAnalysis,organisedbyDanielDaners,XuanDuong,Andrew
Hassell, Alan McIntosh and Robert Taggart at the Australian National University
in July 2009. The meeting was sponsored by the Mathematical Sciences Institute
of the Australian National University and the Australian Mathematical Sciences
Institute whose support is gratefully acknowledged.
The workshop covered a variety of topics in spectral theory and harmonic
analysis, and brought together experts, early career researchers, and doctoral stu-
dents from Australia, Canada, China, Finland, France, Germany, Italy, Japan, the
Netherlands, New Zealand, Scotland, Spain and the USA. It is our hope that this
volume reflects the lively research atmosphere of this conference. We are partic-
ularly honoured to open the proceedings with an expository article by Jan van
Neerven, which was based on a series of lectures he presented at the workshop.
We wish to express our appreciation to the authors who contributed to this
volume,toDanielDanersandXuanDuongwhowerefelloworganisersofthework-
shop, and to the CMA and MSI support staff (Annette Hughes and Alison Irvine)
who ensured that the event ran smoothly.
Each article in this volume was peer refereed.
Andrew Hassell, Alan McIntosh and Robert Taggart (Editors)
i

Contents
γ-Radonifying operators – a survey
Jan van Neerven 1
Algebraic operators, divided differences, functional calculus, Hermite
interpolation and spline distributions
Sergey Ajiev 63
A Strichartz estimate for de Sitter space
Dean Baskin 97
A maximal theorem for holomorphic semigroups on vector-valued spaces
Gordon Blower, Ian Doust, and Robert J. Taggart 105
Low energy behaviour of powers of the resolvent of long range perturbations of
the Laplacian
Jean-Marc Bouclet 115
Calder´on inverse problem for the Schr¨odinger operator on Riemann surfaces
Colin Guillarmou and Leo Tzou 129
A note on A estimates via extrapolation of Carleson measures
Steve H∞ofmann and Jose´ Mar´ıa Martell 143
Stability in p of the H -calculus of first-order systems in Lp
∞
Tuomas Hyto¨nen and Alan McIntosh 167
Feynman’s operational calculus and the stochastic functional calculus in
Hilbert space
Brian Jefferies 183
Local quadratic estimates and holomorphic functional calculi
Andrew J. Morris 211
Strichartz estimates and local wellposedness for the Schr¨odinger equation with
the twisted sub-Laplacian
Zhenqiu Zhang and Shijun Zheng 233
Conference photo 244
iii
γ-RADONIFYING OPERATORS – A SURVEY
JANVANNEERVEN
Abstract. Wepresentasurveyofthetheoryofγ-radonifyingoperatorsand
itsapplicationstostochasticintegrationinBanachspaces.
Contents
1. Introduction 1
2. Banach space-valued random variables 4
3. γ-Radonifying operators 7
4. The theorem of Hoffmann-Jørgensen and Kwapien´ 16
5. The γ-multiplier theorem 21
6. The ideal property 23
7. Gaussian random variables 27
8. Covariance domination 29
9. Compactness 33
10. Trace duality 35
11. Embedding theorems 39
12. p-Absolutely summing operators. 44
13. Miscellanea 48
References 57
1. Introduction
Thetheoryofγ-radonifyingoperatorscanbetracedbacktothepioneeringworks
of Gel(cid:48)fand [40], Segal, [111], Gross [42, 43], who considered the following
problem. A cylindrical distribution on a real Banach space F is a bounded linear
operatorW :F∗ →L2(Ω),whereF∗ isthedualofF and(Ω,F,P)isaprobability
space. ItissaidtobeGaussianifWx∗ isGaussiandistributedforallx∗ ∈F∗. IfT
isaboundedlinearoperatorfromF intoanotherrealBanachspaceE,thenT maps
every Gaussian cylindrical distribution W to a cylindrical Gaussian distribution
T(W):E∗ →L2(Ω) by
T(W)x∗ :=W(T∗x∗), x∗ ∈E∗.
The problem is to find criteria on T which ensure that T(W) is Radon. By this
we mean that there exists a strongly measurable Gaussian random variable X ∈
Date:Received23November2009/Accepted27February2010.
2000 Mathematics Subject Classification. Primary: 47B10;Secondary: 28C20,46B09,47B10,
60B11,60H05.
Key words and phrases. γ-Radonifying operators, stochastic integral, isonormal process,
Gaussian random variable, covariance domination, uniform tightness, K-convexity, type and
cotype.
SupportbyVICIsubsidy639.033.604oftheNetherlandsOrganisationforScientificResearch
(NWO)isgratefullyacknowledged.
1
2 JANVANNEERVEN
L2(Ω;E) such that
T(W)x∗ =(cid:104)X,x∗(cid:105), x∗ ∈E∗
(theterminology“Radon”isexplainedbyProposition2.1andtheremarksfollowing
it). The most interesting instance of this problem occurs when F = H is a real
Hilbert space with inner product [·,·] and W :H →L2(Ω) is an isonormal process,
i.e. a cylindrical Gaussian distribution satisfying
EW(h )W(h )=[h ,h ], h ,h ∈H.
1 2 1 2 1 2
Here we identify H with its dual H∗ via the Riesz representation theorem. A
bounded operator T : H → E such that T(W) is Radon is called γ-radonifying.
Here the adjective ‘γ-’ stands for ‘Gaussian’.
Gross[42,43]obtainedanecessaryandsufficientconditionforγ-radonification
in terms of so-called measurable seminorms on H. His result includes the classical
result that a bounded operator from H into a Hilbert space E is γ-radonifying
if and only if it is Hilbert-Schmidt. These developments marked the birth of the
theory of Gaussian distributions on Banach spaces. The state-of the-art around
1975 is presented in the lecture notes by Kuo [69].
γ-Radonifying operators can be thought of as the Gaussian analogues of p-
absolutely summing operators. For a systematic exposition of this point of view
we refer to the lecture note by Badrikian and Chevet [4], the monograph by
Schwartz [109] and the Maurey-Schwartz seminar notes published between
1972 and 1976. More recent monographs include Bogachev [9], Mushtari [84],
and Vakhania, Tarieladze, Chobanyan [118].
In was soon realised that spaces of γ-radonifying operators provide a natural
tool for constructing a theory of stochastic integration in Banach spaces. This
idea, which goes back to a paper of Hoffman-Jørgensen and Pisier [48], was
first developed systematically in the Ph.D. thesis of Neidhardt [93] in the con-
text of 2-uniformly smooth Banach spaces. His results were taken up and further
developed in a series of papers by Dettweiler (see [29] and the references given
there) and subsequently by Brze´zniak (see [11, 13]) who used the setting of mar-
tingale type 2 Banach spaces; this class of Banach spaces had been proved equal,
up to a renorming, to the class of 2-uniformly smooth Banach spaces by Pisier
[99]. Themoregeneralproblemofradonificationofcylindricalsemimartingaleshas
beencoveredbyBadrikianandU¨stu¨nel[5],Schwartz[110]andJakubowski,
Kwapien´, Raynaud de Fitte, Rosin´ski [55].
If E is a Hilbert space, then a strongly measurable function f : R → E is
+
stochastically integrable with respect to Brownian motions B if and only if f ∈
L2(R ;E). IthadbeenknownforalongtimethatfunctionsinL2(R ;E)mayfail
+ +
tobestochasticallyintegrablewithrespecttoB. Thefirstsimplecounterexamples,
for E = (cid:96)p with 1 (cid:54) p < 2, were given by Yor [120]. Rosin´ski and Suchanecki
[105] (see also Rosin´ski [103, 104]) were able to get around this by constructing a
stochastic integral of Pettis type for functions with valued in an arbitrary Banach
space. This integral was interpreted in the language of γ-radonifying operators by
van NeervenandWeis[90]; someoftheideasinthispaperwerealreadyimplicit
in Brze´zniak and van Neerven [14]. The picture that emerged is that the space
γ(L2(R ),E) of all γ-radonifying operators from L2(R ) into E, rather than the
+ +
Lebesgue-BochnerspaceL2(R ;E),isthe‘correct’spaceofE-valuedintegrandsfor
+
the stochastic integral with respect to a Brownian motion B. Indeed, the classical
Itˆo isometry extends to the space γ(L2(R ),E) in the sense that
+
(cid:13)(cid:90) ∞ (cid:13)2
E(cid:13)(cid:13) φdB(cid:13)(cid:13) =(cid:107)φ(cid:101)(cid:107)2γ(L2(R+),E)
0
γ-RADONIFYING OPERATORS – A SURVEY 3
for all simple functions φ : R+ → H ⊗ E; here φ(cid:101) : L2(R+) → E is given by
integrationagainstφ;onthelevelofelementarytensors,theidentificationφ(cid:55)→φ(cid:101)is
givenbytheidentitymappingf⊗x(cid:55)→f⊗x. ForHilbertspaces,thisidentification
sets up an isomorphism
L2(R ;E)(cid:104)γ(L2(R ),E).
+ +
In the converse direction, if the identity mapping f ⊗x (cid:55)→ f ⊗x extends to an
isomorphism L2(R ;E)(cid:39)γ(L2(R ),E), then E has both type 2 and cotype 2, so
+ +
E is isomorphic to a Hilbert space by a classical result of Kwapien´ [70].
Interpreting B as an isonormal process W :L2(R )→L2(Ω) by putting
+
(cid:90) ∞
W(f):= fdB, (1.1)
0
this brings us back to the question originally studied by Gross. However, instead
ofthinkingofanoperatorT :L2(R )→E as‘acting’ontheisonormalprocessW,
φ +
we now think of W as ‘acting’ on T as an ‘integrator’. This suggests an abstract
φ
approach to E-valued stochastic integration, where the ‘integrator’ is an arbitrary
isonormal processes W : H → L2(Ω), with H an abstract Hilbert space, and the
‘integrand’ is a γ-radonifying operator from H to E. For finite rank operators
T =(cid:80)N h⊗x the stochastic integral with respect to W is then given by
n=1
N N
(cid:16)(cid:88) (cid:17) (cid:88)
W(T)=W h⊗x := W(h)⊗x.
n=1 n=1
In the special case H = L2(R ) and W given by a standard Brownian motion
+
through (1.1), this is easily seen to be consistent with the classical definition of the
stochastic integral.
This idea will be worked out in detail. This paper contains no new results;
the novelty is rather in the organisation of the material and the abstract point
of view. Neither have we tried to give credits to many results which are more or
less part of the folklore of the subject. This would be difficult, since theory of
γ-radonifying operators has changed face many times. Results that are presented
here as theorems may have been taken as definitions in previous works and vice
versa, and many results have been proved and reproved in apparently different but
essentially equivalent formulations by different authors. Instead, we hope that the
references given in this introduction serves as a guide for the interested reader who
wants to unravel the history of the subject. For the reasons just mentioned we
have decided to present full proofs, hoping that this will make the subject more
accessible.
Theemphasisinthispaperisonγ-radonifyingoperatorsratherthanonstochas-
tic integrals. Accordingly we shall only discuss stochastic integrals of deterministic
functions. The approach taken here extends to stochastic integrals of stochastic
processesiftheunderlyingBanachspaceisaso-calledUMDspacebyfollowingthe
linesofvan Neerven, Veraar, Weis[88]. Weshouldmentionthatvariousalter-
native approaches to stochastic integration in general Banach spaces exist, among
them the vector measure approach of Brooks and Dinculeanu [10] and Din-
culeanu [32], and the Dol´eans measure approach of Metivier and Pellaumail
[83]. Asweseeit,thevirtueoftheapproachpresentedhereisthatitistailor-made
for applications to stochastic PDEs; see, e.g., Brze´zniak [11, 13], Da Prato and
Zabczyk [27], van Neerven, Veraar, Weis [86, 89] and the references therein.
For an introduction to these applications we refer to the author’s 2007/08 Internet
Seminar lecture notes [85].