Selected Papers on Differential Equations and Analysis [illustrated] 0821839276, 9780821839270

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Table of contents :
Cover
Contents
Foreword
Regularity in free boundary problems
Behavior of spatial critical points and zeros of solutions of diffusion equations
Martin boundary and boundary Harnack principle for non-smooth domains
Singular integrals and Littlewood-Paley functions
Convergence of metric measure spaces and energy forms
Subfactor theory and its applications: Operator algebras and quantum field theory
Brownian motion and value distribution theory of holomorphic maps and harmonic maps
Statistical inference in two-stage sampling
Back_Cover
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Selected Paper s o n Differential Equation s and Analysi s

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http://dx.doi.org/10.1090/trans2/215

American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 21 5

Selected Paper s o n Differential Equation s and Analysi s

•s\\& America n Mathematica l Societ y Providence, Rhod e Islan d

2000 Mathematics Subject Classification. Primar y 46L37 , 81 T05 , 60J65 , 31 A05 , 31 B05 , 31B25, 31 C35 , 58Jxx , 62D05 , 35K57 , 35R35 , 42B25 .

Library o f Congres s Cataloging-in-Publicatio n D a t a Selected paper s o n differentia l equation s an d analysis . p. cm . (America n Mathematica l Societ y translations , ISS N 0065-929 0 ; ser. 2 , v . 21 5 ) "[Translations o f articles ] originall y publishe d i n th e journa l Sugaku"—Foreword . Includes bibliographica l references . ISBN 0-821 8-3927- 6 (acid-fre e paper ) 1. Differentia l equations . 2 . Functiona l analysis . 3 . Operato r algebras . 4 . Potentia l theor y (Mathematics). I . Sugaku . II . Series . QA3.A572 ser . 2 , vol . 21 5 [QA370] 510 s—dc2 2 [515'.35]

2005052407

C o p y i n g an d reprinting . Materia l i n thi s boo k ma y b e reproduce d b y an y mean s fo r edu cational an d scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y services tha t collec t fee s fo r deliver y o f document s an d provide d tha t th e customar y acknowledg ment o f th e sourc e i s given . Thi s consen t doe s no t exten d t o othe r kind s o f copyin g fo r genera l distribution, fo r advertisin g o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercial us e o f materia l shoul d b e addresse d t o th e Acquisition s Department , America n Math ematical Society , 20 1 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n also b e mad e b y e-mai l t o [email protected] . Excluded fro m thes e provision s i s materia l i n article s fo r whic h th e autho r hold s copyright . I n such cases , request s fo r permissio n t o us e o r reprin t shoul d b e addresse d directl y t o th e author(s) . (Copyright ownershi p i s indicate d i n th e notic e i n th e lowe r right-han d corne r o f th e firs t pag e o f each article. ) © 200 5 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 21 1

0 09 0 8 07 0 6 0 5

Contents

Foreword Regularity i n fre e boundar y problem s GEORG S . W E I S S

Behavior o f spatial critica l point s an d zero s o f solutions o f diffusion equation s SHIGERU SAKAGUCH I

Martin boundar y an d boundar y Harnac k principl e fo r non-smoot h domain s HlROAKI AlKAW A

Singular integral s an d Littlewood-Pale y function s SHUICHI SAT O

Convergence o f metri c measur e space s an d energ y form s ATSUSHI KASU E

Subfactor theor y an d it s applications : Operato r algebra s an d quantu m field theory YASUYUKI KAWAHIGASH I

Brownian motio n an d valu e distributio n theor y o f holomorphi c map s an d harmonic map s ATSUSHI ATSUJ I

Statistical inferenc e i n two-stag e samplin g M A K O T O AOSHIM A

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Foreword

This i s a collection o f several paper s originall y publishe d i n th e journal Sugaku. Normally thes e translate d article s woul d hav e appeare d i n th e America n Mathe matical Societ y journa l Sugaku Expositions. T o spee d u p publication , th e AM S decided t o publis h the m a s a volum e i n it s serie s America n Mathematica l Societ y Translations Serie s 2 . The volum e begin s wit h th e articl e "Regularit y i n fre e boundar y problems " b y Georg S . Weiss , devote d t o partia l differentia l equation s wit h fre e boundary . Th e author describe s thre e particula r fre e boundar y problems : th e obstacl e problem , the hea t equatio n wit h stron g absorption , an d a singula r limi t proble m i n combus tion theory . Fo r eac h o f thes e problem s th e autho r present s recen t result s relate d to regularit y condition s o f wea k solution s o f th e correspondin g equation s and , i n particular, t o condition s ensurin g th e smoothnes s o f the boundar y betwee n region s corresponding t o differen t phases . The articl e "Behavio r o f spatial critica l points an d zero s of solutions of diffusio n equations" b y Shigeru Sakaguch i i s devoted t o properties o f solutions o f Cauchy an d Neumann-type problem s fo r th e hea t equation s wit h N spatia l variables . Defin e the se t o f ho t spot s M(t) o f a solutio n u(y, i) o f suc h a n equatio n b y M(t) = {x e R N : u(x, t) = ma x u(y, t)} . y oo. I n particular, th e autho r describe s th e case s whe n th e origi n i s a stationar y ho t spo t of a solutio n o f th e Dirichle t proble m i n a bounde d domai n i n 1 SL N. The articl e "Marti n boundar y an d boundar y Harnac k principl e fo r non-smoot h domains" b y Hiroaki Aikaw a presents result s generalizing th e classica l Poisson inte gral representation o f a harmonic functio n o f the uni t bal l in the n-dimensiona l bal l to domains wit h non-smoot h boundaries . Fo r suc h domains, th e ordinar y boundar y of a smooth domai n i s replaced b y the so-calle d minima l Marti n boundary , an d th e classical Herglotz' s theore m give s th e Martin integral representation, whic h i s a n analog o f th e Poisso n integral . Th e autho r studie s a clas s o f non-smoot h domain s for whic h the Martin boundar y ca n b e described explicitl y an d clarifie s th e relation ship betwee n th e geometri c structur e o f a domai n an d th e propertie s o f harmoni c functions i n it . In the articl e "Singula r integral s an d Littlewood-Pale y functions" , Shuich i Sat o studies oscillatin g singula r integra l operator s wit h kernel s satisfyin g certai n growt h and smoothnes s condition s an d survey s som e result s o n th e boundednes s o f suc h operators i n variou s L p spaces . vii

viii F O R E W O R

D

In th e articl e "Convergenc e o f metri c measur e space s an d energ y forms" , Atsushi Kasu e consider s th e Dirichle t proble m fo r harmoni c function s an d stud ies convergenc e o f th e direc t calculu s o f variation s method s fo r obtainin g solution s using th e genera l convergenc e propertie s o f Riemannia n manifold s withi n th e gen eral theor y o f Hausdorff-Gromo v distanc e betwee n metri c spaces . Th e autho r als o studies th e analogou s convergenc e o f Riemannia n vecto r bundles . The article "Subfacto r theor y an d its applications: Operato r algebra s and quan tum field theory" b y Yasuyuki Kawahigash i provide s a review of recent application s of the theor y o f operator algebra s i n geometry an d physics . Th e autho r start s wit h the presentatio n o f result s b y Alai n Conne s o n classificatio n o f factor s an d o n th e use of operator algebra s as one of the main ingredient s of Connes's approac h t o non commutative geometry . The n h e describe s th e application s o f operato r algebra s i n the classificatio n o f knots (subfacto r theor y o f V. F. R . Jones ) an d i n quantum field theory. In th e articl e "Brownia n motio n an d valu e distributio n theor y o f holomorphi c maps an d harmoni c maps" , Atsush i Atsuj i give s a surve y o f result s relatin g th e Brownian motio n on a Riemannian manifol d M t o the function theor y o n M. Thes e properties ca n b e viewed a s a generalization o f the so-calle d conforma l invarianc e of the Brownia n motion . Th e articl e start s wit h a presentatio n o f th e Davi s theore m about a typica l homotopi c clas s o f a Brownia n pat h o n the plane . The n th e autho r describes recen t result s o n th e relationshi p betwee n Picard-typ e theorem s o n M and martingales , a s well as relation s betwee n th e Nevanlinn a theor y an d stochasti c calculus. Finally, th e articl e "Statistica l inferenc e i n two-stag e sampling " b y Makot o Aoshima discusses the two-stage sampling method i n hypothesis testing. Th e articl e starts b y describin g th e genera l two-stag e schem e o f hypothesi s testin g an d it s asymptotic an d statistica l properties . The n th e autho r continue s b y presentin g examples o f ho w suc h a schem e i s used i n practica l problem s o f hypothesi s testing . American Mathematica l Societ y

Amer. Math . Soc . Transl . (2) Vol . 21 5 , 200 5

http://dx.doi.org/10.1090/trans2/215/01

Regularity i n fre e b o u n d a r y problem s Georg S. Weiss

A "fre e boundar y problem " i s a proble m wher e on e solve s a give n partia l dif ferential equatio n i n a domai n tha t i s no t know n a priori , meanin g tha t ther e i s no wa y t o kno w th e domai n before knowing th e solution . Th e boundar y o f suc h a domain i s calle d a "fre e boundary " (se e Figur e 1 ) . T o avoi d a n underdetermine d problem-setting, i t i s necessar y t o ad d a n additiona l boundar y conditio n o n th e free boundar y t o th e usua l boundar y condition . A s th e fre e boundar y depend s o n the solution , th e proble m a s a whole i s going t o b e a nonlinea r proble m eve n i f on e starts wit h a linea r differentia l equatio n i n eac h phase . For motivatio n le t u s consider th e proces s o f melting ic e into water . Unles s on e does a n experiment , on e ha s i n thi s cas e n o mean s o f knowin g th e positio n o f th e boundary betwee n ic e an d water . Moreove r th e boundar y i s movin g i n time . W e may assume that th e temperature i s in each phase (soli d and liquid ) a solution o f the heat equation , bu t finding a n adequat e conditio n o n th e fre e boundar y separatin g the tw o phase s i s no t a n eas y task . I f on e assume s fo r exampl e tha t th e fre e boundary i s the 0-leve l se t o f the temperature , on e is thereby excludin g phenomen a like undercooling . I n th e cas e o f th e Stefa n proble m wit h Gibbs-Thomso n la w ([28]), i t i s possibl e t o mode l undercooling , bu t th e proble m become s harde r t o handle fro m th e mathematica l poin t o f vie w sinc e th e boundar y betwee n ic e an d water i s not a leve l se t an y more . The industria l application s o f fre e boundar y problem s ar e s o numerou s tha t a mere overvie w woul d b e beyon d th e scop e o f this article . Her e w e confine ourselve s to th e explanatio n o f a fe w example s i n som e detail . Solidificatio n an d meltin g processes a s wel l a s phas e separatio n processe s ar e ofte n mentione d a s classica l applications o f fre e boundar y problems , bu t ther e ar e man y othe r application s a s various a s flui d jets , tumo r growth , Black-Schole s marke t model s an d others .

This articl e originall y appeare d i n Japanes e i n Sugak u 5 4 (3 ) (2002) , 225-234 . 2000 Mathematics Subject Classification. Primar y 35R35 . ©2005 America n Mathematica l Societ y 1

2

GEORG S . WEIS S

Phase 1 PDE 1 Free Boundary , Boundary Conditio n Phase 2 PDE 2 FIGURE

1 . Fre e boundar y proble m

Let u s mentio n tha t fro m th e poin t o f vie w o f th e engineer , classica l solution s of free boundar y problem s - meanin g solution s wit h fre e boundarie s tha t ar e locall y smooth surface s suc h tha t th e solution s ca n b e extende d t o smoot h function s o n the closur e o f their respectiv e phase s - ar e to b e desired. Reason s fo r thi s desir e ar e that numerica l analysi s an d th e compactnes s o f fre e boundarie s whic h i s necessar y in invers e problem s becom e muc h easie r i n th e cas e o f classica l solutions . Fro m the theoretica l poin t o f view , however , th e situatio n i s different : th e existenc e of classica l solution s o f fre e boundar y problem s ca n rarel y b e show n i n a direc t way. I f al l goe s well , on e obtain s a wea k solutio n (e.g . a solutio n i n th e sens e o f distributions o r a viscosit y solution) , an d on e ha s a prior i n o informatio n o n th e regularity o f that solutio n o r th e regularit y o f the fre e boundary . I n th e cas e o f th e classical Stefa n problem , fo r example , a prior i on e doe s no t eve n kno w whethe r th e weak solution i s continuous an d whethe r th e fre e boundar y i s a relatively close d set . Even whe n on e ha s prove d tha t th e wea k solutio n i s continuous , th e boundarie s o f the phases , d{u > 0 } an d d{u < 0} , may hav e positiv e volum e (thin k fo r exampl e of a generalize d Canto r set) . Althoug h th e fre e boundar y ma y b e a leve l set , it i s usuall y no t possibl e t o appl y undergraduat e tool s lik e th e implici t functio n theorem directly , a s th e solutio n o r derivative s ar e discontinuous . I n th e exampl e of th e classica l Stefa n proble m th e firs t derivative s ar e discontinuous . In thes e circumstances , regularit y theor y play s a vital role . Th e regularit y the ory o f free boundar y problem s i s - apar t fro m th e obviou s relatio n t o th e regularit y theory o f nonlinear partia l differentia l equation s - deepl y relate d t o geometri c mea sure theor y an d harmoni c analysis . Muc h progres s ha s bee n achieve d her e durin g the las t fe w decades , th e startin g poin t bein g th e "flatnes s implie s regularity " re sults tha t hav e bee n prove d i n th e tim e fro m th e seventie s t o th e earl y nineties . Roughly speaking , "flatnes s implie s regularity " mean s tha t i f th e fre e boundar y i s in th e neighborhoo d o f a certai n poin t sufficientl y clos e t o a plane , the n i t ha s t o be a smooth surfac e i n a certain smalle r neighborhoo d o f that point . Base d o n thi s success i n provin g regularit y i n th e neighborhoo d o f point s tha t ar e i n th e abov e sense "flat" , researc h i s now focusing o n the stud y o f singular point s (fre e boundar y points tha t ar e no t flat), fre e boundarie s movin g i n time , an d systems . I t i s t o b e hoped tha t th e method s develope d i n fre e boundar y problem s wil l lea d als o t o a framework tha t ca n be applied t o other problems , lik e for exampl e geometri c partia l differential equation s an d nonparametri c minima l submanifolds . In wha t follow s w e wil l discus s thre e selecte d fre e boundar y problems : th e obstacle problem , th e hea t equatio n wit h stron g absorption , an d a singula r limi t problem i n combustio n theory .

3

R E G U L A R I T Y I N F R E E BOUNDAR Y P R O B L E M S

1. T h e o b s t a c l e p r o b l e m In orde r t o motivat e th e ter m "obstacl e problem" , le t u s conside r th e proces s of spannin g a n elasti c membran e ove r a give n obstacl e (se e Figur e 2) . In a simpl e mode l o f th e equilibriu m situatio n w e ma y assum e th e elasti c mem brane t o b e th e grap h o f a functio n v an d th e obstacl e t o b e th e subgrap h o f a function g, wher e v minimize s th e energ y functiona l

/ |Vt f Jn on th e clas s {(/> G i7 1 , 2 (£l) : g i n ft, o} in th e sens e o f distributions ; her e c = —Ag i s a smoot h positiv e coefficient . Let u s mentio n som e mathematica l result s concernin g th e obstacl e proble m t h a t ar e relevan t i n thi s context . B y th e Harnac k inequalit y argumen t o f H . Brezi s and D . Kinderlehre r ([7]) , th e secon d derivative s o f u ar e bounded . Thi s regularit y is shar p a t eac h fre e boundar y point , an d u ha s quadrati c growt h a t eac h fre e boundary poin t b y th e resul t o f L.A . Caffarell i an d N.M . Rivier e i n [1 7] . In th e landmar k pape r [1 0] , L.A . Caffarell i prove d i n 1 97 7 t h a t th e fre e bound ary i s a smoot h surfac e i n a neighborhoo d o f eac h fre e boundar y poin t a t whic h th e coincidence se t {u — 0} ha s positiv e Lebesgu e density . Th e resul t i s base d o n blow up argument s (argument s usin g blow-u p sequence s o f th e for m ( u^x™ 2 r )meN ) and Harnac k inequalit y arguments . Step 1 . B y studyin g th e blow-u p limit s (tha t is , th e limit s o f th e abov e blow-u p sequences) an d th e secon d blow-u p limit s (th e blow-u p limit s o f t h e blow-u p limits) , one obtain s th e following : th e fre e boundar y i s "flat " i n a neighborhoo d B r(xo) o f each poin t xo a t whic h th e coincidenc e se t {u = 0 } ha s positiv e Lebesgu e density . This mean s t h a t fo r eac h give n e > 0 on e ca n choos e p smal l enoug h (dependin g o n e) s o t h a t u = 0 o n {x G

B P(XQ) :

x-v >

ep} an d u

> 0 o n {x G

B P(XQ) :

for a certai n directio n v (se e Figur e 3) . Th e directio n v depend s o n p.

Free Boundar y

^—V Elastic Membran e

Obstacle F I G U R E 2 . T h e obstacl e proble m i n on e dimensio n

x-v
—C\ log(dist(x, {u — 0}))| _ e\x — xo\} an d u > 0 i n

{x G Bp(xo) : x-v < — e\x — xo\} fo r a certain directio n v (se e Figure 4) . I n contrast to flatness , whic h doe s not imply regularit y o f the free boundary , thi s cone-flatnes s directly implie s th e desired regularit y o f the free boundary . Let u s point ou t tha t fo r a genera l functio n c ( a general obstacle ) ther e may occur singularities , tha t is , points xo such tha t {u > 0} D Br(xo) i s not a subgrap h of a C 1 -function, n o matter ho w small on e chooses r > 0. For c = 1 the solution s u(x) = \x\ 2 an d v(x) = ^;|^| 2 giv e example s o f singularities . I n genera l i t i s possible to find singular set s of dimension k for each / c E { 0 , l , . . . , n — 1 } . Moreover , D. Schaeffe r constructe d a n analytic c that give s ris e t o a cusp (se e [33]). Fo r c of class C°° the situation i s even worse : fo r each relativel y ope n subse t U of {x\ = 0} there exis t a c and boundar y dat a suc h tha t {u > 0 } Pi {x\ = 0 } coincides wit h U ([33]) . Consequentl y i t i s possible t o construc t a singula r se t whose boundar y

R E G U L A R I T Y I N F R E E BOUNDAR Y P R O B L E M S

5

(relative t o th e fre e boundary ) i s a se t o f dimensio n n — 1 . Thi s mean s tha t i t i s not possibl e t o deriv e a partia l regularit y resul t fo r th e fre e boundar y o r fo r th e singular set . What ca n b e shown i s the following . I n a certain neighborhoo d o f each singula r point, th e singula r se t i s contained i n a smooth surface . Thi s was first prove d i n two dimensions b y L.A . Caffarell i an d N.M . Rivier e vi a Harnac k inequalit y argument s (see [1 8]) . Fo r analyti c c , M . Saka i obtaine d a mor e precis e resul t usin g method s of comple x analysi s ([32]) : i n th e cas e o f analyti c c , th e relativ e boundar y o f th e singular se t consist s o f isolate d points . Fo r th e higher-dimensiona l case , L.A . Caf farelli prove d i n 1 99 8 vi a th e monotonicit y formul a i n [9 ] tha t th e singula r se t i s contained i n a smoot h surface . I worke d o n th e obstacl e proble m myself , whe n I developed m y epiperimetri c inequalit y approac h t o th e regularit y i n fre e boundar y problems wit h th e obstacl e proble m a s mode l proble m (se e [36]) . Th e ai m wa s with application s t o two-phas e problems , system s an d time-dependen t problem s i n mind - t o replac e th e existin g method s base d o n Harnac k inequalit y argument s b y methods usin g energ y functionals , whic h ar e give n i n man y problem s i n a natura l way. Th e epiperimetri c inequalit y i s a n abstract , quantitativ e estimat e i n a give n function space . I too k th e ter m "epiperimetri c inequality " fro m th e inequalit y de rived b y E . Reifenber g fo r th e perimete r usin g completel y differen t method s (se e [30] an d [35]) . I n th e case o f th e obstacl e proble m on e doe s no t wor k wit h th e perimeter, bu t wit h th e energ y functiona l M(w):= f

(|VH

2

+max(w,0))-2 / w

1 2 n

dH ~ .

This energ y satisfie s fo r al l non-negativ e function s h G iif1,2(jBi(0)) tha t ar e homo geneous o f degre e 2 the followin g epiperimetri c inequality : i f \\h- -max(x-z/,0)

2

||j / i,2 ( j B l ( 0 ) ) < S

for som e v < E B\(0), the n ther e exist s a functio n w G i71,2 (i?i(0)) tha t satisfie s M(w) < ( 1 - K,)M(h) + ftM(- max( x • v, 0) 2 ) as wel l a s th e boundar y conditio n w = h o n dBi(0). A possible interpretatio n o f this theore m i s that i n a neighborhood o f the "half plane solutions" | max(x-z/ , 0)2 (solution s o f the obstacle problem fo r th e coefficien t c = | ) , th e energ y satisfie s a deca y estimate . Th e proo f doe s no t us e th e Harnac k inequality o r th e maximu m principle , bu t i s base d o n standar d technique s i n rea l and abstrac t analysis . Thu s i t i s possibl e t o thin k o f extension s t o othe r problem s like system s etc . (actuall y th e metho d ha s bee n extende d t o th e time-dependen t heat equatio n wit h stron g absorption , whic h ha s a mor e complicate d nonlinearit y than th e obstacl e proble m (se e [41 ])) . Fo r th e obstacl e proble m th e techniqu e ha s been successful i n providing a n alternative proof fo r th e "flatnes s implie s regularity " result. Moreover , derivin g a simila r epiperimetri c inequalit y a t singula r solution s of th e for m \\x • v\ 2 an d ~^\x\ 2 •> I recovere d b y completel y differen t method s th e result [1 8 ] o f L.A . Caffarell i an d N.M . Rivier e - whic h I di d no t kno w a t th e time ; ironically, althoug h th e autho r o f [32 ] i s working i n Japa n I hear d o f [1 8 ] an d [32 ] for th e first tim e whe n attendin g a meetin g i n Sweden . Apart fro m th e application s o f th e obstacl e proble m mentione d above , ther e are numerou s rea l lif e application s in , say , filtration, elastic-plasti c torsion , optima l

6

GEORG S . WEIS S

control, mathematica l finance, an d s o on (i n [23 ] man y application s ar e presented) . Also, various two-phas e problem s relate d t o the obstacl e proble m enjo y th e interes t of mathematicians studyin g th e regularit y o f free boundarie s (se e for exampl e [1 3] , [39], [34 ] an d [5]) . Let u s als o poin t ou t th e relatio n betwee n th e obstacl e proble m an d th e Stefa n problem mentione d above : i f w e assum e tha t u i s a non-negativ e solutio n o f th e classical Stefa n proble m dt{u + X{u>o}) - A u = 0 and tha t it s tim e derivativ e satisfie s d tu > 0, then w e obtain b y integratio n i n tim e that v(t,x) : = J 0 u(s^x) ds i s a solutio n o f th e time-dependen t obstacl e proble m dtv(t, x) - Av(t, x) = -X{t;>o } (*, x) + f(x) . In a n ope n neighborhoo d o f ((0 , oo) x R n ) H {v = 0 } the functio n / = 0 , so that on e can trea t thi s proble m a s a perturbatio n o f th e stationar y obstacl e proble m (se e [10]). 2. Th e hea t equatio n wit h stron g absorptio n The hea t equatio n wit h stron g absorptio n u > 0, d

tu

— Au = —u 1 ,

where 7 G (0,1) i s a give n parameter , i s a proble m o f th e sam e typ e a s th e time dependent obstacl e problem , bu t fo r physica l reason s on e i s force d i n thi s case to dro p th e assumptio n tha t d tu > 0 . Thi s result s i n a bi g differenc e concernin g the dynamic s o f th e equation . Apar t fro m singularitie s o f th e stationar y problem , we encounte r her e singularitie s peculia r t o th e time-dependen t problem , lik e fo r example extinctio n points , i.e . fre e boundar y point s i n a spatia l neighborhoo d o f which th e solutio n i s 0 . The hea t equatio n wit h stron g absorptio n ha s bee n use d b y L . K . Martinso n and b y Ph . Rosena u an d S . Kami n a s a mode l fo r th e transpor t o f therma l energ y in plasm a (se e [29 ] an d [31 ]) . Besides , i t ha s bee n derive d b y C . Bandl e an d I. Stakgol d i n [6 ] a s th e asymptoti c limi t o f a simpl e reaction-diffusio n process . In th e cas e o f on e spac e dimensio n th e solution' s behavio r nea r th e extinctio n point ha s bee n extensivel y studie d (se e fo r exampl e [24] , [20] , [25 ] an d [26]) . Le t us take a closer loo k a t tw o of these results : i n [24] , A. Friedman an d M.A . Herrer o analyzed th e solution' s spatia l profil e nea r extinctio n points . I n [20] , X.-Y. Chen , H. Matan o an d M . Mimur a estimate d th e numbe r o f extinctio n point s i n th e cas e of initia l dat a wit h compac t support . Th e author s als o prove d continuit y o f the se t {u > 0} i n tim e wit h respec t t o th e Hausdorf f distance . In th e higher-dimensiona l case , H.W . Al t an d D . Phillip s obtaine d a resul t concerning th e regularit y o f th e solutio n an d tha t o f th e fre e boundar y ([3]) : th e proof o f th e solution' s regularity , tha t is ,

relies - a s i n th e cas e o f th e obstacl e proble m - o n Harnac k inequalit y arguments . The method s use d i n th e proo f o f th e regularit y o f th e fre e boundary , however , have bee n adopte d fro m a differen t problem , namel y th e proble m wit h Bernoull i free boundar y conditio n i n [1 ] . W e ar e goin g t o discus s thes e method s i n mor e detail i n sectio n 3 for th e exampl e o f a Bernoull i typ e fre e boundar y problem . A s

7

R E G U L A R I T Y I N F R E E BOUNDAR Y P R O B L E M S

in th e obstacle problem , th e singular se t of the heat equatio n wit h stron g absorp tion ca n b e o f an y intege r dimensio n betwee n 0 an d n — 1 . Th e analysi s o f the structure o f homogeneous solutions , however , i s more difficul t tha n tha t i n the case of the obstacle problem . Th e reason i s that th e positive homogeneou s solution s of degree 2 of the obstacle proble m i n R n — {0 } are second orde r polynomials , bu t a corresponding fac t i s not known fo r the equation Au = u 1 . Concerning th e higher-dimensiona l cas e i n th e time-dependen t problem , H. Cho e an d the author obtaine d i n [21 ] regularity o f the solution, finit e propaga tion spee d o f the set {u > 0}, a Hausdorff measur e estimat e o f the free boundar y and th e asymptotic behavio r nea r "horizonta l fre e boundar y points" . "Horizonta l free boundar y points " ar e free boundar y point s a t whic h th e behavio r i n tim e i s 1—7

dominant. Regularit y o f the solutio n mean s i n this case that u~~^~ i s Lipschitz con tinuous wit h respec t t o the space variables an d Holder continuou s wit h exponen t \ with respec t t o the time variable . Th e proof relies , roughl y speaking , o n the fac t that |0} JRn{s=t}

rt

8

G E O R G S . WEIS S

On th e othe r hand , th e fac t t h a t th e singula r par t i s ignore d b y integratio n b y part s does no t necessaril y mea n t h a t E i s a smal l set : th e stationar y solutio n

presents her e a counterexampl e (compar e t o th e obstacl e problem) . Higher regularit y o f R i s a n ope n problem . 3. A s i n g u l a r limi t p r o b l e m i n c o m b u s t i o n t h e o r y In thi s sectio n w e conside r th e reaction-diffusio n equatio n (1) d

tue

- Au

e n

= -(3

ue > 0 i n (0 , oo) x R , u

e(ue),

e (0, • )

= u° e in R n ,

and it s singula r limi t a s e - » 0 ; her e e G (0 , l),/? e (z) = \fi(\ ),/? G C£([0, l]),/ 3 > 0 in (0,1 ) an d J (5 = | . Formall y eac h limi t u wit h respec t t o a subsequenc e i s a solution o f th e fre e boundar y proble m (2) d

tu

- Au = 0 i n {u > 0 } n (0 , oo) x R n ,

\\7u\ = 1 o n d{u > 0 } H (0, oo) x R n . This limi t proble m ha s variou s applications : th e stationar y proble m ha s bee n ap plied t o fluid jet s an d cavitie s ([2]) , electro-chemica l machinin g ([27]) , th e construc tion o f optima l hea t conductor s ([22]) , etc . Th e singula r limi t proble m (1 ) ha s bee n introduced a s a mode l fo r th e propagatio n o f premixe d flame s i n th e cas e o f hig h activation energ y ([8] , [1 9]) . I n thi s mode l u = X(T C — T ) , wher e T c i s th e flam e temperature, whic h i s assume d t o b e constant , T i s th e temperatur e outsid e th e flame, an d A is a constan t normalizatio n factor . Let u s shortl y summariz e th e know n mathematica l results , beginnin g wit h th e limit problem . I n th e brillian t pape r [1 ] , H.W . Al t an d L.A . Caffarell i showe d existence o f a stationar y solutio n o f (2 ) i n th e sens e o f distribution s b y minimizin g the energ y functiona l J(\Vu\ 2 - f X{u>o})- The y furthermor e prove d regularit y o f the fre e boundar y outsid e a se t o f n — 1-dimensional Hausdorf f measur e 0 . A s th e technique i n thei r pape r ha s sinc e bee n applie d t o othe r problem s lik e th e hea t equation wit h stron g absorptio n (se e [3]) , w e ar e goin g t o discus s thei r method s in som e detail . I n thi s Bernoulli-typ e fre e boundar y problem , too , th e Harnac k inequality play s a ke y role . I t lead s t o Lipschit z continuit y o f th e solution , an d i t i s important whe n provin g th e followin g vita l property : i f th e fre e boundar y i s flat o n the sid e o f {u = 0} , then i t mus t als o b e flat o n th e sid e o f {u > 0} . More precisely , u = 0 i n B r(xo) f l {(x — XQ) • v > e } = > u > 0 i n B cr(xo) D

{(x — XQ) • v < —Ce} .

Here x o i s a fre e boundar y point , C an d c ar e constants , an d v i s a uni t vector . The poin t i s t h a t th e scalin g o f th e left-han d sid e an d t h a t o f th e right-han d sid e in e ar e th e same . Other valuabl e idea s ar e "partitio n o f energy " an d non-positiv e mea n curvatur e of th e fre e boundar y o f blow-u p limits . "Partitio n o f energy " mean s her e t h a t th e Dirichlet par t o f th e energ y i s asymptoticall y les s t h a n o r equa l t o th e volum e par t

9

REGULARITY I N FRE E BOUNDAR Y PROBLEM S

t= 1

{u>0} /

{u

= 0} \

{u>0}

t = 0x= 0 x FIGURE

= 1

5 . Collidin g travelin g wave s

of th e energy , i.e . 1

/ \Vu\

2

o};

Br(x0) JBr(x0) JB here, too , x 0 i s assume d t o b e a fre e boundar y point . I n tw o spac e dimension s th e inequality becomes an equality, that is , an "equipartitio n o f energy". "Equipartitio n of energy" wit h respec t t o a different energ y functiona l ha s late r playe d a vital rol e in th e contex t o f mea n curvatur e flow . "Non-positive mea n curvatur e o f the fre e boundar y o f blow-up limits " refer s t o the fac t tha t eac h limi t UQ of th e blow-u p sequenc e ( r +rm'^ satisfie s fo r every open tes t se t D th e inequalit y H n~1 (Dnd{u0 > 0} ) < %nn-1 (dDn{u0 > 0}) . Let u s poin t out , however , tha t th e ide a behin d thi s fac t ha s i n [1 ] bee n applie d to inhomogeneou s blow-u p limit s an d no t t o th e homogeneou s blow-u p limit s fro m above. Bette r regularit y fo r inhomogeneou s blow-u p limit s i s use d i n [1 ] t o deriv e a quantitativ e estimate , whic h eventuall y lead s t o regularit y o f th e fre e boundary . Existence o f a classica l solutio n i n thre e dimension s i s stil l a n ope n problem , but i t woul d follo w fro m m y resul t i n [40 ] an d [37] , if non-existenc e o f minimizin g singular cone s coul d b e shown . Non-minimizin g singula r cone s d o i n fac t exis t (se e [1, Exampl e 2.7]) . Moreover , th e solutio n o f th e stationar y Dirichle t proble m i s i n general no t uniqu e (se e [1 , Example 2.6]) . Let u s return t o th e time-dependen t proble m (2) . Ther e i s no uniquenes s here , either. First , th e positiv e solutio n o f th e hea t equatio n i s alway s anothe r solutio n of th e problem . Eve n i f w e exclud e thi s trivia l exampl e o f non-uniqueness , ther e are othe r counterexamples . Also , eve n i f on e start s wit h flawles s initia l data , a classical solutio n o f (2 ) wil l in genera l develo p singularitie s afte r a finit e time-span . Consider, e.g. , th e exampl e o f tw o collidin g travelin g waves ,

u(t, x) = X{x+t>\] (exp(x + t - 1 ) - 1 ) + X{-*+t>i}(exp(- x + t - l ) - l ) f o r t

e [0,1 )

(cf. Figur e 5) . Let u s proceed wit h result s concerning the singula r limi t proble m (1 ) . Concern ing th e stationar y problem , H . Berestycki , L.A . Caffarell i an d L . Nirenber g prove d in [4 ] uniform estimate s fo r th e solutio n u € (concernin g unifor m estimate s se e als o [12]). Assumin g th e existenc e o f a minima l solution , th e author s o f [4 ] obtaine d further results . I n [1 9] , L.A. Caffarell i an d J.L . Vazque z prove d th e correspondin g uniform estimate s i n th e cas e o f time-dependenc e (se e als o [1 4]) . Moreove r the y obtained a convergenc e result : i f th e initia l dat a i s strictl y mean-concav e i n it s

10

GEORG S . WEIS S

FIGURE

6 . A singularit y i n th e stationar y limi t proble m

support, the n eac h limi t o f solution s o f proble m (1 ) i s a solutio n o f proble m (2 ) i n the sens e o f distributions . Let u s als o mentio n result s concernin g th e correspondin g two-phas e problem . In [1 5 ] an d [1 6] , L.A . Caffarelli , C . Lederma n an d N . Wolansk i sho w tha t i f th e limit functio n u satisfie s {u = 0} ° = 0 , the n i t mus t b e a sor t o f barrie r solution . Knowing th e resul t o f L.A . Caffarell i an d J.L . Vazquez , on e migh t conjectur e that eac h limit o f ue i s always a solution in the sense of distributions. This , however , is not true : fo r arbitrar y 6 G [0,1] ther e i s u e convergin g t o th e limi t u(t,x) — 0\x\\ which is not a solution of (2) in the sense of distributions. Th e same limit functio n u also appear s a s blow-u p limi t i n th e stationar y proble m an d i n th e time-dependen t problem (se e Figure s 6 an d 5) . Fo r thi s reason , a clas s o f solution s containin g th e function u(t,x) — 6\x\\ i s desirable . My contributio n t o thi s proble m i s a s follow s ([42]) . First , eac h limi t functio n is a solutio n i n the sens e o f domain variations : u i s smooth i n {u > 0} and satisfie s for ever y f G C£,:L((0,oo) x R n ; R n ) th e equatio n /•oo

(4) / / JO «/R

r

/»oo

[-2d n

tuVu-^\Vu\

2

divi-2VuDiVu) =

JO

-/ /



^vdH^dt.

JR(t)

Here R(t) : = {x G d{u(t) > 0 } : there i s v(t,x) G dBi(0) suc h tha t u

r(s,y)

2

u{t + r s,x + ry) —> ma x - y • v(t, x) , 0) r locally uniforml y i n (5 , y) G Rn + 1 a s r — > 0 } is fo r a.e . t G (0 , 00) a countabl y n — 1-rectifiable subse t o f th e fre e boundar y (that is , a unio n o f countabl y man y C 1 manifold s an d a se t o f ?Y n_1 -measure 0) . Moreover, R(t) i s "almost " ope n relativ e t o d{u(t) > 0}. Furthermore, th e fre e boundar y ca n fo r a.e . t G (0 , oo) an d u p t o a se t o f vanishing 7^ n_1 -measure b e decompose d a s d{u(t) > 0 } = R(t) U £**(£) U T, z(t).

REGULARITY IN FREE BOUNDARY PROBLEMS

11

Here th e non-degenerate singula r set £**(£) : = {x e d{u(t) > 0} :there i s 0(t,x) G (0,1] and £(t,x) G dB^O) suc h that ur(8,y) = "f t + ^ +

H/) - > 6 l ( t , x ) | y - ^ , x ) |

locally uniforml y i n (s,y) G R n + 1 a s r—> • 0} is a countabl y n — 1 -rectifiabl e subse t o f th e fre e boundary , an d th e degenerat e singular set Ez(t) : = {x G d{u(t) > 0} : r~ n ~ 2 / \Vu\

2

- > 0 as r - > 0} ,

•/Q r (t,z)

where Q r{t,x) = (t — r 2,t - f r2 ) x B r(x). Tha t mean s tha t w e can decompose th e free boundar y int o a countabl y n — 1 -rectifiabl e par t an d the degenerate singula r set. Th e crucial too l in the proof of this resul t i s the following estimat e of the mea n frequency: 2

2 ( / -r^-u \JT- (t)

l

~

S

J

G

(ttX)

) /

\Vu\

JT~

2

GltiX) > 1.

(t)

I was surprised whe n I obtained thi s estimate , whic h hold s only on the singular set. The dimensio n o f the degenerate singula r se t an d the influence o f the degenerat e singular se t on the limit equatio n pos e ope n question s whic h ar e deeply relate d t o harmonic analysi s (se e [42], [43] ) and constitute a challenge t o future research . Thus th e beaut y o f fre e boundar y problem s lie s i n th e fac t tha t i t i s no t a closed field but takes it s position i n between partia l differentia l equations , measur e theory, geometry , harmoni c analysis , etc . I n th e futur e w e may expec t fruitfu l joint researc h no t only wit h th e classical relate d fields bu t also wit h area s suc h a s geometric equations , minima l submanifolds , an d stochastic analysis . References [1] Alt , H.W . & Caffarelli , L.A. , Existenc e an d regularit y fo r a minimu m proble m wit h fre e boundary, J . Rein e Angew . Math . 32 5 (1 981 ) , 1 05-1 44 . [2] Alt , H.W., Caffarelli, L.A . & Friedman, A. , Axially symmetri c jet flows, Arch . Rationa l Mech . Anal. 8 1 (1 983) , 97-1 49 . [3] Alt , H.W. & Phillips, D. , A free boundar y proble m fo r semilinear ellipti c equations , J . Rein e Angew. Math . 36 8 (1 986) , 63-1 07 . [4] Berestycki , H. , Caffarelli , L.A . & Nirenber g L. , Unifor m estimate s fo r regularizatio n o f free boundary problems , i n Analysis and Partial Differentia l Equations , Marce l Dekker , Ne w York , 1990, pp . 567-619. [5] Apushkinskay a D.E. , Shahgholian H . & Uraltseva N.N. , O n the global solution s o f the para bolic obstacle problem , Algebr a i Analiz 1 4 (2002), no. 1, 3-25; Englis h transl. , St . Petersbur g Math. J . 1 4 (2003) , 1 -1 7 . [6] C . Bandl e & I . Stakgold , Th e formatio n o f th e dea d cor e i n paraboli c reaction-diffusio n problems, Trans . Amer . Math . Soc . 286 (1 984) , 275-293 . [7] Brezis , H . & Kinderlehrer D. , The smoothness o f solutions t o nonlinea r variationa l inequali ties, Indian a Univ . Math . J . 2 3 (1 974) , 831 -844 . [8] Buckmaster , J.D . & Ludford, G.S.S. , Theor y o f laminar flames, Cambridg e Universit y Press , Cambridge-New York , 1982. [9] Caffarelli , L.A. , A monotonicity formul a fo r hea t function s i n disjoint domains , i n Boundar y value problem s fo r partia l differentia l equation s an d applications , Masson , Paris , 1 993 , pp . 53-60. [10] Caffarell i L.A. , Th e regularit y o f fre e boundarie s i n highe r dimensions , Act a Math . 1 3 9 (1978), 1 55-1 84 .

12

GEORG S . WEIS S Caffarelli, L.A. , Th e obstacl e proble m revisited , J . Fourie r Anal . Appl. , 4 (1 998) , 383-402 . Caffarelli, L.A. , Unifor m Lipschit z regularit y o f a singula r perturbatio n problem , Differentia l Integral Equation s 8 (1 995) , 1 585-1 590 . Caffarelli, L.A. , Karp , L . & ; Shahgholian, H. , Regularit y o f a fre e boundar y wit h applicatio n to th e Pompei u problem , Ann . Math . 1 5 1 (2000) , 269-292 . Caffarelli, L.A . & Kenig, C.E. , Gradien t estimate s fo r variabl e coefficien t paraboli c equation s and singula r perturbatio n problems , Amer . J . Math . 1 2 0 (1 998) , 391 -439 . Caffarelli, L . A. , Lederman , C . & Wolanski, N. , Pointwis e an d viscosit y solution s fo r th e limi t of a tw o phas e paraboli c singula r perturbatio n problem , Indian a Univ . Math . J . 4 6 (1 997) , 719-740. Caffarelli, L . A. , Lederman , C . & ; Wolanski, N. , Unifor m estimate s an d limit s fo r a tw o phas e parabolic singula r perturbatio n problem , Indian a Univ . Math . J . 4 6 (1 997) , 453-489 . Caffarelli, L.A . & ; Riviere, N.M. , Smoothnes s an d analyticit y o f fre e boundarie s i n variationa l inequalities, Ann . Scuol a Norm . Super . Pis a CI . Sci . I V 3 (1 976) , 289-31 0 . Caffarelli, L.A . & Riviere , N.M. , Asymptoti c Behavio r o f Fre e Boundar y Point s a t thei r Singular Points , Ann . Math . 1 0 6 (1 977) , 309-31 7 . Caffarelli, L.A . & ; Vazquez, J.L. , A fre e boundar y proble m fo r th e hea t equatio n arisin g i n flame propagation . Trans . AM S 34 7 (1 995) , 41 1 -441 . Chen, X.-Y. , Matano , H . & ; Mimura, M. , Finite-poin t extinctio n an d continuit y o f interface s in a nonlinea r diffusio n equatio n wit h stron g absorption , J . Rein e Angew . Math . 45 9 (1 995) , 1-36. Choe H.J . & Weiss , G.S. , A Semilinea r Paraboli c Equatio n wit h Fre e Boundary , Indian a Univ. Math . J . 5 3 (2003) , 1 9-50 . Flucher, M. , A n asymptoti c formul a fo r th e minima l capacit y amon g set s o f equal area , Calc . Var. Partia l Differentia l Equation s 1 (1 993) , 71 -86 . Friedman, A. , Variationa l principle s an d free-boundar y problems . Joh n Wile y & Sons , Inc. , New York , 1 982 . Friedman, A . & Herrero, M.A. , Extinctio n propertie s o f semilinea r hea t equation s wit h stron g absorption. J . Math . Anal . Appl . 1 2 4 (1 987) , 530-546 . Herrero, M.A . & ; Velazquez, J . J.L., O n th e dynamic s o f a semilinea r hea t equatio n wit h stron g absorption, Comm . Partia l Differentia l Equation s 1 4 (1 989) , 1 653-1 71 5 . Herrero, M.A . & Velazquez, J.J.L. , Approachin g a n extinctio n poin t i n one-dimensiona l semi linear hea t equation s wit h stron g absorption , J . Math . Anal . Appl . 1 7 0 (1 992) , 353-381 . Lacey, A.A . & Shillor , M. , Electrochemica l an d electro-discharg e machinin g wit h a threshol d current, IM A J . Appl . Math . 3 9 (1 987) , 1 21 -1 42 . Luckhaus, S. , Solution s fo r th e two-phas e Stefa n proble m wit h th e Gibbs-Thomso n la w fo r the meltin g temperature , Europea n J . Appl . Math . 1 (1 990) , 1 01 -1 1 1 . Martinson, L.K. , Th e finit e velocit y o f propagatio n o f therma l perturbation s i n medi a wit h constant therma l conductivity , U.S.S.R . Comput . Math , an d Math . Phys . 1 6 (1 976) , 1 41 -1 49 . Reifenberg, E.R. , A n epiperimetri c inequalit y relate d t o th e analyticit y o f minima l surfaces , Ann. Math . 8 0 (1 964 ) 1 -1 4 . Rosenau, Ph . & Kamin , S. , Therma l wave s i n a n absorbin g an d convectin g medium , Physic a D 8 (1 983) , 273-283 . Sakai, M. , Regularit y o f a boundar y havin g a Schwar z function , Act a Math . 1 6 6 (1 991 ) , 263-297. Schaeffer, D.G. , Som e example s o f singularitie s i n a free boundary , Ann . Scuol a Norm . Super . Pisa CI . Sci . I V 4 (1 977) , 1 33-1 44 . Shahgholian, H . & Uraltseva, N. , Regularit y propertie s o f a fre e boundar y nea r contac t point s with th e fixe d boundary , Duk e Math . J . 1 1 6 (2003) , 1 -34 . Taylor, J.E. , Regularit y o f th e singula r set s o f two-dimensiona l area-minimizin g flat chain s modulo 3 i n R 3 , Invent . Math . 2 2 (1 973) , 1 1 9-1 59 . Weiss, G.S. , A homogeneit y improvemen t approac h t o th e obstacl e problem , Invent . Math . 138 (1 999) , 23-50 . Weiss, G.S. , Partia l regularit y fo r wea k solution s o f a n ellipti c fre e boundar y problem , Com mun. Partia l Differ . Equation s 2 3 (1 998) , 439-457 . Weiss, G.S. , Self-simila r Blow-u p an d Hausdorff-dimensio n Estimate s fo r a Clas s o f Paraboli c Free Boundar y Problems , SIA M J . Math . Anal . 3 0 (1 999) , 623-644 .

REGULARITY I N FRE E BOUNDAR 1 Y PROBLEM S

3

[39] Weiss , G.S. , A n Obstacle-Problem-Lik e Equatio n wit h Tw o Phases : Pointwis e Regularit y o f the Solutio n an d a n Estimat e o f th e Hausdorf f Dimensio n o f th e Fre e Boundary , Interface s and Fre e Boundarie s 3 (2001 ) , 1 21 -1 28 . [40] Weiss , G.S. , Partia l Regularit y fo r a Minimu m Proble m wit h Fre e Boundary . J . Geom . Anal . 9 (1 999) , 31 7-326 . [41] Weiss , G.S. , Th e Fre e Boundar y o f a Therma l Wav e i n a Strongl y Absorbin g Medium . J . Diff. Equations . 1 6 0 (2000) , 357-388 . [42] Weiss , G.S. , A Singula r Limi t arisin g i n Combustio n Theory : Fin e Propertie s o f th e Fre e Boundary, Calc . Var . Partia l Differentia l Equation s 1 7 (2003) , 31 1 -340 . [43] Wolff , T. , Plan e harmoni c measure s liv e o n set s o f ^--finit e length , Ark . Mat . 3 1 (1 993) , 137-172. GRADUATE SCHOO L O F MATHEMATICA L SCIENCES , UNIVERSIT Y O F T O K Y O , 3-8- 1 KOMABA

,

MEGURO, T O K Y O 1 53-891 4 , JAPA N

Translated b y GEOR G S . WEIS S

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Amer. Math . Soc . Transl . (2) Vol . 21 5 , 200 5

http://dx.doi.org/10.1090/trans2/215/02

Behavior o f spatia l critica l point s an d zero s of solution s o f diffusio n equation s Shigeru Sakaguch i

1. Introductio n Since solution s o f partia l differentia l equation s ar e functions , i t i s natura l t o want t o kno w thei r shapes . T o know the shape s o f graphs o f functions, w e begin b y investigating thei r critica l points . Le t u s conside r th e Cauch y proble m fo r th e hea t equation a s a plai n example . I n th e TV-dimensiona l Euclidea n spac e R N (N ^ 1 ) , let u = u(x, t) b e th e uniqu e bounde d solutio n o f th e proble m fo r bounde d initia l data u 0 G L°°(RN). The n u satisfie s (1.1) d

tu

= Au i n R N x (0 , oo) an d u(x,0) = u o(x) i n R^ .

Here, x = ( x i , . . . , £TV) £ R N, d tu — | |, an d Au = Ylj=i f^ f • Of course , w e hav e the integra l representatio n o f u vi a Gaussia n kernel : (1.2) u{x,t)

= (47rt)~ ^ / e~

Here \x — y\ = (X^7=i( x j~ ~ Vj) 2) > UQ satisfie s th e following :

ano

1 SL L

d 'uo(y) dy.

^ &V ~ dy\dy2 • • • dy^. Le t u s assum e tha t

(1.3) UQ

i s nonnegativ e an d bounded , an d uo =£ 0 ;

(1.4) th

e suppor t o f UQ (supp uo) i s a compac t set .

For eac h t > 0, defin e tw o set s M(i) an d C(t) b y = {xeR N :

(1.5) M(t)

u(x,t) = max u(y,t)}, yeRN

N

(1.6) C(t)

= {xeR :

Vu(x,t) =

0}.

Here Vu = ( | ^ , . . . , ^ - ) . \ ox i ' '

OXN

'

Note tha t M(t) C C(i). M(t) i s calle d th e se t o f ho t spots , an d C(t) i s calle d the se t o f spatia l critica l points . Fro m th e integra l representatio n (1 .2 ) o f ti , w e obtain th e followin g fact s ([ChaK] , [JS]) . (a) Fo r eac h t > 0, C(t) i s contained i n th e close d conve x hul l o f supp UQ. This articl e originall y appeare d i n Japanes e i n Sugak u 5 4 (3 ) (2002) , 249-264 . 2000 Mathematics Subject Classification. Primar y 35K57 ; Secondar y 35B05 . ©2005 America n Mathematica l Societ y 15

16

SHIGERU SAKAGUCH I

(b) Ther e exist s a tim e T > 0 suc h t h a t C(t) consist s o f on e poin t fo r eac h t^.T. Precisely , fo r eac h t ^ T , ther e exist s a poin t x(t) G M^ satisfyin g C(t) = M(t) = {*(*)} . (c) A s t — » oo , x(t ) tend s t o th e Euclidea n cente r o f mas s o f ito Let u s conside r a n analogou s proble m fo r th e initial-Dirichle t problem . W e consider nonnegativ e initia l d a t a uo G L°°(ft ) satisfyin g UQ ^ 0 , wher e ft i s a bounded domai n i n R N (N ^ 1 ) . Le t u = u(x,t) b e th e uniqu e solutio n o f th e following initial-Dirichle t proble m fo r th e hea t equation : (1.7) d

= Au i

n fix(0

(1.8) u

= 0 o

n 0 th e firs t eigenvalue yield s (1.10) —

A(/?i = \\tpi i

n ft, (pi > 0 i n ft, an d c(fi(x) i

Furthermore, w e se e t h a t th e convergenc e i n (1 .1 1 ) i s unifor m o n ft, an d b y th e regularity theor y o f solution s o f paraboli c partia l differentia l equation s an y second order partia l derivativ e o f e Xltu wit h respec t t o spac e variable s converge s t o t h a t o f c(fi uniforml y o n an y compac t se t i n ft. I n conclusion , ther e exis t a tim e T > 0 an d a poin t x(t) ett (t^T) suc h t h a t M(t) = {x(t)} fo r eac h t ^ T an d a s t - > o o x(t) tends t o th e poin t XOQ at whic h cpi attain s it s maximum . Next, le t u s assum e t h a t ft i s a bounde d conve x domai n an d UQ = 1 . The n it follow s fro m th e maximu m principl e fo r th e hea t equatio n t h a t 0 < u(x,t) < 1 (x £ ft, t > 0) . Moreover , w e se e t h a t , fo r eac h tim e t > 0 , ever y isothermi c surface o f u i s conve x an d smooth , an d th e se t o f ho t spot s M(t) consist s o f on e point. Precisely , a resul t o f [BL ] (se e als o [Ko] ) show s t h a t \ogu{x,t) i s concav e i n x, which , togethe r wit h th e analyticit y o f w in x , implie s t h a t fo r eac h tim e t > 0 there exist s a uniqu e poin t x(t) G ft satisfyin g (1.12) C(£

) = M(t) =

{x(t)}.

Define th e functio n d = d(x) b y (1.13) d{x)=

dis

t (ir, +0 ,

G ft),

C R I T I C A L P O I N T S AN D ZERO S O F D I F F U S I O N E Q U A T I O N S

17

since th e functio n —4£log( l — u(x,t)) attain s it s maximu m a t x = x(t) fo r eac h t > 0 an d a resul t o f Varadha n [V ] (se e als o [FW] ) show s t h a t (1.15) - 4 £ l o g (

l - u(x,t)) —

» d(x) 2 a

s t — • +0 uniforml y o n ft.

In conclusion , th e ho t spo t x(i) start s fro m / an d a s t — > oo i t tend s t o th e poin t XOQ at whic h th e firs t eigenfunctio n pi attain s it s maximum . Concerning th e behavio r o f th e ho t spo t x(t), le t u s giv e a n exampl e wher e x(t) really moves . Le t N = 2 an d conside r a semi-dis k Q = {(x\,X2) G l 2 : x\ - f x\ < 4, xi > 0} . T h e n I = {(1 ,0)} . P u t ft* = {{x 1 ,x2) e £1 : x x > 1 } , an d conside r the functio n v{x\^X2) — 0 o n dQK Moreover, v ^ 0 , which , togethe r wit h th e monotonicit y o f th e first eigenvalu e with respec t t o th e domai n perturbation , implie s t h a t v ^ 0 i n Q.K (Thi s follow s also fro m th e followin g argument . Fo r th e solutio n u(x,t), conside r th e functio n w(xi,X2,t) = u(2 — Xi,x2,t) — u(xi,X2,t). Then , w satisfie s th e hea t equatio n i n ft* x (0 , 00) , w ^ 0 o n dVfi x (0,oo) , an d w(x,0) = 0 . Henc e i t follow s fro m th e maximum principl e fo r th e hea t equatio n t h a t w ^ 0 . Therefore , th e convergenc e (1.11) yield s t h a t v ^ 0. ) Furthermore , th e minimu m principl e fo r superharmoni c functions implie s t h a t v > 0 i n Q.K Consequently , b y usin g Hopf' s boundar y poin t lemma (se e [GT] , Lemm a 3.4 , p . 34 ) a t th e boundar y poin t (1 ,0 ) G («i,... , KN-I) is constant o n S can be converted into th e condition tha t g satisfies a secon d orde r ellipti c partia l differentia l equa tion, sinc e J ^ > 0 (i = 1 , • • • , N — 1 ) . In the case wher e $ i s given b y (4.6), thi s equation i s a secon d orde r ellipti c partia l differentia l equatio n o f Monge-Amper e type.

How abou t th e case wher e the domain Q is unbounded? B y collaborating wit h R. Magnanin i (Universit a d i Firenze), th e author recentl y obtaine d th e followin g theorem b y a proof almos t simila r t o that o f Theorem 4.2.

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3

T H E O R E M 4. 6 (Magnanini an d Sakaguchi) . Let Ct be an exterior domain in 1^ (N ^ 2 ) satisfying the exterior sphere condition, and suppose that D is a domain satisfying the interior cone condition, and such that D C £1 . Assume that the boundary dD is a stationary isothermic surface of the solution u of problem (3.1 ) (3.3); that is, condition (4.3 ) holds true for some function a : (0 , oo)— > (0, oo). Then dVt must be a sphere. That is, Q must be the exterior of a ball.

Since th e boundary dQ is compact, th e proof o f this theore m i s based o n th e fact tha t w e can use arguments simila r t o those use d i n the proof o f Theorem 4.2 (see [Sa k 6]) . Messoud A . Efendie v gav e the following conjectur e t o the author a t the conference title d "Nonlinea r diffusiv e system s an d related topics " a t Kyot o i n October , 2001. CONJECTURE 4. 7 (Efendiev). Consider a domain Q whose boundary dQ is not compact. In particular, for a locally Lipschitz continuous function (p = (p(x') [x' = (xi,...,Xiv-i) GlR^- 1 ), set Q = {x G 1^ : x N > (p(x')}. If there exists a stationary isothermic surface of u of problem (3.1 )-(3.3) ; then dQ must be a hyperplane. Our answe r t o this conjectur e i s the following theorem . THEOREM

(4.7) Vp(x

4.8 . Let the function p satisfy f

)=

o(|x ; |2) near infinity,

and suppose that ft satisfies the uniform exterior sphere condition. Assume that there exists a domain D with D e l] such that dD is a stationary isothermic surface of the solution u of problem (3.1 )-(3.3) , that is, condition (4.3 ) holds true for some function a : (0 , oo)—- > (0, oo). Furthermore, assume that the mean curvature of dD with respect to the exterior normal direction to dD is nonnegative everywhere. Then dfl must be a hyperplane. Here, O is said t o satisfy th e uniform exterior sphere condition i f there exist s r > 0 such tha t fo r every y G dQ we hav e a ball B r(z) wit h B r(z) n i ] = {y}. Sinc e the domai n ft i s given b y an unbounded domai n abov e a locall y Lipschit z graph , by usin g th e maximum principl e w e see that -J^r- > 0 i n H x (0 , oo). Thus , i n Theorem 4. 8 the boundary dD i s analytic b y condition (4.3) , an d hence th e mean curvature o f dD i s well-defined i n Theorem 4.8 . With th e help o f a Bernstei n typ e theore m du e to Caffarelli , Nirenberg , and Spruck (se e Theorem 2" in [CNS]), w e can prove thi s theorem . Th e details wil l be given i n a forthcoming paper . Every theore m (se e Theorem s 4.2 , 4.6 , and 4.8) conclude s tha t dVt must b e an isoparametric hypersurfac e i n RN. Thus , w e may have a theorem whic h conclude s that dtt mus t b e a spherical cylinder . Also , i t is conjectured tha t w e cannot hav e such a theorem wher e th e surface conclude d i s not an isoparametric hypersurface . These ar e problems t o be solved. Note adde d i n translation . Whe n thi s translatio n wa s being mad e b y th e author himself , in the process he noticed tha t Theore m 4. 8 needs one more assump tion concernin g th e mean curvatur e o f dD. Thi s correctio n ha s been incorporate d into th e present translation .

24

SHIGERU SAKAGUCH I

5. Balanc e la w an d it s rol e In this section, w e explain the balance law of the temperature aroun d stationar y critical point s an d zero s o f th e hea t flow , whic h i s mentione d i n Section s 3 and 4 . THEOREM 5. 1 (Balanc e law) . Let G be a domain in R N (N ^ 2) . Fix a point XQ G G, and set d* = dist(#o,dG) . Suppose that v = v(x,t) satisfies the heat equation in G x (0 , oo). Then the following hold: (i) v(xo Jt) = 0 (t > 0) if and only if (5.1) /

v{x,t)dS

x

=

0 ((r,£ ) G (0,d0 x (0,oo)) ,

dBr(x0)

where dS x denotes the area element of the sphere dB (ii) V^(xo , t) = 0 (t > 0 ) if and only if (5.2) /

(x-x

0)v(x,t)

dS

x

r(xo).

= 0 ((r,t ) G (0,d*) x (0,oo)) .

96 r (a:o)

This balanc e la w resemble s th e mea n valu e theore m fo r harmoni c functions , and i t ma y b e a kin d o f theorem i n th e functio n theor y fo r function s satisfyin g th e heat equation , (ii ) wa s prove d i n [M S 1 ] , an d i t wa s state d i n [M S 2 ] tha t (i ) can b e prove d similarly . I t wa s show n i n [Sa k 4 ] tha t th e balanc e la w simila r t o (ii) hold s tru e fo r th e hea t flo w i n TV-dimensiona l spher e S ^ an d TV-dimensiona l hyperbolic spac e M N. Also , i n [M S 4] , anothe r proof o f (i ) an d a shorte r proo f o f (ii) wit h th e ai d o f (i ) wer e given . Her e le t u s mentio n thes e proofs . P R O O F O F THEORE M 5.1 . I n (i) , sinc e sufficienc y i s obvious , le t u s sho w ne cessity. Fo r (r , i) G [0, d*) x (0 , oo), defin e th e functio n p = p(r, t) b y

(5.3) p(r,t)

= / v(xo

+ ruj.t) dto,

where S N~X i s th e uni t spher e i n R N centere d a t th e origin , uu = (CJI , . .. ,LJN) £ SN~X i s a uni t vecto r i n R^ , an d duo denote s th e are a elemen t o f S N~1 . Then , p i s analytic wit h respec t t o r o n [0 , d*) an d satisfie s TV- 1 (5.4) p

t=prr-\

(5.5) p(0,t)=p

p r(0,t)=0

r

i n (0,d* ) x (0 , oo),

(£>0)

.

Hence, (5.1 ) follow s fro m th e fac t tha t ^ | ( 0 , * ) = 0 ( t > 0 , fc = 0 , l , 2 , . . . ) by inductio n o n th e intege r k. (I n [Sa k 4 ] a simila r argumen t wa s give n i n detail. ) Next, le t u s deduc e (ii ) fro m (i) . Sinc e ever y componen t o f th e vecto r Vv(x,i) also satisfie s th e hea t equation , Vv(xo,t) — 0 (t > 0) i s equivalent t o Vv(x, t) dS x = 0 ((r , t) G (0, d*) x (0 , oo)). dBr(x0)

C R I T I C A L P O I N T S AN D ZERO S O F D I F F U S I O N E Q U A T I O N S 2

5

Furthermore, b y integratin g th e latte r formul a wit h respec t t o r , w e se e t h a t i t i s equivalent t o Vv(x, t) dx = 0 ((r , t) G (0, d*) x (0 , oo)). Br(x0)

Therefore, b y th e divergenc e theorem , thi s formul a i s equivalen t t o (5.2) . Thu s (ii ) is proved . • Concerning th e Cauch y proble m fo r th e hea t equatio n (1 .1 ) , wit h th e hel p o f the integra l representatio n o f th e solutio n (1 .2) , th e initia l d a t a satisfie s th e balanc e law (5.2 ) wit h respec t t o a poin t xo G R^ i f an d onl y i f th e poin t XQ i s a stationar y critical poin t o f th e solution . Henc e th e followin g theore m holds : THEOREM 5. 2 ([M S 1 ]) . Fix a point xo G R ^ . Let u be the unique solution of the Cauchy problem (1 .1 ) for initial data uo G L°°(R N). Then, Vu(xo,t) = 0 (t > 0 ) if and only if (5.6) /

(x

— xo)uo(x) dS

x

— 0 (a.e

. r > 0) .

JdBr(x0)

R E M A R K 5.3 . Whe n N = 1 i n particular , i t i s eas y t o characteriz e stationar y critical point s o f th e hea t flow . Fo r simplicity , w e conside r th e Cauch y proble m (1.1). Le t XQ G R b e a stationar y critica l poin t o f th e solutio n u. Namely , suppos e t h a t d xu(xn,t) — 0 (t > 0) . Defin e th e functio n v = v(x,t) b y

(i () i » ( * , < ) - ' " (2^0 ' — X, t) i

f x 0, and conside r th e domain f J satisfying B$(0) C Q , where B§(0) denote s a n open bal l centere d a t the origin an d with radiu s S > 0. W e consider th e following initial-boundar y valu e problem : (6.1) d

tu

= Au inftx(0,oo)

(6.2) ( l - a ) f ^ + c

m= 0o

(6.3) u(x,0)

= uo(x) i

,

n C C 8. T H E O R E M 6. 2 ( [ M S 2]) . For any initial data UQ G $ c the solution u of the initial-boundary value problem (6.1 )-(6.3 ) always satisfies Vit(0,t ) = 0 (t > 0), if and only if Q is centro-symmetric with respect to the origin. We hav e simila r theorem s i f w e conside r spatia l zer o point s instea d o f spatia l critical points , o r i f w e conside r th e wav e equatio n instea d o f th e hea t equatio n (se e [MS 2j) . Also , i t wa s show n i n [Sa k 4 ] t h a t simila r theorem s hol d tru e fo r th e heat flo w i n t h e TV-dimensiona l spher e S ^ an d i n TV-dimensiona l hyperboli c spac e MN. (A s fa r a s Theore m 6. 1 goes , onl y th e cas e wher e a = 1 was deal t wit h fo r th e heat flow i n S ^ an d U N.) Consider a subgrou p G o f th e orthogona l grou p O(N), an d defin e t h e famil y 3>(G) o f initia l d a t a UQ by $(G) = {w 0GC0oo(^(0)) :

u 0(x)=u0(gx) f o r a n

y (*,(/)G^(0)xG} .

Then, concernin g th e notio n "essential " mentione d i n Sectio n 3 , w e hav e th e fol lowing proposition . P R O P O S I T I O N 6. 3 ( [ M S 3]) . G is essential if

and only if ${G) C

$.

Hereafter, w e suppos e t h a t N = 2 , £1 i s a bounde d an d simpl y connecte d domain, an d a = 1 (tha t is , th e initial-Dirichle t problem) . Fo r an y intege r d ^ 2 , let Gd b e t h e cycli c grou p generate d b y rotatio n throug h a n angl e ^ . Sinc e Gd i s essential, Propositio n 6. 3 implie s t h a t $ ( G d ) C $ . Not e t h a t $ c = $ ( G 2 ) . The n the followin g theore m hold s true : T H E O R E M 6. 4 ( [ M S 3]) . Let u be the solution of (6.1)-(6.3) with a = 1 . Then:

the initial-Dirichlet problem

(1) Let d = 2 or d = 3 . Then, Vu(Q,t) = 0 (t > 0 , u 0 G $(G d)) if and only if Q is invariant under the action of Gd(2) Let d = 4 . Then, Vu(0,t) = 0 (t > 0 , u 0 G $ ( G 4 ) ) if and only if fi is invariant under the action of G 2 (that is, ft is centro-symmetric with respect to the origin). The proo f wa s base d o n th e m e t h o d o f comple x analysi s wit h th e ai d o f th e Riemann mappin g theore m (se e [M S 3 ] fo r details) . T h e cas e wher e d = 5 i s no t known, a s i n t h e conjectur e mentione d a t th e en d o f Sectio n 3 . B y includin g this , w e write on e o f t h e conjecture s i n [M S 3 ] regarde d a s a generalizatio n o f Theore m 6.4 . C O N J E C T U R E 6. 5 ( [ M S 3]) . Let d^ 2 be an arbitrary integer, and let u be the solution of the initial-Dirichlet problem (6.1 )-(6.3 ) with a = 1 . Then, Vu(0,£ ) = 0 (t > 0 , UQ ^ \x — £|eos#} b e the con e with verte x a t £ and aperture 0 , where ( £ — x, £) stand s fo r th e inne r product . W e say tha t a functio n / on 5(0,1 ) ha s a non-tangential limi t ^ if lim f(x)

= £ fo r ever y 0 , 0 < 0 < TT/2 .

Fatou Theorem . Every positive harmonic function on the unit ball JB(0, 1 ) has a non-tangential limit at almost all boundary points on 5(0,1 ) . It ma y b e natura l t o exten d th e Herglot z an d Fato u theorem s t o mor e genera l domains. Suppos e D i s a bounde d smoot h domain . Le t G(x,y) b e th e Gree n function fo r D , i.e. , —AG(-,y) = 5 y i n D i n th e distributio n sens e an d G(-,y) vanishes o n th e boundar y fo r eac h fixe d y G D. I f a harmoni c functio n h o n D is continuou s u p t o th e boundary , the n h i s represente d b y th e Poisso n integra l ——G(x,y)h(y)da(y). Mor e generally , i f h i s positiv e o n D, instea d o f bein g / .3D OUy _ continuous o n D, the n ther e exist s a uniqu e measur e fih on 3D suc h tha t h(x) = o

——G(x1y)dnh{y)- Thi s Poisso n integra l representation , no t s o explici t a s fo r 3D driy

the ball , yield s th e Fato u theore m fo r positiv e harmoni c function . The situatio n i s muc h mor e complicate d fo r genera l domains . A Lipschit z domain ma y hav e wedge s an d it s boundar y ma y admi t point s havin g n o normals . A fracta l domain , suc h a s a snowflak e domain , ma y hav e boundar y o f Hausdorf f dimension bigge r than n — 1 and th e 'surfac e measure ' ma y los e its meaning. Marti n [57] established th e integral representation o f positive harmonic functions o n general domains wit h th e minimu m requirement : th e existenc e o f th e Gree n function . H e introduced a n idea l boundary , no w calle d th e Martin boundary, an d succeede d in establishin g th e integra l representatio n o f positiv e harmoni c functions . Le t u s give a n outlin e o f hi s idea . Fo r th e precis e definitio n th e reade r i s referre d t o th e bibliography, e.g. , [23] . Let h b e a positiv e harmoni c functio n o n a genera l domai n D wit h Gree n function G(x,y). Tak e a n exhaustio n {Dj} o f D. Le t RhJ (x) = mi{u{x) : u > 0 is superharmoni c o n D wit h u > h o n Dj}. The lowe r regularizatio n R hJ o f R hJ i s a superharmoni c functio n o n D. Thi s i s called th e balayage or regularize d reduce d functio n o f h t o Dj. B y definitio n R^ J vanishes o n th e boundar y an d i s harmoni c i n D \ dDj, s o tha t th e balayag e i s a Green potentia l / G(-,y)di/j(y) o f som e measur e Vj o n dDj. Le t u s fi x a poin t XQ £ D. Sinc e R hJ = h o n Dj, w e obtain a n integra l representatio n /

G{x,y)dv0(y) =

[ G(x v) / ' G(x

0,y)diSj(y)

fo

rx G D

r

MARTIN B O U N D A R Y A N D B O U N D A R Y H A R N A C K PRINCIPLE

35

Let u s conside r a sequenc e o f measure s d(ij{y) = G(xo,y)dvj(y). Substitutin g x — XQ in th e abov e integra l representation , w e observ e tha t th e tota l mas s o f /J,J is identically equa l t o h(xo). Thus , a subsequenc e ha s a wea k limi t /z^ , which give s an integra l representatio n

Wl^**""

for al l x G D.

However, thi s i s no t sufficient . Th e rati o o f th e Gree n function s G(x,y)/G(xo,y) may no t exten d continuousl y t o th e boundary . We , thus , conside r th e minimu m ideal boundary A wher e the ratio extends continuously. The n th e extension K(x,y) of G(x,y)/G(xo,y) t o D x A give s th e integra l representatio n h(x] = / K(x,y)dfi h(y). JA The idea l boundar y A an d th e kerne l K(x,y) ar e calle d th e Martin boundary an d the Martin kernel. Not e tha t K(x,y) i s a positiv e harmoni c functio n o n D wit h K(xo,y) = 1 . Th e Marti n kerne l ca n b e identifie d wit h a Marti n boundar y point . Unfortunately, th e measur e /i^ i n th e abov e integra l representatio n nee d no t b e unique. Thi s lead s to a minimal boundar y point , o r a n essentia l point , amon g Mar tin boundar y points . I n general , a positiv e harmoni c functio n u i s calle d minima l if ever y non-negativ e harmoni c functio n majorize d b y u coincide s wit h a constan t multiple o f u. I f th e Marti n kerne l K(-,y) i s minimal , the n y i s calle d a minima l Martin boundar y point . Th e se t o f al l minima l Marti n boundar y point s i s calle d the minimal Martin boundary an d i s denote d b y Ai . Th e remainin g boundar y i s called th e non-minima l Marti n boundar y an d i s denoted b y Ao - Herglotz' s theore m is now generalize d a s follows : Martin Integra l Representation . For each positive harmonic function h on D there exists a unique measure fih on A i such that h{x) = / K(x,y)dfi

h(y).

J Ax

Based o n th e Marti n integra l representation , Nair n [64 ] extende d th e Fato u theorem. W e sa y tha t E C D i s minimally thin a t y G Ai i f R^t \ ^ K(-,y). W e say that a function f on D ha s a minimal fine limit a t y i f the limi t o f / , excep t fo r a certai n se t minimall y thi n a t y, exists . Minima l thinnes s ma y b e considere d t o be a natura l analogu e o f mor e familia r thinness. I n general , a se t E C D i s thin a t y G D i f an d onl y i f R^t \ ^ G(-,y). Irregularit y o f a domai n wit h respec t t o th e Dirichlet proble m i s equivalent t o th e thinnes s o f th e complement . Th e thinnes s i s completely characterize d b y th e th e non-existenc e o f a barrie r an d b y th e Wiene r criterion. Fatou-Nairn-Doob Theorem . Let u and v be positive harmonic functions on D with Martin representation measures \i u and \i V} respectively. Then, for \i va.e. y G Ai, the ratio u/v has minimal fine limit at y, which coincides with the Radon-Nikodym derivative dfih/dfi v(y). Armitage an d Gardine r [23 ] i s a goo d referenc e fo r th e abov e accounts . Th e monograph cover s almos t al l classical result s i n potential theory , fro m introductor y

36

HIROAKI AIKAW A

materials t o advance d subjects , suc h a s minima l fine limits . Helm s [48 ] i s als o a nice introduction t o potential theory . Constantinesc u an d Corne a [37 ] is classical in connection wit h Rieman n surfaces . Potentia l theor y i s closely related t o probabilit y theory. I n thi s regar d [42] , [70] , and [26 ] ar e nic e references . A simpl e expositio n of Nairn' s theore m i s included i n th e appendi x o f [8]. Now le t u s stud y th e abov e genera l theorem s fo r specifi c domains . I f D i s sufficiently smooth , the n G(xo,y) i s comparabl e t o th e distanc e functio n Sjj(y) = dist(2/, dD)\ th e Marti n kerne l K(x, y) i s the inwar d norma l derivativ e o f the Gree n kernel time s a positiv e functio n o f y. Includin g thi s positiv e functio n i n th e rep resentation measure , w e ma y identif y th e Marti n representatio n an d th e Poisso n representation. Sinc e K(-,y) vanishe s continuousl y o n 3D \ {y}, i t follow s tha t K(-,y) i s minimal , an d th e Marti n boundar y coincide s wit h 3D an d consist s o f only minima l points . Th e argument s o f Brelo t an d Doo b [34 ] giv e a correspon dence betwee n minima l fine limit s an d non-tangentia l limits . For a genera l domain , however , th e Marti n boundar y ma y diffe r fro m th e Eu clidean boundary ; non-minima l point s ma y appear . N o explicit criteri a fo r minima l thinness ar e known . Th e mor e genera l domain s w e consider , th e les s precis e prop erties w e obtain. However , ther e ar e severa l interestin g classe s o f domain s betwee n smooth domains and general domains, for which the Martin boundaries an d minima l fine topologie s ca n b e described . I n th e stud y o f suc h domains , w e recogniz e tha t properties o f harmoni c function s i n smoot h domains , sometime s take n fo r granted , may heavil y depen d o n th e geometri c natur e o f domains . Clarifyin g th e relation ship betwee n th e geometri c structur e o f the domai n an d th e propertie s o f harmoni c functions o n i t ma y hel p ou r intrinsi c understandin g o f harmoni c functions . Ho w can w e stud y suc h a relationship ? Th e boundar y Harnac k principl e state d i n th e next sectio n wil l provid e a powerfu l tool . 2. Boundar y Harnac k principl e Hereafter, A stand s fo r a constan t bigge r tha n 1 , whos e valu e ma y chang e from on e occurrenc e t o th e next . W e sa y tha t tw o positiv e quantitie s / an d g ar e comparable, an d writ e / ~ g, i f A -1 f < g < Af wit h som e A > 1 . Le t u s recal l the classica l Harnac k principle . Le t K b e a compac t subse t o f a domai n D. The n there exist s A > 1 , dependin g onl y o n K an d D , suc h tha t fo r al l pair s o f positiv e harmonic function s u an d i ; o n D , u(x)/v(x) ^ A £ r^ ) 7 ) i < A forx,yeK. Let us , now , remov e th e assumptio n tha t K i s a compac t subse t o f D. Wha t happens i f K touche s th e boundary ? N o comparison o f positive harmoni c function s u an d v wil l b e available , becaus e o f th e boundar y value s o f u an d v. T o get around this , w e impose th e requiremen t u = v — 0 on the intersectio n o f K an d th e boundary. Mor e precisely , th e boundar y Harnac k principl e i s formulated a s follows . Boundary Harnac k Principle . Let K be a compact set intersecting 3D. Suppose V C R n is an open set containing K. Then, there exists A > 1 , depending only on K, V and D, such that u(x)/v(x) u(y)/v(y)

0 . Th e boundar y conditio n o f th e Gree n functio n i s understoo d t o b e G(-,y) = 0 q.e . o n dD. Wit h thi s interpretation , th e Gree n functio n alway s ex ists an d th e Marti n boundar y i s defined , wheneve r th e domai n ha s a non-constan t positive harmoni c function . 3. Generalizatio n o f domain s Let u s giv e a surve y o f generalization s o f domain s t o whic h th e Marti n theor y and th e BH P ar e applicable . Thi s surve y ma y als o offe r som e (incomplete ) histor y of understandin g o f harmonicit y withou t relyin g o n explici t formulae . 3.1. Lipschit z domain . Suppos e tha t a domai n enjoy s som e smoothnes s as sumption. Fo r example , suppos e tha t a t eac h boundar y poin t ther e exis t tw o ball s tangent a t tha t point , on e of which lie s in the domai n an d th e othe r outsid e th e do main. Then , th e compariso n o f the Gree n functio n fo r th e domai n an d thos e fo r th e ball an d th e complemen t o f th e close d bal l give s tha t a positiv e harmoni c functio n vanishing o n a portio n o f th e boundar y i s comparabl e t o th e distanc e functio n t o the boundar y nea r th e portion . Thu s th e BH P readil y follow s i n this cas e fro m thi s observation. However , problem s aris e fo r non-smoot h domains . A domai n whos e boundary i s represente d locall y a s th e grap h o f a Lipschit z continuou s functio n i s called a Lipschitz domain . Th e BH P wa s recognized fo r th e first tim e i n connectio n with Lipschit z domains . Even fo r smoot h domains , suc h a s a bal l an d a hal f space , i t i s necessar y t o study Lipschit z domains , i f w e conside r precis e propertie s o f harmoni c functions . The followin g loca l Fato u theore m i s a typica l example . I n Sectio n 1 w e state d the Fato u theorem , whic h ca n b e strengthe n t o th e followin g loca l form . Le t u s illustrate i t fo r th e uppe r hal f spac eM ™ for simplicity . Local Fato u Theorem . Let h be a harmonic function on W^ non-tang entially bounded at E C dW+. Then, h has non-tangential limits at almost all points x G E. Here, w e sa y tha t h i s non-tangentiall y bounde d a t E i f fo r eac h £ G E, ther e is a truncate d non-tangentia l con e T(£ ) wit h verte x a t £ o n whic h h i s bounded . An elementar y rea l analysi s show s tha t th e apertur e an d th e siz e o f T(£ ) ma y b e considered t o b e constant . Moreover , addin g a suitabl e constant , w e ma y assum e that h i s positiv e o n T(£) . W e observe tha t th e unio n o f non-tangentia l cone s T(£ ) is a Lipschit z domain . Thu s th e abov e loca l Fato u theore m reduce s t o th e globa l Fatou theore m o n a Lipschit z domain .

38

HIROAKI AIKAW A

D = u^ eEno

A FIGURE

1 . Unio n o f non-tangentia l cone s i s a Lipschit z domain .

Carleson [35 ] proved this local Fatou theorem by showing the following Carleso n estimate relate d t o th e BHP . Carleson Estimate . Let £ G dD, and let r > 0 be small If u is a positive harmonic function in the union D of non-tangential cones, vanishing on 3D P i B(£,Ar), then u(x) < Au(^ r) forxeDH

B(f

, r),

where £ r G S(£,r) H D is a non-tangential point, i.e., Sr)(^ r) ~ r. Hunt an d Wheede n [50] , [51 ] applie d th e method s o f Carleso n t o Lipschit z domains an d argued , furthermore , t o prov e th e Fato u theore m an d t o sho w tha t the Marti n boundar y coincide s wit h th e Euclidea n boundary . I n the lat e 1 970s , th e BHP fo r a Lipschit z domai n wa s formulate d explicitly . Ancon a [1 2] , Dahlberg [41 ] and W u [77 ] prove d th e BH P almos t a t th e sam e tim e b y differen t methods . W e distinguish tw o type s o f BHP's . Scale Invarian t Boundar y Harnac k Principle . Let £ G dD, and let r > 0 be sufficiently small. If u and v are positive harmonic functions in D D £?(£, Ar) vanishing on dD n -B(£ , Ar), then u(x)/v(x) . u{y)/v{y)



^



_.

where A > 1 is independent of £, r, u and v. See Figure 6 . This scal e invarian t BHP , o r unifor m BHP , i s importan t fo r applications . Th e BHP o f Dahlber g wa s no t scal e invariant , thoug h h e employe d method s o f rea l analysis an d prove d som e remarkabl e results : th e harmoni c measur e i s absolutel y continuous wit h respec t t o th e surfac e measur e wit h densit y satisfyin g th e Muck enhoupt A^ condition . Hi s work playe d a n importan t rol e i n harmoni c analysi s o n Lipschitz domains . O n th e othe r hand , Ancon a studie d no t onl y harmoni c func tions bu t als o positiv e solution s t o genera l uniforml y ellipti c partia l differentia l equations. Hi s investigatio n proceede d fro m Lipschit z domain s t o mor e genera l non-smooth domain s i n Euclidea n space , manifold s o f negativ e curvatur e an d dis crete graph s ([1 3] , [1 4] , [1 6] , [1 7] , [1 8]) . W u studie d harmoni c function s ver y deeply. He r precis e estimate s o f the harmoni c measur e o f a box influenced th e wor k of Jeriso n an d Keni g an d th e bo x argumen t o f Bas s an d Burdzy . Se e Sectio n 5 .

M A R T I N BOUNDAR Y AN D B O U N D A R Y H A R N A C K P R I N C I P L E 3

9

3.2. N T A domain . Jeriso n an d Keni g [52 ] extende d Lipschit z domain s t o NTA (Non-Tangentiall y Accessible ) domains . The y establishe d th e scal e invari ant BHP , the identificatio n o f the Martin boundary , th e doubling propert y o f the harmonic measure , th e loca l Fato u theorem , an d so on. Thei r method s ar e reminiscent o f those fo r a Lipschit z domain . I t wa s crucial fo r a Lipschit z domai n tha t at eac h boundar y point , ther e ar e interior an d exterior cone s o f the same apertur e and siz e wit h verte x a t th e boundar y point . Fro m th e potential-theoretica l poin t of view , th e cones ma y be twisted, o r even disconnected . Jeriso n an d Kenig calle d a twiste d con e a corkscrew. Le t 0 < c < 1 . Fo r x,y G D, a Harnack chain fro m x t o y i n D i s a sequenc e {B(XJ,C5D(XJ))}JLO suc h tha t x 0 = x, x^ = y an d B(XJ-I,CSD(XJ-I)) f

l B(XJ,CSD(XJ)) ^

0 fo r j — 1, . . ., N. Th e numbe r N i s re -

ferred t o as the length o f the Harnack chain . Th e value o f c is not importan t an d we ma y take c = 1 /2 . Jeriso n an d Kenig sa y that a domai n i s NTA if there exis t constants A an d ro with th e following properties : (i) Corkscrew condition. Fo r an y £ G 3D an d 0 < r < ro , ther e exist s a point x G D n £(£ , r) suc h tha t 5D(X) « r. (ii) Th e complement o f D satisfie s th e corkscrew condition , (hi) Harnack chain condition. I f e > 0, and x,y G D with 8JJ(X) > £, 5jj(y) > e an d \x — y\ < Ce, then ther e exist s a Harnack chai n fro m x t o y whos e length depend s o n C, but not e. Under thes e assumptions , Jeriso n an d Keni g succeede d i n obtainin g th e BHP for an NT A domai n i n the same way as for a Lipschitz domain . A s a result, th e Martin boundary is homeomorphic to the Euclidean boundary . Nevertheless , NTA domain s can b e much mor e complicate d tha n Lipschit z domains . Ther e i s an NTA domain whose boundar y ha s Hausdorf f dimensio n bigge r tha n n — 1, s o tha t th e surfac e measure o f the boundary ma y lose it s meaning . Thus , Dahlberg' s resul t doe s not extends to an NTA domain. Harmoni c analysi s on NTA domains is developed base d on harmoni c measur e i n place o f surface measure . 3.3. Negativel y curve d manifold . Anderso n an d Schoe n [20 ] proved tha t the Marti n boundar y associate d wit h th e Laplace-Beltram i equatio n o f a simpl y connected complet e Riemannia n manifol d i s a sphere attache d a t infinity , provide d the sectiona l curvatur e i s bounde d b y a negativ e constant . Thei r proo f relie d o n a specia l superharmoni c functio n an d complicated calculations . Ancon a [1 6 ] gave an alternativ e proo f base d o n techniques i n potential theor y state d a s above. Hi s method i s easie r an d mor e applicable . Le t C b e a uniforml y ellipti c operato r o n a manifold . Suppos e C i s weakly coercive, i.e. , ther e i s a positiv e superharmoni c function wit h respec t t o C -h el fo r som e e > 0 . Unde r thi s assumption , Ancon a showed th e BHP and identified th e Martin boundary . A domain i n Euclidean spac e is weakly coerciv e i f and only i f it has a strong barrier s. i.e., not only i s s positiv e and superharmoni c bu t als o i t i s more superharmonic , i n the sense tha t s As(x) + g(x z dD{x)

) < 0 fo r som e e > 0.

The existenc e o f a strong barrie r i s equivalent t o Hardy's inequality:

L

0 fo r x e dD an d 0 < r < r 0 .

Here, C a p ^ ^ r ) ^ ) i s th e Gree n capacit y o f E wit h respec t t o B(x,2r) 1 ([ 5]) . Recall t h a t th e famou s Wiene r criterio n fo r th e regularit y o f a boundar y poin t wit h respect t o th e Dirichle t proble m i s ^CapB(a:i21-J)(£>cnB(a:,2-J)) , t S Ca

P B ( x , 2 i - * ) ( £ ( * , 2-*) )

We easil y observ e t h a t C&p B(x2i-j)(Dc f

l B(x, 2

-J

oo.

')) ca n b e replace d b y

CapB(Xi21-J)(9DnB(a:)2-J)). Hence, i f th e boundar y satisfie s th e CDC , the n th e domai n i s regula r an d ha s a uniform barrier . For a plana r domain , th e CD C o f the boundary , th e existenc e o f a strong barrier , the Hard y inequality , an d th e unifor m perfectnes s o f th e boundar y ar e equivalen t to eac h other . I n particular , i f a domai n ha s a stron g barrier , the n i t i s regular . For a domai n i n a highe r dimensiona l space , th e CD C o f th e boundar y implie s th e existence o f a stron g barrier , bu t th e convers e doe s no t hold . A puncture d bal l i s irregular an d doe s no t satisf y th e CD C an d ye t ha s a stron g barrie r ([1 5]) . Char acterization o f domain s havin g a stron g barrie r i s ope n i n th e highe r dimensiona l case. Ancona's metho d i s widel y applicable . Ancon a himsel f applie d i t t o th e Marti n boundary o f a discret e grap h ([1 7]) . Ara i [21 ] , [22 ] extende d i t t o degenerat e elliptic equations . Bonk , Heinonen , an d Koskel a [33 ] studie d i t i n connectio n wit h Gromov hyperbolicity , a s w e shal l se e i n th e nex t section . One ma y wonde r whethe r condition s suc h a s coercivit y an d existenc e o f a stron g barrier ar e essentia l fo r th e identificatio n o f th e Marti n boundar y an d th e BHP . I n fact, i t turn s ou t t h a t ther e i s a certai n famil y o f domains , withou t suc h conditions , for whic h th e B H P holds . 3.4. Irregula r d o m a i n s . Bas s an d Burdz y [27 ] rathe r recentl y establishe d the B H P fo r a Holde r domain 1 an d a Joh n domai n b y usin g probabilisti c tech niques. I n vie w o f th e "Lebesgu e spine" , a Holde r domai n nee d no t b e regula r wit h respect t o th e Dirichle t proble m fo r th e highe r dimensiona l case . T h e hear t o f Bas s and Burdzy' s argumen t require s n o exterio r conditions ; onl y interio r condition s yield th e BHP . Fro m thi s poin t o f vie w th e Bass-Burdz y metho d i s remarkable . Unfortunately, thei r B H P i s no t scal e invarian t an d i s no t sufficien t fo r th e identi fication o f th e Marti n boundary . W e no w rais e a question : I s ther e a domai n mor e general t h a n a n NT A domai n fo r whic h th e scal e invarian t B H P hold s an d whos e Martin boundar y coincide s wit h th e Euclidea n one ? I n th e succeedin g sections , w e shall, indeed , prov e t h a t a unifor m domai n i s required . I n th e procedure , w e shal l understand th e Bass-Burdz y probabilisti c argumen t i n a purel y analyti c way . Th e 1 Here, a Holde r domai n i s a domai n whos e boundar y i s locall y give n b y th e grap h o f a Holde r continuous function . Smit h an d Stegeng a [73 ] sai d tha t a domai n i s Holder i f i t satisfie s th e quasi hyperbolic boundar y condition . A Joh n domai n i s a Holde r domai n i n th e sens e o f Smit h an d Stegenga. Ver y complicatedly , severa l relate d paper s us e thi s notatio n ([74] , [24] , [47]) .

MARTIN B O U N D A R Y A N D B O U N D A R Y H A R N A C K PRINCIPLE

41

crux o f thei r argumen t ca n b e interprete d a s th e estimat e o f th e Gree n functio n b y a decompositio n o f a domain . 4. U n i f o r m d o m a i n a n d J o h n d o m a i n A unifor m domai n i s a domai n closel y relate d t o quasiconforma l mappings . I t has a n intimat e connectio n wit h th e extendabilit y o f B M O function s an d Sobole v functions a s well . Se e [46] , [53] , [54] , [45] , [75] , [69] , an d s o on . Ther e ar e severa l equivalent definition s fo r a unifor m domain . Her e w e adop t th e following : W e sa y t h a t D i s a uniform domain i f ever y pai r o f point s x,y G D ca n b e connecte d b y a curve 7 C D suc h t h a t e(>y) < A\x - y\, mm{t('y{x,z)),t{'Y(z,y))}


CJ£(J(X,Z)) fo

rz G 7 .

We ma y replac e th e cente r xo b y a fixed compac t se t o f D. I n general , th e Joh n constant cj i s les s t h a n 1 . I f cj i s gettin g clos e t o 1 , w e ma y conside r t h a t D i s getting smooth . T h e definitio n o f a Joh n domai n implie s t h a t fo r ever y x G D, there i s a twiste d con e wit h verte x a t x an d fixed apertur e towar d XQ. Becaus e of it s geometri c nature , th e conditio n i s referre d t o a s a carrot condition. I t i s known t h a t a ciga r conditio n withou t bounde d turnin g condition , £(7 ) < A\x — y\, is equivalen t t o th e carro t condition . Balogh an d Volber g [25 ] introduce d a uniformly John domain, a domai n in termediate betwee n a Joh n domai n an d a unifor m domain . Thi s i s give n b y re laxing th e bounde d turnin g conditio n ^(7 ) < A\x — y\ fo r a unifor m domai n t o ^(7) < Apo(x,y), wher e po(x 1 y) i s th e interna l metric , o r th e inne r diamete r met ric, betwee n x an d y define d b y th e infimu m o f t h e diameter s o f curve s connectin g

42

HIROAKI AIKAW A

x an d y i n D. Th e inne r lengt h distanc e \rj(x,y) betwee n x an d y i s define d b y the infimu m o f the length s o f curves connectin g x an d y i n D. Bonk , Heinone n an d Koskela [33 ] say that D i s an inner uniform domain i f every pai r o f points x,y G D can b e connecte d b y a curv e 7 C D suc h tha t e(-y)(x,y) fo r a Joh n domai n (Vaisal a [76 , Theore m 3.4]) . In term s o f th e quasihyperbolic metric kD{x,y) =

in f / —— -

with th e infimu m bein g taken ove r al l curves xy C D connectin g x an d y, a unifor m domain i s characterized b y X y kD{x,y) < Alog\mm{6 ( . s }(x),6 ~ [ (y)} , Ju + l) D

D

(Gehring an d Osgoo d [46]) . W e see tha t a Joh n domai n satisfie s a quasihyperbolic boundary condition: kD(x,xo) 0 bein g independent o f r . Applyin g th e maximu m principl e an d thi s inequalit y repeatedly , we obtain th e following . LEMMA 3 . There are positive constants A\ and A2, depending only on the domain, such that if 0 < Ar < R, then

exp(^i - A 2 —) for y E B(x,2r)nU r. r For the abov e lemma, B(x, Ar)\U r nee d no t includ e a ball of radius r . Instead ,

uj(y1S(x,R)r)Ur,B(x,R)r\Ur)
0 such tha t SD(x) Rn'2GR(x^R)>A R For j > 0 w e pu t D 3 = {x G D : exp(-2^ + 1 ) < R n~2GR(x^R) < exp(-2-?) } and Uj = {x G D : R n-2GR{x,£R) < exp(-2^')} . The n U 3 C {x G D : R)'

Then th e lemm a follow s fro m sup J > 0 d 3 < A < oo. Apply th e maximu m principl e ove rLX, - H i?(£, R3-i) t o obtai n on Uj H B(£, Rj-i). Divid e bot h side s b y R n~2GR(-,£)R) an d tak e th e supremu m over Dj n B(£,Rj). The n Lemm a 3 implie s ^ < Aexpl 2

J +1

- v4j" ^ exp

'2J' '

T

+ do -

Since V • exp ( 2 J + 1 — v4 j 2 exp(2 J /A)) i s convergent , w e have sup J > 0 d 3 < A < oo. The lemm a i s proved . n P R O O F O F THEORE M 1 . Sinc e u i s a positiv e bounde d harmoni c functio n o n D r\B(£, AR) vanishin g q.e . on dDnB(£, AR), w e obtain a potential representation :

u(x)= /

G

R(x,y)dii{y)

fo

r x G D n £(f , 6R).

JDnS(£,6R)

Take x G L> H B(f, i ? ) a n d i / E i ) n S(f , 6#) . Moreover , le t x* e D n S(f , R) an d y* £ D D S(£,6i?) b e nontangentia l points . Observ e tha t G R(x,y) wit h eithe r x

MARTIN B O U N D A R Y A N D B O U N D A R Y H A R N A C K PRINCIPLE

45

or y fixed i s a harmoni c functio n o f th e othe r argumen t fo r x ^ y vanishin g o n the boundary . Multipl y th e trivia l inequalit y G R(x,y) < AK 2~n b y th e harmoni c measure an d appl y th e maximu m principle . The n w e hav e fro m Lemm a 4 ^ ^R [X^ V \ GR{x*,y*) where G R(x*,y*) ~ R 2~n i s used . Th e revers e inequalit y i s derive d rathe r easily . Integration wit h respec t t o \i yield s GR(x,y)
l \

x

\

This theore m an d th e monotonicit y o f the harmoni c measur e /3E giv e th e nex t corollary.

MARTIN B O U N D A R Y A N D B O U N D A R Y H A R N A C K PRINCIPLE

47

COROLLARY 8 (Monotonicity) . Let E c F be closed sets on the hyperplane. If the Denjoy domain W 1 \ E has two minimal Martin boundary points at infinity, then so does the Denjoy domain W 1 \ F.

If ther e ar e tw o minima l Marti n boundar y point s a t infinity , the n ther e exis t non-minimal Marti n boundar y point s b y the connectednes s o f the Marti n bound ary. Benedicks ' criterio n give s a necessary an d sufficien t conditio n i n terms o f the harmonic measure , whos e estimat e i s not obvious . Segaw a [71 ] , [72 ] an d Gardine r [44] giv e mor e explici t condition s i n terms o f the Lebesgu e measure . The Kelvi n transfor m convert s th e Benedicks theore m t o a criterio n fo r th e number o f minimal Marti n boundar y point s ove r th e origi n o f a bounded domain . Let E b e a compact se t on a Lipschit z surfac e E and suppose B i s an open bal l containing E. W e cal l B \ E a Lipschit z Denjo y domain . I f E i s smooth, the n an analogu e o f Benedicks' theore m hold s (Chevallie r [36]) . Th e firs t assertio n of the Benedick s theore m i s valid fo r a genera l Lipschit z Denjo y domain , i.e. , th e number o f minimal Marti n boundar y point s ove r a Euclidea n boundar y poin t i s at mos t tw o (Ancon a [1 4 ] and Chevallie r [36]) . However , th e secon d assertio n no longer holds . On e migh t think , intuitively , tha t th e large r th e boundary , th e mor e minimal Marti n boundar y points . Thi s i s not th e case . Ancon a [1 8 ] constructe d a counterexample t o Corollary 8 for a Lipschitz Denjo y domain , i.e. , ther e ar e close d sets E C F C E suc h tha t B \ F ha s one minima l Marti n boundar y poin t ove r the origin , wherea s B\E ha s two . Thus , ther e i s no Benedick s typ e criterio n fo r a Lipschitz Denjo y domain . The followin g weak boundary Harnack principle i s used fo r th e proo f o f the fac t that th e number o f minimal Marti n boundar y point s i s one o r two: Le t E be a Lipschitz surfac e containin g th e origin. Fo r smal l r > 0 there ar e points x+ an d x~ o n #(0 , r) whos e distance s to E are comparabl e t o r an d E lies between x+ an d x~. Suppos e ho, h\ an d h2 are positive harmoni c function s o n D = B \ E suc h that ho = hi — h 2 = 0 Q-e . on dD and bounde d apar t fro m th e origin . The n

ho(x) < A (^llhi(x) +

^0h2(x)) fo

r xe D

\ B^ r )'

Let £ G dD. We say that a positive harmonic functio n h on D is a kernel function at £ if h vanishes q.e . o n dD, is bounded apar t fro m £ , and h(xo) = 1 . Here xo G D is a referenc e point . Th e weak BH P assert s tha t amon g thre e kerne l function s at th e origin, ther e i s one kerne l functio n bounde d b y the sum o f the other s up to a multiplicative constant . Thi s observatio n readil y implie s tha t th e number of minimal Marti n boundar y point s ove r th e origi n i s at most two . 7.2. Sectoria l domai n an d John domain . Cransto n an d Salisbury [40] called a planar domai n whos e boundary lie s on finitely man y ray s leaving the origi n a sectorial domain. Se e Figure 5 . The y prove d a result correspondin g t o Benedicks ' theorem an d relate d probabilisti c result s fo r a sectorial domain . Lomke r [56 ] calle d a higher dimensiona l analogu e a quasisectorial domain an d give s similar result s wit h respect t o the Schrodinger equation . Sinc e a Denjo y domain , a sectoria l domai n and a quasisectorial domai n ar e al l John domains , i t may b e natural t o study the minimal Marti n boundar y point s o f a John domain . A s Ancona' s counterexampl e illustrates, precise results similar t o the last assertio n o f the Benedicks theorem nee d not hol d withou t stric t assumption s o n the locatio n o f the boundary . Nonetheles s we ca n prov e th e followin g resul t [9 ] by establishin g a weak BHP .

48

H I R O A K I AIKAW A

0

*

FIGURE 5 . Denjo y domai n (harmoni c measur e (3E) an d sectoria l domain . T H E O R E M 9 . The number of minimal Martin boundary points over a Euclidean boundary point of a bounded John domain is finite and estimated by the John constant cj. Moreover, if the John constant cj is bigger than A / 3 / 2 , then the number of minimal Martin boundary points over a boundary point is at most two.

The theore m sound s reasonabl e fro m th e poin t o f vie w tha t th e smoothe r th e domain is , th e close r t o 1 the Joh n constan t cj is . However , th e mos t interestin g phenomenon, tha t ther e i s one Marti n boundar y poin t a t eac h Euclidea n boundar y point an d th e Marti n boundar y coincide s wit h th e Euclidea n boundary , canno t b e derived fro m th e theorem . I n thi s direction , Ancon a [1 3 ] showe d tha t th e unio n D of a family o f open ball s with th e sam e radiu s p o n a s o n e minima l Marti n boundar y point a t a Euclidea n boundar y poin t £ i f wheneve r D include s tw o ope n ball s B\ and B2 with radiu s p o tangential t o eac h othe r a t £ , D include s a truncated circula r cone To(^y) H £?(£,r) fo r som e 0 > 0 , r > 0 an d y i n th e hyperplan e tangen t t o Bi a t £ . Ancona' s argumen t i s based o n subtl e estimate s o f the Gree n function , fo r which i t i s crucia l tha t D i s th e unio n o f ball s wit h th e sam e radius . Applyin g th e study o n th e minima l Marti n boundar y point s o f a Joh n domain , w e can prov e tha t the unio n o f non-shrinkin g conve x set s ha s on e minima l Marti n boundar y poin t a t each Euclidea n boundar y poin t an d tha t th e Marti n boundar y coincide s wit h th e Euclidean boundary , provide d D satisfie s a n appropriat e conditio n correspondin g to Ancona' s con e conditio n ([9]) . 8. Application s an d relate d topic s 8.1. Minimall y thi n sets . Le t u s giv e a n explici t criterio n fo r minimall y thin sets , simila r t o th e Wiene r criterio n fo r thi n sets , i n orde r t o understan d th e Fatou-Naim-Doob theore m i n a mor e concret e fashion . Th e usua l Wiene r crite rion i s give n a s th e convergenc e o f th e serie s o f capacitie s o f th e intersection s o f concentric spherica l shell s an d th e se t unde r consideration . A simila r Wiene r typ e criterion ca n b e give n fo r minimall y thi n sets . I n plac e o f th e logarithmi c capacit y and th e Newtonia n capacity , th e Gree n capacity , define d i n term s o f th e Gree n function, i s involve d i n th e serie s a s lon g a s th e sam e intersectio n i s taken . Sinc e the Gree n functio n i s implicit , s o i s the Gree n capacity . Mor e explicitly , quantitie s are wante d t o describ e minima l thinness . Thi s i s possible . Takin g th e intersec tion wit h Whitne y cube s instea d o f concentri c spherica l shells , w e can describ e th e Wiener typ e criterio n fo r minima l thinnes s i n terms o f the logarithmi c capacit y an d the Newtonia n capacity . I n general , capacitie s ar e subadditive , i.e. , th e capacit y of a se t i s dominate d b y th e summatio n o f th e capacitie s o f th e partition s o f th e set. Fo r th e Gree n capacit y an d th e Whitne y decomposition , a n opposit e inequal ity u p t o a multiplicativ e constan t hold s unde r a certai n conditio n o n th e domai n

MARTIN B O U N D A R Y A N D B O U N D A R Y H A R N A C K PRINCIPLE

49

(quasiadditivity). I n eac h Whitne y cube , th e Gree n capacit y ca n b e estimate d b y the logarithmi c capacit y an d th e Newtonia n capacity ; an d henc e a Wiene r typ e criterion fo r minimall y thi n set s ca n b e state d i n term s o f th e logarithmi c capacit y and th e Newtonia n capacity . Fo r simplicit y w e shal l restric t ourselve s t o a smoot h domain D i n th e Euclidea n spac e o f dimensio n greate r tha n o r equa l t o three . Le t {Qj} b e th e Whitne y decompositio n o f D. The n E C D i s minimall y thi n a t x G 3D i f an d onl y if ^distCQ,,^)2^_

£ dist(4, ^

c

_

gn

.

»p( Qi>

^" 1 |Q J -|.

On th e othe r hand , ther e i s a positiv e constan t a , dependin g o n th e Lipschit z constant k, suc h tha t g(x) > ASD^)"- Henc e

/ hf-

l

\Vg\2dx >

A- lh{y3)8D{yjT^-2\Q3\ «

h{y

2 3)8D{y3T^- ^.

J

Qj

Since /

r

\

1 / p

h(y3)SD(y3y/\ it follow s tha t i f p < n/(a — 2 + n) = p/e , the n I/P

[/h

\JQ3 J

p

dx) 1 for a genera l Joh n domai n remain s open . Lindqvis t [55 ] showed th e integrability o f nonlinear superharmoni c functions . Goto h [47 ] studie d the integrabilit y o f BMO function s o n a domain wit h the quasihyperbolic conditio n and derive d th e integrability o f nonlinear superharmoni c functions . Th e author [4] studied th e integrability, relyin g no t on the BHP but on the Cranston-McConnell inequality relate d to intrinsic ultracontractivity ([24 ] and [28]), the lifetime estimat e ([39] an d [11]) and the perturbation of the Green function ([1 9 ] and [3]). Thes e subjects hav e intimat e connection s t o th e parabolic boundary Harnack principle and th e logarithmic Sobolev inequality, whic h canno t b e touched i n this exposition . Beyond th e subject, ther e seem s t o be a fascinatin g an d still activ e researc h fiel d where rea l an d complex analysis , partia l differentia l equations , differentia l geome try an d probability interact . Th e author wishe s tha t th e survey coul d she d a ligh t on som e wor k in such a n attractive area . Th e classica l stud y o f harmonic function s in a unit bal l ma y lea d t o the frontier o f analysis. 9. Convers e o f the boundary Harnac k principl e Recently, th e author [7 ] proved tha t Joh n domains , unifor m domain s an d inner uniform domain s (equivalently , uniforml y Joh n domains ) ar e characterized b y some potential-theoretic properties . Th e conditions ar e necessary an d sufficient, provide d the domai n satisfie s th e CDC. Th e 'if parts' o f Theorems 1 2 and 13 below ma y b e regarded a s the converse of the BHP. Recall tha t th e CDC holds i f and only i f there exis t constant s f3 > 0, A > 1 and r o > 0 such tha t (1) u(x,

D fl S(£, r),D n B(f, r)) < A (lE^lQ j

for

x

G D n B(f, r),

whenever £ G dD and 0 < r < TQ. Thi s is a uniform uppe r estimat e of the harmoni c measure. Se e also [6 ] for the connection t o the Dirichlet problem . A John domai n is characterized b y a uniform lowe r estimate . T H E O R E M 1 1 . Let D satisfy the CDC. Then D is a John domain if and only if there exist constants a > 0, A > 1 and r$ > 0 such that

(2) ^ , i ) n S ( ( , r ) , J ) n B ( { , r ) ) > ^

W /orxGDnB(^)

,

whenever £ G dD and 0 < r < r$. A unifor m domai n i s characterize d b y (2 ) an d th e scal e invarian t BHP , o r uniform BHP . T H E O R E M 1 2 . Let D satisfy the CDC. Then D is a uniform domain if and only if the scale invariant BHP and (2) hold.

52

HIROAKI AIKAW A

FIGURE 6 . Scal e invarian t BH P an d BH P wit h respec t t o th e in ternal metric .

An inne r unifor m domai n i s characterize d b y (2 ) an d th e unifor m BH P wit h respect t o th e interna l metric . A bal l o f cente r a t a boundar y poin t wit h respec t to th e interna l metri c become s a connecte d componen t o f th e intersectio n o f a Euclidean bal l an d th e domai n ([1 0 , Lemm a 2.2]) . So , w e arriv e a t th e followin g version o f the unifor m BH P wit h respec t t o the interna l metric , whic h i s a propert y weaker tha n th e unifor m BHP . Se e Figur e 6 . DEFINITION. W e say tha t a domain D enjoy s th e uniform BHP with respect to the internal metric o r th e scale invariant BHP with respect to the internal metric if there exis t constant s A > 1 and r o > 0 , dependin g onl y o n D, wit h th e followin g property: Le t £ 2, unless otherwise noted. Firs t w e consider a singula r integra l i n (1.3) which i s defined b y a homogeneous kerne l o f the for m

We assum e tha t th e function ft G L1( 5 n _ 1) satisfie s (2.1) /

n(x')da(x')

=

0,

where S 71 "1 denote s th e unit spher e i n W1 and da the Lebesgue measur e o n S n~l. In thi s case , w e also writ e T = Tfo . I f we denote b y K th e Fourier transfor m o f the principa l valu e distributio n define d b y the kernel K, the n K i s a homogeneou s function o f degree 0 and can be expressed a s follows:

(2.2) K(0 = - f fi(0) \£ n 1L

8ea((?,0))

+

]og\(Z',9)\\ do{6) (£

' e 5"- 1 ),

l J

Js ~ where (• , •) denotes the inner produc t i n IRn. Th e definition o f the Fourie r transfor m we are using i s (for / G L1(M n ))

M) = I f(x)e- 2^^ dx; we als o writ e / = 3 r(f). B y usin g (2.2 ) we can show tha t T i s bounde d o n L 2 if ft G L\ogL(Sn~1). Her e L log L(Sn~1) i s the space o f all those function s ft o n 71 1 S " tha t satisf y J ^ ^ |ft | log(2 + |ft| ) da < oo. Furthermore, b y the method o f rotations o f Caldero n an d Zygmund (se e [4] ) we can show th e following: (2.3) I f ft G L1^ 7 1- 1) an d ft is odd, then T n : Lp - • Z7 for all 1< p < oo. (2.4) I f ft G LlogL(S'n - 1), the n T n : Lp -^ L p fo r all 1< p < oo. Indeed, whe n ft i s odd, we can write Tn(f)(x)=1- f

fi(0)[p.v.

[°°

f(x-tO)^

AJ(0);

therefore, b y usin g th e L p boundednes s o f th e Hilber t transform , w e can prov e (2.3). Her e w e recall tha t th e Hilbert transfor m i s a singular integra l o n R define d by th e principal valu e integral :

# ( / ) ( * ) = p . v .- / f(x-t)-. Next, let ft G L log L(Sn~l) b e an even function. Then , i f Rj i s the Riesz transfor m defined b y ( i ^ / H O = H f j / l f l ) / ( 0 U = 1 ,2,.. . ,n), w e can see that i^-T h = Th^ fo r som e od d functio n ftj G L 1 (5 n ~ 1 ), an d hence , sinc e / = —^2 nz^1 R2 (I denotes th e identity operator) , w e have T Q = — Y^j=\ RjTsiy Thu s w e get the L p boundedness o f TQ b y (2.3) and the Lp boundednes s o f Rj. T o prove (2.4 ) w e write ft a s a su m of an odd function an d an eve n functio n i n LlogL(S n~1 ). The n we

SINGULAR INTEGRAL S AN D LITTLEWOOD-PALE Y FUNCTION S

59

apply the abov e results to the two singular integral s arisin g from thi s decompositio n of Q , whic h complete s th e proo f o f (2.4) . We ca n us e th e Hard y spac e H 1 (Sn~1 ) t o improv e (2.4) . First , w e recal l th e definition o f i / 1 ( 5 ' n _ 1 ) . Th e Poisso n kerne l o n th e uni t spher e S ^ - 1 i s define d b y 11 ±ry' yE

00 J\x-y\>e

60

SHUICHI SAT O

By applyin g th e Littlewood-Pale y decompositio n an d usin g orthogonalit y vi a Fourier transfor m estimates , Duoandikoetxe a an d Rubi o d e Francia [1 6 ] develope d methods whic h ca n b e use d t o stud y variou s interestin g operator s i n harmoni c analysis, includin g th e singula r integral s wit h inhomogeneou s kernel s mentione d above. Now , w e present a resul t whic h i s a specia l cas e o f a mor e genera l theore m in [1 6] . Le t {cr/ c }^_ 0 0 b e a sequenc e o f Bore l measure s o n R n . Le t a G (0,1]. W e assume th e following : (2.5) a k * f- Defin e a singula r measur e r r o n W 1 whic h i s concentrated o n {\x\ = r} b y (r r , /) = / f(rO)Q(0)

da(0) (f

G C0°°(Rn)),

where Co°(IR. n) denote s th e spac e o f al l thos e function s o n R n tha t ar e infinitel y differentiable an d compactl y supported . The n th e Schwar z inequalit y implie s / f 2k+1 \ 1

2

1^(01 0 a.e. ( a weight function) . W e say w e Ai if there exist s a non-negative constan t C suc h tha t M(w)(x) < C w(x) a.e. Her e M denote s th e Hardy-Littlewood maxima l operator : M(w)(x) =

su p | Q |_ 1 / w(x)dx xeQ Jo xeQ JQ

1

where the supremum i s taken ove r all cubes Q in W1 such that x G Q. Th e weighted space L vw is the vector spac e o f all measurable function s / suc h tha t II/IIL& = ( 7

\f(x)\Mx)dx)

P

A}) 1 /p < oo, A>0

where w(E) = J Ew(x) dx. B y Varga s [57] , Theorem 5 has bee n extende d t o th e weighted Lebesgu e space s a s follows . THEOREM

6 . Let w e A\. Then

IIW)|| Li

> - y\)f(y) dy.

PutS(/) = £f =1 T,(/). Put E = (JQ* , where Q * is a cub e wit h th e same cente r a s Q and sid e lengt h 100 time s tha t o f Q (w e assum e th e side s o f Q * t o b e paralle l t o th e coordinat e

67

SINGULAR INTEGRAL S AN D LITTLEWOOD-PALE Y FUNCTION S

axes). The n w e not e tha t w(E) < CX 1 Proposition 1 is to sho w tha t

||/||L^-

Th e principa l par t o f th e proo f o f

•({x€Rn\E:\S(b)(x)\>\})

0

68

SHUICHI SAT O

LEMMA 3 . Let v be a locally integrable, non-negative function. Then for every positive integer s we have

< CA j>s

Ei3ij^ M ( i ; ) ( j : ) Q

By usin g a modifie d versio n o f th e interpolatio n metho d o f A . Varga s [57 ] between th e estimate s i n Lemma s 2 an d 3 , w e ca n ge t Lemm a 1 . Th e proo f o f Lemma 3 i s easy . W e no w outlin e th e proo f o f Lemm a 2 . Fo r integer s /c,r a > 1 , put (4.4)

Hkm(x, y) = J e-* p^+w^Lk(z -

x)L rn(z - y) dz.

Then w e not e tha t G* kGm(f){x) = J Hkm{x iy)f(y) dy, Gfc. LEMMA

wher e G £ i s th e adjoin t o f

4 . Let j , s be integers such that 0 < s < j . Then we have V su p / ~l^Rn \J

Bi-.

s(y)Hji(x,y)dy

< C\2~

s

.

Let (•,• ) denot e th e inne r produc t i n L 2 (R n ). I f w e assum e Lemm a 4 , the n Lemma 2 can b e prove d a s follows : 2

z2Gj(B3-s)\

s i—s

3>s

S Z

=S

j>s \_i=s

s>0,~f>0. Suppose that deg(P ) = N >l, \\P\\ = 1 . Let a e R n . Then

\{xeRn : where C UJN^ is

\a-x\
0, then a 5(f) and v 0(f) ( / e S(R n)) are pointwise equivalent] that is, there are two positive constants A and B such that for all x £ R n, 2 and n/(n — 1 ) < p . T/ie n the inequality

I \M(f)(

x)\p\x\-a dx

jRn

< C

[ l/( x)\p\x\-adx

holds for n — p(n — 1 ) < a < n — 1 and does not hold for a > n — 1 . The affirmativ e par t wa s partly prove d b y Rubio d e Francia [35] . W e note tha t the inequalit y fo r a = 0 and p > n/{n — 1 ) wa s prove d b y Stei n [53 ] fo r n > 3 an d by Bourgai n [2 ] for n — 2 (se e als o Sogg e [50]) . I t i s known tha t Propositio n 4 ca n be use d t o prov e th e following . 5 . Suppose that (3 > 3/2 - n/ 2 and 0 < a < 1 . Then

PROPOSITION

[ \Mt 71

JR '

'

1 /2

(f)(x)\2\x\-«dx 3 , 0 < a < n — 1 and p > n/(n — 1 ) ca n b e give n by usin g Propositio n 5 and a n interpolation .

76

SHUICHI SAT O

References 1. A . Benedek , A . P . Caldero n an d R . Panzone , Convolution operators on Banach space valued functions, Proc . Nat . Acad . Sci . U.S.A . 4 8 (1 962) , 356-365 . 2. J . Bourgain , Averages in the plane over convex curves and maximal operators, J . Analys e Math. 4 7 (1 986) , 69-85 . 3. A . P . Caldero n an d A . Zygmund , On the existence of certain singular integrals, Act a Math . 88 (1 952) , 85-1 39 . 4. A . P . Caldero n an d A . Zygmund , On singular integrals, Amer . J . Math . 7 8 (1 956) , 289-309 . 5. A . Carbery , J . L . Rubi o d e Franci a an d L . Vega , Almost everywhere summability of Fourier integrals, J . Londo n Math . Soc . (2 ) 3 8 (1 988) , 51 3-524 . 6. S . Chanill o an d M . Christ , Weak (1 ,1 ) bounds for oscillatory singular integrals, Duk e Math . J. 5 5 (1 987) , 1 41 -1 55 . 7. L . K . Chen , On a singular integral, Studi a Math . 8 5 (1 987) , 61 -72 . 8. M . Christ , Weak type (1 ,1 ) bounds for rough operators, Ann . o f Math . 1 2 8 (1 988) , 1 9-42 . 9. M . Chris t an d J . L . Rubi o d e Francia , Weak type (1 ,1 ) bounds for rough operators, II, Invent . Math. 9 3 (1 988) , 225-237 . 10. M . Chris t an d C D . Sogge , The weak type L 1 convergence of eigenfunction expansions for pseudodifferential operators, Invent . Math . 9 4 (1 988) , 421 -453 . 11. R . R . Coifma n an d Y . Meyer , Au deld des operateurs pseudo-differentiels, Asterisqu e no . 57 , Soc. Math . France , 1 978 . 12. R . R . Coifma n an d G . Weiss , Extensions of Hardy spaces and their use in analysis, Bull . Amer. Math . Soc . 8 3 (1 977) , 569-645 . 13. W . Connett , Singular integrals near L 1 , Proc . Symp . Pur e Math . 35 , Par t 1 (1 979) , 1 63 165. 14. Y . Ding , On Marcinkiewicz integral, Proc . o f th e conferenc e "Singula r integral s an d relate d topics, III " (Osak a Kyoik u University , Japan , Januar y 27-29 , 2001 ) , 28-38 . 15. Y . Ding , D . Fa n an d Y . Pan , Weighted boundedness for a class of rough Marcinkiewicz integrals, Indian a Univ . Math . J . 4 8 (1 999) , 1 037-1 055 . 16. J . Duoandikoetxe a an d J . L . Rubi o d e Francia , Maximal and singular integral operators via Fourier transform estimates, Invent . Math . 8 4 (1 986) , 541 -561 . 17. J . Duoandikoetxe a an d L . Vega , Spherical means and weighted inequalities, J . Londo n Math . Soc. (2 ) 5 3 (1 996) , 343-353 . 18. D . Fa n an d Y . Pan , Singular integral operators with rough kernels supported by subvarieties, Amer. J . Math . 1 1 9 (1 997) , 799-839 . 19. D . Fa n an d Y . Pan , A singular integral operator with rough kernel, Proc . Amer . Math . Soc . 125 (1 997) , 3695-3703 . 20. D . Fan an d S . Sato, Weak type (1 ,1 ) estimates for Marcinkiewicz integrals with rough kernels, Tohoku Math . J . 5 3 (2001 ) , 265-284 . 21. D . Fa n an d S . Sato , Remarks on Littlewood-Paley functions and singular integrals, J . Math . Soc. Japa n 5 4 (2002) , 565-585 . 22. R . Fefferman , A note on singular integrals, Proc . Amer . Math . Soc . 7 4 (1 979) , 266-270 . 23. L . Grafako s an d A . Stefanov , L p bounds for singular integrals and maximal singular integrals with rough kernels, Indian a Univ . Math . J . 4 7 (1 998) , 455-469 . 24. S . Hofmann , Weak (1 ,1 ) boundedness of singular integrals with nonsmooth kernel, Proc . Amer. Math . Soc . 1 0 3 (1 988) , 260-264 . 25. S . Hofmann , Weighted weak-type (1 ,1 ) inequalities for rough operators, Proc . Amer . Math . Soc. 1 0 7 (1 989) , 423-435 . 26. L . Hormander , Estimates for translation invariant operators in L p spaces, Act a Math . 1 0 4 (1960), 93-1 39 . 27. M . Kanek o an d G . Sunouchi , On the Littlewood-Paley and Marcinkiewicz functions in higher dimensions, Tohok u Math . J . 3 7 (1 985) , 343-365 . 28. J . E . Littlewoo d an d R . E . A . C . Paley , Theorems on Fourier series and power series, J . London Math . Soc . 6 (1 931 ) , 230-233 . 29. J . E . Littlewoo d an d R . E . A . C . Paley , Theorems on Fourier series and power series (II), Proc. Londo n Math . Soc . 4 2 (1 936) , 52-89 . 30. J . E . Littlewoo d an d R . E . A . C . Paley , Theorems on Fourier series and power series (III), Proc. Londo n Math . Soc . 4 3 (1 937) , 1 05-1 26 .

SINGULAR INTEGRAL S AN D LITTLEWOOD-PALE Y FUNCTION S 7

7

31. S . Z . L u an d Y . Zhang , Criterion on L p-boundedness for a class of oscillatory singular integrals with rough kernels, Rev . Mat . Iberoamerican a 8 (1 992) , 201 -21 9 . 32. J . Namazi , On a singular integral, Proc . Amer . Math . Soc . 9 6 (1 986) , 421 -424 . 33. F . Ricc i an d E . M . Stein , Harmonic analysis on nilpotent groups and singular integrals, I, J. Func . Anal . 7 3 (1 987) , 1 79-1 94 . 34. F . Ricc i an d G . Weiss , A characterization of H 1 (Yin-i), Proc . Symp . Pur e Math . 35 , Par t 1 (1979), 289-294 . 35. J . L . Rubi o d e Francia , Weighted norm inequalities for homogeneous families of operators, Trans. Amer . Math . Soc . 27 5 (1 983) , 781 -790 . 36. J . L . Rubi o d e Francia , Transference principles for radial multipliers, Duk e Math . J . 5 8 (1989), 1 -1 9 . 37. J . L . Rubi o d e Francia , F . J . Rui z an d J . L . Torrea , C alderon-Zygmund theory for operatorvalued kernels, Adv . Math . 6 2 (1 986) , 7-48 . 38. S . Sato , Spherical summability and a vector-valued inequality, Bull . Londo n Math . Soc . 2 7 (1995), 58-64 . 39. S . Sato, A weighted vector-valued weak type (1 ,1 ) inequality and spherical summation, Studi a Math. 1 0 9 (1 994) , 1 59-1 70 . 40. S . Sato , Divergence of the Bochner-Riesz means in the weighted Hardy spaces, Studi a Math . 118 (1 996) , 261 -275 . 41. S . Sato , Some weighted weak type estimates for rough operators, Math . Nachr . 1 8 7 (1 997) , 211-240. 42. S . Sato , Remarks on square functions in the Littlewood-Paley theory, Bull . Austral . Math . Soc. 5 8 (1 998) , 1 99-21 1 . 43. S . Sato , Weighted weak type (1 ,1 ) estimates for oscillatory singular integrals, Studi a Math . 141 (2000) , 1 -24 . 44. S . Sato , Weighted weak type (1 ,1 ) estimates for oscillatory singular integrals with Dini kernels, Bull . Fac . Ed . Kanazaw a Univ . Natur . Sci . 4 9 (2000) , 1 -22 . 45. S . Sato , Some weighted estimates for Littlewood-Paley functions and radial multipliers, J . Math. Anal . Appl . 27 8 (2003) , 308-323 . 46. S . Sato an d K . Yabuta, Multilinearized Littlewood-Paley operators, Sci . Math. Jpn . 5 5 (2002) , 447-453. 47. A . Seeger , Endpoint estimates for multiplier transformations on compact manifold, Indian a Univ. Math . J . 4 0 (1 991 ) , 471 -533 . 48. A . Seeger , Singular integral operators with rough convolution kernels, J . Amer . Math . Soc . 9 (1 996) , 95-1 05 . 49. A . Seege r an d T . Tao , Sharp Lorentz space estimates for rough operators, Math . Ann . 32 0 (2001), 381 -41 5 . 50. C . D . Sogge , Fourier integrals in classical analysis, Cambridg e Universit y Press , 1 993 . 51. E . M . Stein , On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans . Amer . Math. Soc . 8 8 (1 958) , 430-466 . 52. E . M . Stein , Singular Integrals and Differentiability Properties of Functions, Princeto n Univ . Press, 1 970 . 53. E . M . Stein , Maximal functions: Spherical means, Proc . Nat . Acad . Sci . U.S.A . 7 3 (1 976) , 2174-2175. 54. E . M . Stein , Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeto n Univ . Press , 1 993 . 55. Q . Sun , Two problems about singular integral operators, Approx . Theor y Appl . 7 (1 991 ) , 83-98. 56. T . Tao , The weak-type (1 ,1 ) of L l o g L homogeneous convolution operator, Indian a Univ . Math. J . 4 8 (1 999) , 1 547-1 584 . 57. A . Vargas , Weighted weak type (1 ,1 ) bounds for rough operators, J . Londo n Math . Soc . (2 ) 54 (1 996) , 297-31 0 .

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58. T . Walsh , On the function of Marcinkiewicz. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, III, Studi a Math . 4 4 (1 972) , 203-21 7 . 59. A . Zygmund , Trigonometric series, 2n d ed. , Cambridg e Univ . Press , Cambridge , London , New Yor k an d Melbourne , 1 977 . DEPARTMENT O F MATHEMATICS , FACULT Y O F EDUCATION , KANAZAW A UNIVERSITY , KANAZAW A

920-1192, JAPA N

E-mail address: shuichiOkenroku.kanazawa-u.ac.j p Translated b y SHUICH I SAT O

Amer. Math . Soc . Transl . (2) Vol . 21 5 , 200 5

http://dx.doi.org/10.1090/trans2/215/05

Convergence o f metri c measur e space s and energ y form s Atsushi Kasu e

Introduction Let u s conside r th e Dirichle t proble m fo r harmoni c function s i n a relativel y compact domai n o f a Riemannia n manifold . Th e solutio n i s obtained b y th e direc t method o f the calculu s of variations a s follows. W e first take a sequence of function s with prescribe d boundar y value s whic h minimize s th e Dirichle t energy . The n i t follows fro m th e Poincar e inequalit y tha t th e sequenc e i s bounde d i n th e Sobole v space, and hence , by Rellich's theorem, w e have a subsequence whic h converge s to a function weakl y i n th e Sobole v spac e an d strongl y i n L 2. Th e lowe r semicontinuit y of th e Dirichle t energ y implie s tha t th e limi t functio n minimize s th e energ y an d satisfies th e Laplac e equatio n i n a wea k sense . Second , w e ca n obtai n a n a priori eatimate o n th e Holde r continuit y o f th e solutio n wit h respec t t o th e Riemannia n distance, usin g Moser' s iteratio n method , i n whic h th e volum e estimate s o f metri c balls, th e Sobole v inequality , an d Poincar e inequalit y pla y crucia l roles . Now Riemannia n manifold s ar e metri c space s equippe d wit h thei r Riemann ian distances , an d als o Dirichle t space s wit h th e Dirichle r energ y functional s o n the Sobole v spaces . Fro m th e forme r poin t o f view , th e convergenc e o f Riemann ian manifold s wit h respec t t o th e Gromov-Hausdorf f distanc e ha s bee n intensivel y studied. Ther e ar e surve y articles , fo r instance , [1 6] , [48] , [49 ] o n th e collapsin g phenomena, [39 ] on the convergence of Einstein manifolds , an d [42 ] on Riemannia n manifolds wit h Ricc i curvatur e bounde d below . Als o w e refe r t o th e monograph s [6], [1 8] , [1 9 ] fo r discussion s i n detail . On the othe r hand , Fukay a [1 5 ] studied a family o f compact Riemannia n mani folds wit h sectiona l curvatur e bounde d uniforml y i n absolute value , an d showe d th e continuity o f eigenvalues an d als o eigenspaces i n a certai n sens e wit h respec t t o th e topology o f the measure d Gromov-Hausdorf f convergence . I n a serie s o f papers [9] , J. Cheege r an d T.H . Coldin g proved th e continuity o f the Dirichle t energ y function als i n a certain sens e wit h respec t t o th e sam e topolog y unde r th e assumptio n tha t the Ricc i curvature o f compact Riemannia n manifold s i n a given family i s uniforml y bounded below . This articl e originall y appeare d i n Japanes e i n Sugak u 5 5 (1 ) (2003) , 20-36 . 2000 Mathematics Subject Classification. Primar y 58Jxx ; Secondar y 53Cxx . ©2005 America n Mathematica l Societ y 79

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ATSUSHI K A S U E

From th e viewpoint s o f th e direc t metho d o f calculu s o f variation s mentione d above an d als o th e convergenc e o f th e Gromov-Hausdorf T sense , w e woul d lik e t o discuss th e convergenc e o f th e Dirichle t energ y functional s o n Riemannia n mani folds an d mor e generall y certai n Dirichle t spaces . I n Sectio n 1 , we recall som e basi c facts o n th e Gromov-Hausdorf T distanc e an d stat e a theore m du e t o Cheege r [8 ] on th e differentiabilit y o f Lipschit z function s o n metri c measur e space s satisfyin g the so-calle d volum e doublin g propert y an d th e (scal e invariant ) Poincar e inequal ity. Sectio n 2 i s devote d t o th e introductio n o f regula r Dirichle t space s wit h th e intrinsic distances . I n Sectio n 3 , the convergenc e o f close d form s o n Hilber t space s in th e sens e o f Mosc o an d thei r spectra l structure s ar e discussed , an d som e result s concerning precompactnes s o f a family o f Dirichlet space s ar e given . W e also stud y the convergenc e o f Riemannia n vecto r bundle s an d th e energ y functional s o n th e space o f map s i n Sectio n 4 . 1. Measure d Gromov-Hausdorf f convergenc e To begi n with , w e recal l th e definitio n o f Gromov-Hausdorf f distanc e betwee n metric spaces . Give n tw o metri c space s (X , dx) an d (Y,dy), a (Bore l measurable ) map / : X — > Y o f X t o Y i s calle d a n e-Hausdorf f approximatio n fo r a positiv e number e if it satisfie s \dx(x,x') — dy (/(#), f(x'))\ < e, x,x' G X, an d Y C f(X) £, where f(X) £ stand s fo r th e ^-neighborhoo d o f th e subse t f(X) i n Y. W e denot e by HD(X, Y) th e infimu m o f th e positiv e number s e fo r whic h ther e exist s a n eHausdorff approximatio n o f X t o Y. W e observe that HD(X, X) = 0 for a bounde d metric spac e X an d it s completio n X , an d als o tha t fo r compac t metri c space s X and y , HD(X, Y) = 0 if an d onl y i f the y ar e isometric . Let X = (X , dx) b e a metric space. A subset S o f X i s said to be ^-separated fo r a positiv e constan t e if dx{a, b) > e for al l a, 6 G 5, a ^ b. Give n a continuous curv e c : [a, b]— » X , w e denote b y L(c) th e leas t uppe r boun d fo r ^27=0 dx(c(U), c(^+i)) > where a = to < t\ < • • - < t n — 6 , an d whe n L(c) i s finite , w e sa y tha t c i s rectifiable an d o f lengt h L(c). W e cal l X a lengt h spac e i f fo r an y pai r o f point s p,q G X, th e distanc e dx(p,q) i s equa l t o th e greates t lowe r boun d o f th e length s of al l rectifiabl e curve s joinin g p t o q. I n th e cas e wher e X i s a locall y compac t length space , th e Hopf-Rino w theore m assert s tha t th e followin g ar e equivalent ; (1) X i s complete ; (2 ) an y bounde d close d subse t o f X i s compact; (3 ) an y locall y distance-minimizing curv e c : [0, a) — • X ca n be continuously extende d t o the close d interval [0 , a]; (4 ) ther e exist s a poin t suc h tha t ever y distance-minimizin g curv e c : [0, a) — » X emanatin g fro m p can b e continuously extende d t o the close d interva l [0, a]. Moreover , i f on e o f thes e propertie s holds , the n an y pai r o f point s o f X ca n be joined b y distance-minimizin g curves . The followin g ar e basi c result s b y M . Gromov . 1 .1 . (1 ) LetT be a family of compact metric spaces. Suppose that J 7 is uniformly and totally bounded; that is, the diameters of all X in T are bounded by a constant and for any positive number e, there exists a positive constant N(e) such that the cardinality of an e-separated subset of X is bounded by N(e). Then T is precompact with respect to the Gromov-Hausdorff distance. (2) Let {X n} be a sequence of metric spaces which converges to a complete metric space X with respect to the Gromov-Hausdorff distance. If each X n is a length space, then so is X. THEOREM

CONVERGENCE O F METRI C MEASUR E SPACES AN D ENERG Y FORM S 8 1

Let u s outlin e th e proo f o f th e firs t assertion . Le t {X n } b e a sequenc e i n T. Given a positiv e numbe r e , w e tak e a maxima l e-separate d subse t fro m eac h X n. Then th e sequenc e o f such subspace s i s precompact becaus e o f the assumptions , s o that a subsequenc e o f suc h subspace s converge s t o a finite metri c space . Takin g a sequence o f positive number s {em} tendin g t o zer o an d usin g a diagona l argument , we ca n obtai n a limi t metri c spac e o f th e subspace s whos e completio n belong s t o the boundar y o f th e sequenc e {X n }. This argument suggest s that an y compact lengt h space is a limit o f finite metri c graphs with respec t t o the Gromov-Hausdorf f distance . Se e [6] , [18], [19] for detail s and furthe r result s o n thi s subject . Now w e ar e abl e t o defin e th e Gromov-Hausdorf f distanc e betwee n tw o metri c spaces i f the diameter s ar e bounded. Otherwise , i t i s not clea r whethe r th e distanc e can b e determined . Fo r thi s reason , le t u s introduc e a topolog y o n a se t o f pointe d metric space s a s follows . W e say a sequenc e {X n = (X n , o n , d n)} o f pointe d metri c spaces converge s t o suc h a spac e X = (X , o, d) i n th e Gromov-Hausdorf f sens e i f there exis t sequence s o f positiv e number s {R n} wit h lim n ^oo R n = +o o an d {s n} with lim n ^oo e n — 0 , an d a sequenc e o f map s f n : B(o n,Rn)— » X suc h tha t (1) fn(o n) = o , (2 ) \d n(x,y) - d(f n(x)Jn(y))\ < e ny x,y G B{o n,Rn), an d (3 ) B(o,Rn — e n) C f n(B(on,Rn))£n. I n addition , thes e approximatin g map s allo w us t o defin e th e convergenc e o f function s an d subset s a s follows . Give n a bounde d subset U i n X an d a functio n w on [/ , w e sa y tha t a sequenc e o f bounde d subset s Un o f X n converge s t o U if f n give s rise to a n e n-Hausdorff approximatio n betwee n Un an d Z7 , and a sequenc e o f function s u n define d o n a subse t includin g f~ l(U) i n Xn uniforml y converge s t o u i f sup{\u n(x) — u(f n(x))\ \ x G fn~1(U)} tend s t o zer o as n — » oo . In wha t follows , w e conside r locall y compact , separable , an d complet e metri c spaces (X,d) an d Rado n measure s \i whic h satisf y 0 < fj,(B(x,r)) < +oo , x G X, r > 0 . W e say tha t a sequence {X n = (X n, o n , X vaguel y converge s t o / i a s n — > oo . Now, befor e recallin g som e fundamenta l result s i n [8] , we consider tw o proper ties of a metric measur e spac e (X , d, fi). First , w e say tha t (X , d, ji) ha s th e volum e doubling propert y if , fo r som e positiv e constant s K and R, [D]K v{B(x,

2r) ) < 2 > ( £ ( x , r)) , x

G X, 0 < r < R.

Second, fo r a locall y Lipschit z continuou s functio n u , puttin g dildu(x) =

limsup{\u(y) -

u(z)\/d(y, z) \ y,z G B(x,r), y

^ z},

the L p integra l o f dildU o n a bounded measurabl e subse t A, wher e p G [1, -f oo], ca n be defined an d calle d th e p-Dirichlet energ y o f u over A. The n w e say tha t (X , d, /x) satisfies a wea k p-Poincar e inequalit y if , fo r som e positiv e constant s r an d R an d for an y locall y Lipschit z continuou s functio n n , [P]r 1 VP / \ /P p f \uu x,r \ p dfi] (A) 1

JAud/j, an d u x,r — -f ud\i.

The followin g result s ar e prove d i n Cheege r [8] . T H E O R E M 1 .2 . Let X n — (X n ,o n ,d n ,/x n ) be a sequence of pointed measured metric spaces which converges to a pointed metric measure space (X, o, d, /i). If all Xn satisfy the volume doubling property [D] K, then so does X, and further if the weak p-Poincare inequality [P] r holds on each Xn, it also holds on X.

This theore m i s a n immediat e consequenc e o f th e following . LEMMA 1 .3 . Let X n = ( X n , o n , d n , / i n ) and X — (X, o, d,/z) be as in Theorem 1.2. Suppose that each X n satisfies the volume doubling property [D] K. Let u be a locally Lipschitz continuous function defined on a bounded open subset U in X. Then there exists a sequence of locally Lipschitz continuous functions u n on bounded open subsets U n in X n such that u n uniformly converges to u as n — > oo and moreover

I

{di\du)p d/i = li m / (dil

p

dnun)

dfi

n.

We remark that i n the situation of this lemma, fo r a sequence of locally Lipschit z continuous function s £ n : U n— > R whic h uniforml y converge s t o suc h a functio n £ : U— » R , th e lowe r semicontinuit y o f th e p-Dirichle t energ y functional s doe s no t hold true ; tha t is , the following does not hold in general: [ (dil d O P d/ x < limin f / (dil JU

n

dn

U P dfJL n

^ ° ° JUr,

Now Rademacher's theore m o n the differentiatio n o f Lipschitz function s ca n b e expanded a s follows . T H E O R E M 1 .4 . Let (X , d, /i) be a metric measure space with the volume doubling property [D] K and the weak p-Poincare inequality [P] T for some constants K, T and p G (l,+oo) . Then X can be covered by a union of Borel subsets Ui (i = 1,2,...) and Z with fi(Z) = 0 in such a way that for each i, there exists a linearly independent set {u[ , . . . ,u^lA of Lipschitz continuous functions on Ui with the following properties: (1) The number n(i) of the functions is bounded above by a constant N(K, r,p ) depending only on the given ones. (2) For any (a) — ( a i , . . ., a n ^)) G Rn ( ^ with (a) ^ 0 , the Lipschitz function u

s {a) — ]Cj=i a juj i asymptotically generalized linear at each point x G Ui and diidu(x) > 0; moreover for any locally Lipschitz function u, there exists an R n ^^valued measurable function a{u\x) (x G Ui) such that

u{y) - u(x) - Ysf=i aj{u,x)(u {j\y) uf{x)) lim — r = 0. y^x d(y,x) In thi s theorem , w e sa y tha t a locall y Lipschit z functio n u i s asymptoticall y generalized linea r a t a poin t x i f i t belong s t o th e Lebesgu e point s o f (dil ^ u)p an d lim [ / {dil r

^ ° \J B(x,r)

du)

p

dfi - in

keCy(B(x,r))

f/

(dil J

B

(x,r) J

d(u

+ k)) p d/ x ) = 0 .

C O N V E R G E N C E O F M E T R I C M E A S U R E SPACE S AN D E N E R G Y F O R M S 8

3

Let w b e a locall y Lipschit z continuou s functio n o n A . Accordin g t o Theore m 1.4, th e functio n x G Ui —> • a(u;x) = {CLJ(U\X)) G R n ( ^ ca n b e though t o f a s a section o f th e cotangen t bundl e T * A o f A o n U t, an d w e obtai n a sectio n o f T*X defined almos t everywher e o n AT , which wil l b e denote d b y du. The n w e se e tha t for u,v G Cx"oc(X) an d a,(3 G R , d(au + /3i> ) = adu + /3d ^ an d d(i«; ) = vdu + Wti . In addition , \du\ = dildU define s a canonica l nor m o n T*X t o whic h th e Dirichle t 2-energy J x(dildu)2dfjJ i s associated . Thi s nor m i s no t induce d fro m a quadrati c form o n T* A i n general ; however , w e ca n defin e a n L°° Riemannia n metri c ( , ) on T* A i n suc h a wa y tha t fo r som e positiv e constant s A an d A depending onl y o n the give n K and r , A dildU < \J(du, du) < A dil^i/ fo r an y u G C/o ' c (X). The n o n the spac e o f locall y Lipschit z continuou s function s o f bounde d Dirichle t 2-energy , we hav e a quadrati c for m £(u,v) = I (du,dv)d{i,

Jx

which ca n b e extende d t o a densel y define d close d for m o n th e Hilber t spac e L2(X,n). W e denot e th e for m b y th e sam e lette r an d th e domai n b y D[£]. The n the for m (£ , D[£]) i n fact give s a strongl y local , regula r Dirichle t for m o n L 2(A", /i), and Cfol(X) D D[£] turn s ou t t o b e a cor e o f the form . We recal l her e som e basi c notion s o n Dirichle t spaces . Le tA T be a locall y compact, separabl e metri c spac e an d \i a Rado n measur e o n X wit h supp/ i = X. Given a densely define d close d for m £ o n L 2 (A, /i), we call i t a Dirichle t for m i f fo r any u i n th e domai n D[£] o f £, v = max{min{w , 1},0} als o belong s t o D[£] an d £{v, v) < £(u,u). A subspace C of Co (A) C\D[£] i s called a core of £ i f it i s dense i n D[£] with respec t t o the nor m (£(• , •) + | | • | | | 2 ) 1 / / 2 an d als o dense in Co (AT) equipped with th e L°° norm . Whe n £ admit s a core , w e sa y £ i s regular . I f £{u,v) = 0 fo r any u, v G D[£] such tha t sup p wDsupp v = 0 (resp., u i s constan t o n suppt;) , the n the for m i s said t o b e loca l (resp . strongl y local) . We refer th e reade r t o [1 7] , Part I , fo r basi c propertie s o f Dirichle t spaces , an d also t o [5] , [20] , [21 ] , [22] , [35] , etc. , fo r man y result s an d interestin g example s concerning th e volum e doublin g propert y an d th e wea k Poincar e inequality . Fi nally, w e should mentio n tha t i n [8] , in additio n t o th e extensio n o f Rademarcher' s theorem a s above , th e differentiatio n o f Lipschitz continuou s function s i s expresse d in term s o f tangen t cones . 2. Dirichle t form s an d intrinsi c distance s In thi s section , w e conside r a regula r Dirichle t spac e (A , /i,£). Fo r an y u G D[£] fl L°°, ther e exist s a uniqu e positiv e Rado n measur e /i( u?u) give n b y (f) G D[£) H C0(X) — > £(u, (f)u) - \£{u 2, 0) . We call it th e energy measur e o f u. I f we set (JL{ U^V) = |(/X( u+t;jU+v ) — H(u,u) ~ ^{v,v)) for u,v E D[£]n L°° , the n thi s define s a bounde d signe d measur e o n A . Let u s assum e tha t £ i s strongl y local . I n thi s case , th e energ y measur e o f a function u E D[£] D L°° vanishe s i n a n ope n subse t A i f u i s constan t there . Thi s implies tha t th e energ y measur e restricte d t o A depend s onl y o n th e restrictio n o f the functio n there . Thi s allow s us to defin e a space Di oc[£] whic h consist s o f locall y square summabl e function s suc h tha t fo r an y ope n subse t A, ther e exist s v G D[£] which coincide s wit h u i n A. Fo r a functio n u G Dioc[£], th e energ y measur e /i/ UjU)

84

ATSUSHI K A S U E

on X ca n b e defined . Le t u s denot e b y A[£] th e spac e o f function s i n Di oc[£] suc h that th e /i( n?n) ar e absolutel y continuou s wit h respec t t o th e measur e ji. Namely , we pu t A[£] = {ue D

loc[£]

|

M(u,u)

=

T(U,U)II,

T(u,u)

€ LJ

0C(X)}.

Let u s furthe r introduc e a pseudo-distance , calle d th e intrinsi c (pseudo-)distance , by dg(x, y) = sup{w(x ) — u(y) \ u G A[£] D C(X), T(u, u) < 1 /i-a.e. } (cf. [4] , [1 0]) . Th e followin g i s proved i n Stur m [45] . T H E O R E M 2.1 . Let (X , /i, £) be a strongly local, regular Dirichlet space. Assume the property [C ] : the intrinsic pseudo-distance dg defines a distance on X and induces the same topology of X. Then the following assertions hold: (1 ) (X,ds) becomes a length space, (2) for a compact subset Y, the distance function to Y, pY(x) — mi{ds{x,y) \ y G Y} (x £ X), belongs to A[£] f l C ( I ) and satisfies r(py,py) 0 and R G (0,-f-oo] with respect to the intrinsic distance ds- Then C /o ' c (X, d^) C A[£], and r(ii,u) 1 / 2 < d\\d £u fi-a.e.

for u G C l(£(X, dg)- Moreover, suppose that for some constants r > 0 and R G (0, +oo], the following weak Poincare inequality on the Dirichlet form £ holds:

([P]e,r)

f \u-u

Xir\

J B(x,r) J

2

dfi)

0). The n th e Bishop-Gromov volum e compariso n theore m read s Vol( B(x,D)) Vol( B(x,r)) 1 k- , < rr, . , f0 {sinhK t^-^dt J*(swhKt) - dt D

0 2, A > 0, and B > 0 . Then there exist a subsequence {X^}, a regular Dirichlet space (X,/i , £) with the same properties, and a linear map L 2 (X m ,/i m ) from a core C of £ into L 2 (X m ,/i m ) satisfying [* ] such that £ rn converges to £ in the sense of Mosco as m— » oo. Moreover, {£ m } is asymptotically compact.

The propertie s [I ] an d [II ] enabl e u s t o embe d regula r Dirichle t space s int o a fixed compact metri c space , an d i t follow s fro m [III ] tha t th e algebr a generate d b y the eigenfunction s o f a given regular Dirichle t spac e contains th e uni t elemen t 1 . I n fact, Theore m 3. 2 can be proved a s follows: Le t X = (X , /x, £) b e a regular Dirichle t space a s above . The n i n vie w o f propert y [I] , the semigrou p possesse s a n integra l kernel p(t, x, y) satisfyin g p(t, x , y) < A*r~ vl2 fo r almos t al l x, y G X an d 0 < t < 1 , where A * is a constant dependin g onl y on A an d v. I n addition , propert y [II ] allows us t o hav e a complet e orthonorma l syste m o f eigenfunction s & o f th e eigenvalu e A^ and th e expansio n o f th e kernel , tha t is , p(t,x,y) = YliLo e~ Xit(j>i(x)(pi(y). W e remark tha t A 2 > CB~ 1 (i + l)2/" fo r X { > 1 and i + 1 < BC fo r A * < 1 ; in addition , El°^o Ke~ tXzi{x)2 < C(t-* vl2 + t~ v) fo r al l x G X an d 0 < t < -hoc , wher e C i s a positiv e constan t dependin g onl y o n v an d A. B y settin g lx{t) = (e- ( t + 1 / t ) / 2 e- A ^/ 2 ^(x)) 2 = 0 ,i,2,..., x G l , 0 < K + o o

,

2

we can assig n point s o f X continuou s loop s in the Hilber t spac e £ — {(a^=o,i,2,... | X ^ o a i2 < +cxo} and assum e tha t X i s a subse t o f th e Banac h spac e consistin g o f continuous loop s 7 a t th e origi n o f l 2 endowe d wit h th e nor m INI = SU

p \j(t)\

0 0 0 wit h respec t t o th e Hausdorf f distanc e i n thi s metri c space . Thu s ther e exist a sequenc e o f positiv e number s £ m tendin g t o 0 a s m — > 00 , Bore l measur able map s fm : X m— > X, an d a nonnegativ e Rado n measur e [i on X suc h tha t s ec

88

ATSUSHI K A S U E

\ds^c(x,y) - d^(f m{x),fm{y))\ < £m for all x,y G Xm, X C (f m(Xm))£m, an d fm+Hm vaguely converge s t o ji a s m— > oo. Moreove r ther e exis t a sequenc e of continuous function s ^ ( z = 0,1 , 2,...) o n X an d a sequence o f numbers {A ^ | 0 = Ao < A i < A 2 < • • • (< 00) } such tha t eac h elemen t 7 o f X ca n be expressed a s 7 = (e~^ + 1 /^e~ A i t ^(7))i = o,i,2,...- The n i t follow s fro m propert y [III ] that th e al gebra C generated b y the system {^ } separate s point s o f X an d contains th e unit element 1 . Thi s implie s tha t C is dens e i n the Banac h spac e C(X). No w if we denote th e support o f the limi t measur e / / by X, the n w e can deduce tha t {^ } i s a complete orthonorma l syste m o f eigenfunctions o f the eigenvalues A * with respec t to a regular Dirichle t for m £ o n L2(X, /i) . Th e linear ma p $m : C—» L2 (X m ,/i m ) in Theore m 3. 2 is now given by $m{u) = u°fm, ueC\ furthermore, th e approximation f m : Xm— > X satisfie s supe~{t+1/t)\pXm(t,x,y) -

p(tjm(x)jm(y))\
0

where p(t, x, y) denotes th e continuous kerne l o f the semigrou p associate d wit h the form £. I n this sense , th e map fm i s spectral approximatio n betwee n X m an d X. We refe r th e reader t o [26] , Part I , [28 ] or [30] for details o n Theorem 3.2 . I n [30], the arguments are based on a distance introduced on the set of regular Dirichle t spaces satisfyin g [I] , [II], and [III]. Se e also [3] , which motivate s th e introduction of the distanc e i n [30] . Let u s now consider strongl y local , regula r Dirichle t space s whic h satisf y [I] , [II], [III] , an d also [C ] in Theorem 2.1 . The n w e have

d8(x,y) oo. (2) The sequence of the Dirichlet forms £ n converges to £ in the sense of Mosco via the approximating maps f n. (3) The limit space X also satisfies properties [IV] and [VI], and moreover

C d s 2 (k* — dimU(k)) an d A' whic h depen d onl y o n A an d k. Henc e w e ca n find a subsequenc e {C/ m} suc h that Um converges t o a regula r Dirichle t spac e (Y , /I, £) a s m— > o o i n th e sens e o f Mosco. Moreover , i t turn s ou t tha t th e unitar y grou p U(k) act s o n th e limi t spac e Y i n such a way that th e actio n i s continuous, i t keep s the measur e Ji invariant, an d the approximatio n map s fro m E/ m t o Y ar e asymptoticall y equivariant ; i n addition , the actio n o f U(k) o n Y i s free o n a n ope n subse t Y o f Y . Thu s th e actio n yield s a Radon measur e /ly o n the quotien t spac e Y = Y/U(k) an d a continuou s Hermitia n vector bundl e V ^ ove r Y , an d als o the limi t Dirichle t for m £ induce s a closed for m J7^ o n the Hilbert spac e of L 2 section s of V^. W e note that th e eigensections of the form J 7^ ar e all continuous, an d moreove r w e remark tha t th e sequence of the bas e manifolds M m ma y be assumed t o converge to a regular Dirichle t spac e (X , nx,£x) as m —> • o o i n th e sens e o f Mosco . The n w e hav e a surjectiv e an d continuou s ma p 7r o f Y ont o X suc h tha t th e push-forwar d o f th e measur e Ji coincides wit h \±x and th e pull-back s o f continuou s function s o n X belon g t o th e subspac e o f U(k)invariant continuou s function s o n Y . However , th e convers e i s no t tru e i n general ; for instance , th e continuou s functio n |a | o n Y induce d b y a continuou s sectio n a o f Yoo over Y ca n b e assume d t o b e define d o n Y b y settin g \a\ = 0 outside o f Y , bu t the functio n doe s no t com e fro m continuou s function s o n X i n general . Second , w e consider th e form s {^~v m,Rm}- The n i n view of Kato's inequalit y an d th e condition s

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ATSUSHI K A S U E

on i? m , we can deduce tha t fo r some constant s a G (0,1) and (3 > 0 and for all m , aTVn(a) -

f3\\a\\ L2 < T

Vrn,Rm(a),

a G D[JVm ]. Thi s show s tha t th e forms ar e asymptotically compact , an d hence by taking a subsequence i f necessary, w e can conclude tha t {JV m ,fl m } converge s t o a closed for m T^ wit h D\Too[ C D[^oo ] in the sense o f Mosco. In geometry an d analysi s on manifolds, w e have important operator s associate d with quadrati c form s Ty^R suc h a s the Hodge Laplacia n A = V* V + 1Z acting on differential p-form s o n a compact Riemannia n manifol d M whic h is associated wit h the energ y for m j M \duo\ 2 + \Su\ 2dpM] i n case p — 1, 7Z is the Ricci curvatur e o f M. See [36 ] fo r related results . 4.2. I n this part , w e consider a strongly local , regula r Dirichle t spac e (X,/i,£ ) satisfying propert y [C] , and a complete Riemannia n manifol dT V = (N,gjy), an d w e study th e Sobolev spac e of maps fro m X t o N an d the energy functiona l explaine d below. Le t us first assum e fo r convenience tha t N i s isometrically embedde d int o some Euclidea n spac e R D , an d denote th e embedding b y t = ( V , . . . , L D) : N—> R D . Fo r a ma p u : X— > AT , we write t o u = (u 1 ,..., w D ). Th e Sobolev spac e W1,2(X, N) consist s of maps u such tha t eac h componen t u l (i = 1 , . . . , D) belong s to th e domain D[£], an d u(x) = (w 1 (x),..., uD(x)) G N fo r almos t al l x G X. Furthermore, give n w G Wrl ' 2 (X, iV), the energy measur e pF^ an d the energy o f u can b e defined b y p^u) = ^ = i M { ^ , ^ ) an d £ X,N{U) = notice tha t i f w G W1, 2 (X, AT) is continuous, the n £X,N{V>)

=

2

J i m / / yi(^(a;)-^(2/)) t

~^u J JXxX

-

/ x ^ , respectively . W e

- p ( t , x , 7 / ) dp(x)dp{y)

r 1 =

ds{x,y)2 t where p(£ , x, y) an d djv respectively stan d fo r the (heat) kerne l o f £ an d the Riemannian distanc e o f N. In wha t follows , w e assume, unles s otherwis e stated , tha t X an d N ar e both compact, an d we consider homotop y classe s o f continuous map s C(X , N) fro m X to N. The n C(X,N) 1( W^ 2(X,N) i s dense i n C(X,N) an d any u G C(X,N) i s homotopic t o some v G C(X , A r) n W 1 , 2 (X, A 7"). Fo r thi s reason , w e set CJX,N(A) — inf{£x,vv(tO | u G A (1 W 1 ,2(X, N)} fo r a homotop y clas s A, an d we denote by 1 2 7 %X,N(A) th e set of u G A n VF ' (X, A ") satisfying £ X,N(U) = p^(A) ^ 0 (see als o [1 3] , [1 4]) . I n fact, for such a spac e P , ther e exis t positiv e constant s i? , K, and r suc h tha t i n B(x,R), the volum e doublin g propert y [D] K an d the weak Poincar e inequalit y [P]s :r hold , and further , an y u G Tip^{A) i s a Holder continuou s ma p wit h exponen t a G (0,1 ) satisfying 1/2 /

/

/* )

J B(«,fl ) /

\

ds{x,y) ( - fl

C O N V E R G E N C E O F M E T R I C M E A S U R E SPACE S AN D E N E R G Y F O R M S 9

3

for al l x, y G B(z, R/1 0) C P , wher e C an d a ar e positiv e constant s dependin g onl y on th e give n K and r. In th e sequel , w e assum e t h a t th e Dirichle t spac e X = (X , /i, £) als o possesse s properties [D] K an d [P]s jT, an d th e targe t manifol d T V is nonpositivel y curved . I n this case , fo r u, v G C ( X, TV ) n W 1 , 2 ( X , TV) , if i t an d v ar e homotopic , the n the y ar e nomotopic i n C ( X , TV ) n W lj2(X, iV) , an d fo r a homotop y clas s A, i f E x , i v ( ^ ) ^ 0 , then th e geodesi c homotop y {u t | 0 < t < 1 } C i joinin g tw o map s UQ,UI G A stays i n EX,A^(^4) , an d /x ^ v = t/i| ^ v + ( 1 — ^ A^ \ - Thes e fact s ar e consequence s of th e definitio n o f th e energ y o f continuou s map s a s abov e an d th e convexit y o f the universa l coverin g spac e o f TV . Not e howeve r t h a t th e Euler-Lagrang e equatio n for u G EX,AK^4 ) m u s t b e expresse d i n term s o f th e signe d measures . We woul d lik e t o as k whethe r th e existenc e o f th e universa l coverin g spac e o f the domai n X woul d b e sufficien t t o asser t t h a t EX,JV(-A ) 7 ^ 0 fo r al l homotop y classes A. Let u s no w restric t ou r attentio n t o a famil y T' o f compact , admissibl e Rie mannian polyhedr a uniforml y satisfyin g [D] K an d [P]^, r . Moreover , w e assum e t h a t /j,(B(x,R)) > 9 fo r som e positiv e constan t 9 an d fo r al l x G X , X G T. The n fo r X G T', i t ca n b e deduce d fro m [H ] t h a t th e cardinalit y o f th e homotop y classe s A G [X, N] wit h (J(A) < A is finite fo r eac h A > 0 , s o t h at th e se t o f homotopy classe s [X, N] ca n b e p u t i n orde r a s 0 = CTX,N{AO)
AM, calle d th e Albanes e m a p , fro m M t o a fla t toru s AM o f dimensio n b\ (M), calle d th e Albanes e toru s o f M , wher e b\(M) denote s th e first Bett i numbe r o f M. T H E O R E M 4.3 . Let {M n} be a sequence of compact Riemannian manifolds satisfying the same properties described above with given positive constants R, K, T, and 9. Suppose that it converges to a regular Dirichlet space X in the sense of Mosco as n — » 0 0 and that bi(M n) is uniformly bounded from above. Then the

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ATSUSHI KASU E

sequence of the Albanese tori Au n converges to a flat torus T of dimension k, 0 < k < liminf n_>00 b\{M n), both in the topology of the Gromov-Hausdorff distance and also in the sense of Mosco. Moreover, the Albanese map OLM n '• M n— > Au uniformly converges to a Holder continuous map aoc : X— > T, and a ^ has the least energy in the homotopy class. A family o f compact Riemannia n manifold s o f the same dimension suc h that th e diameters ar e uniformly bounde d fro m abov e and the Ricci curvatures ar e uniforml y bounded fro m belo w satisfies al l the abov e properties. W e note that fo r limi t space s X o f thi s family , map s i n Y>X,N{A) a r e a ^ Lipschit z continuous . I n fact , w e ca n apply th e Bochne r technique . Let (X , /i, £) b e a strongly local , regular Dirichle t spac e satisfying th e propertie s [C], [D]^ , and [P]£, r> a n d ft a relativel y compac t ope n se t o f X. W e ar e concerne d with th e Dirichle t problem , th e existenc e an d regularit y o f map s o f leas t energ y with give n boundar y values . Le t u b e a ma p i n W l,2(ft1 N). W e assum e tha t th e sectional curvatur e N i s bounde d fro m abov e b y a nonnegativ e constan t K 2 an d the injectivit y radiu s o f N a t a poin t p i s greate r tha n a positiv e constan t i. The n if u(ft) C B(p,r) fo r som e r , 0 < r < min{i , ir/2K}), w e hav e a uniqu e solutio n of the Dirichle t proble m suc h tha t (j)(fl) C B(p, r) an d i t i s Holder continuou s i n ft. In fact , i n thi s case , w e ca n appl y th e direc t metho d o f th e calculu s o f variation s and th e ellipti c regularit y argument s t o weak solutions , becaus e of the give n map a s boundary value s being bounded i n a convex region. However , a s in Theorem 4.2 , so far a s th e convergenc e o f suc h solution s an d thei r energie s ar e concerned , w e nee d certain restriction s o n th e convergenc e o f domain s an d boundar y values . W e refe r the reade r t o [25 ] an d [26] , Par t II , fo r th e detail s o n Theorem s 4. 2 an d 4.3 . Se e also [27] , [33] , and [40 ] fo r relate d results . To en d thi s exposition , w e briefl y mentio n som e expansion s o f domain s an d target space s o f maps . I n [1 2] , [1 3] , an d [1 4] , th e targe t space s ar e take n t o b e complete Riemannia n polyhedr a o f uppe r bounde d curvature . Anothe r approac h to th e extensio n o f the domain s an d target s o f map s an d t o th e introductio n o f th e notion o f Mosc o convergenc e o n th e space s o f maps ha s bee n trie d recentl y i n [44] . We als o refe r th e reade r t o [43 ] an d th e reference s therei n fo r recen t progres s o n geometric analysi s o n Alexandro v spaces . References [1] K . Akutagawa , Convergenc e fo r Yamab e metric s o f positiv e scala r curvatur e wit h integra l bound o n curvature , Pacifi c J . Math . 1 7 5 (1 996) , 239-258 . [2] A . Bella'iche , Th e tangen t spac e i n sub-Riemannia n geometry , i n Sub-Riemannina Geometry (ed. A . Bella'ich e an d J.-J . Risler) , Progres s i n Mathematic s 1 44 , Birkhauser , Boston , 1 -78 . [3] P . Berard , G . Besso n an d S . Gallot , O n embeddin g Riemannia n manifold s i n a Hilber t spac e using thei r hea t kernels , Prepublicatio n d e l'lnstitu t Fourier , 1 09 , 1 988 ; Embedding Riemann ian manifold s b y thei r hea t kernels , Geom . Funct . Anal . 4 (1 994) , 373-398 . [4] M . Birol i an d U . Mosco , A Saint-Venau t principl e fo r Dirichle t form s o n discontinuou s media , Ann. Mat . Pur a Appl . (4 ) 1 6 9 (1 995) , 1 25-1 81 . [5] M . Bourdo n an d H . Pajot , Poincar e inequalitie s an d quasiconforma l structur e o n th e bound ary o f som e hyperboli c buildings , Proc . Amer . Math . S o c , 1 2 7 (1 999) , 231 5-2324 . [6] D . Burago , Y . Burag o an d S . Ivanov , A Cours e i n Metri c Geometry , GS M 3 3 , Amer . Math . S o c , Providence , R.I. , 2001 . [7] P . Buser , A not e o n th e isoperimetri c constant , Ann . Sci . Ecol e Norm . Sup . 1 5 (1 982) , 21 3 230. [8] J . Cheeger , Differentiabilit y o f Lipschit z function s o n metri c spaces , Geom . Funct . Anal . 9 (1999), 428-51 7 .

n

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5

J. Cheege r an d T.H. Colding, O n the structure o f spaces wit h Ricc i curvatur e bounde d below , I, J. Differentia l Geom . 4 6 (1 997) , 406-48 0 ; — II, ibid. 5 4 (2000) , 1 3-3 5 : — III, ibid., 37-74 . E.B. Davies , Hea t Kernel s an d Spectra l Theory , Cambridg e Universit y Press , Cambridge , 1989. G. Dal Maso, An Introduction t o r-Convergence, Progres s in Nonlinear Differentia l Equation s and Thei r Application s 8 , Birkhauser , Boston , 1 993 . J. Eell s an d B . Fuglede, Harmoni c Map s betwee n Riemannia n Polyhedra , Cambridg e Univer sity Press , Cambridge , 2001. B. Fuglede , Holde r continuit y o f harmoni c map s fro m Riemannia n polyhedr a t o space s o f upper bounde d curvature , Calc . Var . P . D . E . 1 6 (2003) , 375-403 . B. Fuglede , Th e Dirichlet proble m fo r harmoni c map s fro m Riemannia n polyhedr a togeodesi c spaces o f uppe r bounde d curvature , Preprin t Serie s 2001 , No. 1 8 , Dec . 2001 , University o f Copenhagen. K. Fukaya , Collapsin g Riemannia n manifold s an d eigenvalues o f the Laplac e operator , Invent . Math. 8 7 (1 987) , 51 7-54 7 K. Fukaya , Hausdorf f convergenc e o f Riemannia n manifold s an d it s applications , i n Recent Topics in Differential and Analytic Geometry, Adv . Stud , i n Pur e Math . 1 8 (1 990) , North Holland, Amsterdam , an d Kinokuniya , Tokyo , 1 43-238 . M. Fukushima , Y . Oshima an d M. Takeda, Dirichle t Form s an d Symmetric Marko v Processes , Walter d e Gruyter , Berlin-Ne w York , 1 994 . M. Gromov , Structure s metrique s pou r le s variete s riemanniennes , redig e pa r J . LaFontain e et P . Pansu , Cedi c Fernand-Nathan , Pari s 1981. M. Gromov , Metri c Structure s fo r Riemannia n an d non-Riemannia n space s (wit h appendice s by M . Katz, P . Pansu, an d S. Semmes, ed. by J. LaFontain e an d P. Pansu, Englis h translatio n by S . M . Bates) , Progres s i n Mathematic s 1 52 , Birkhauser , Boston , 1 999 . P. Hajlas z an d P . Koskela , Sobole v me t Poincare , Mem . Amer . Math . Soc . 1 4 5 (2000) , No. 688. B. Hanso n an d J . Heinonen , A n n-dimensiona l spac e tha t admit s a Poincar e inequalit y bu t has n o manifol d points , Proc . Amer . Math . Soc . 1 2 8 (2000) , 3379-3390 . J. Heinonen , Lecture s on Analysis on Metric Spaces , Universitext , Springer-Verlag , Ne w York, 2001 H. Hess , R . Schrade r an d D . A . Uhlenbrock , Kato' s inequalit y an d th e spectra l distribution s of Laplacian s o n compac t Riemannia n manifolds , J . Differentia l Geom . 1 5 (1 980) , 27-38 . K. Ishig e an d M . Murata , Uniquenes s o f nonnegativ e solution s o f th e Cauch y proble m fo r parabolic equation s o n manifold s o r domains , Ann . Scuol a Norm . Sup . Pis a 3 0 (2001 ) , 17 1223. A. Kasue , Convergenc e o f Riemannia n manifold s an d harmoni c maps , Math , preprin t series , Osaka Cit y University , 2000 . A. Kasue , Convergenc e o f Riemannia n manifold s an d Laplac e operators , I , Ann . Institu t Fourier 5 2 (2002) , 1 21 9-1 257 ; II , t o appear . A. Kasue , Convergenc e o f Riemannia n manifolds , Laplac e operator s an d energ y forms , i n Proceedings of the Fifth Pacific Rim Geometry Conference (ed . S. Nishikawa), Tohok u Math . Publ. 2 0 (2001 ) , 75-97 . A. Kasue , Geometri c analysi s o n metri c measur e space s -differentiatio n o f Lipschit z functions- (Japanese) , t o appea r i n Riemannia n manifold s an d thei r limits , Sugak u Memoire , Math. Soc . of Japan . A. Kasue , Spectra l convergenc e o f Riemannia n vecto r bundle s (Japanese) , Lectur e Note s Series i n Mathematic s 1 7 (2002) , Osak a University , 69-92 . A. Kasu e an d H . Kumura , Spectra l convergenc e o f Riemannia n manifolds , Tohok u Math . J . 46 (1 994) , 1 47-1 7 9 ; II , Tohok u Math . J . 4 8 (1 996) , 71 -1 20 . A. Kasu e an d H . Kumura , Spectra l convergenc e o f conformall y immerse d surface s wit h bounded mea n curvature , J . Geom . Anal . 1 2 (2002) . A. Kasue , H . Kumur a an d Y . Ogura , Convergenc e o f hea t kernel s o n a compac t manifold , Kyushu J . Math . 5 1 (1 997) , 453-524 . [33] M . Kotan i an d T . Sunada , Albanes e map s an d off-diagona l lon g tim e asymptotic s fo r th e heat kernel , Comm . Math . Phys . 20 9 (2000) , 633-670 . [34] K . Kuwa e an d T . Shioya , A convergenc e o f spectra l structures : a functiona l analyti c theor y and it s application s t o spectra l geometry , t o appea r i n Comm . Anal . Geom .

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ATSUSHI KASU E T. Laakso , Ahlfor s Q-regula r space s wit h arbitrar y Q admittin g wea k Poincar e inequality , Geom. Funct . Anal . 1 0 (2000) , 1 1 1 -1 23 . J. Lott , Collapsin g an d th e differentia l for m Laplacian : th e cas e o f a smoot h limi t space , Duke Math . J . 1 1 4 (2001 ) , 267-306 . X. Menguy , Noncollapsin g example s wit h positiv e Ricc i curvatur e an d infinit e topologica l type, Geom . Funct . Anal . 1 0 (2000) , 600-627 . U. Mosco , Composit e medi a an d asymptoti c Dirichle t forms , J . Funct . Anal . 1 2 3 (1 994) , 368-421. H. Nakajima, A convergence theorem fo r Einstei n metric s an d AL E space s (Japanese) , Sugak u 44 (1 992) , 1 33-1 46 ; Englis h transl. , Amer . Math . Soc . Transl . (2 ) 1 6 0 (1 994) , 79-94 . Y. Ogura , Wea k convergenc e o f law s o f stochasti c processe s o n Riemannia n manifolds , Probab. Theor y Relat . Field s 1 1 9 (2001 ) , 529-557 . L. Saloff-Coste , A not e o n Poincare , Sobole v an d Harnac k inequality , Duk e Math . J . Internat . Math. Res . Notice s 2 (1 992) , 27-38 . T. Sakai , Riemannia n manifold s wit h bounde d Ricc i curvatur e an d thei r limit s (Japanese) , to appea r i n Riemannia n manifold s an d thei r limits , Sugak u Memoire , Math . Soc . o f Japan . T. Shioya , Analysi s o n Alexandro v space s (Japanese) , t o appea r i n Riemannia n manifold s and thei r limits , Sugak u Memoire , Math . Soc . o f Japan . T. Shioya , Convergenc e o f metri c space s an d (nonlinear ) variationa l functional s (Japanese) , in th e abstract s o f Conferenc e o n Differentia l Geometry , Tsukub a University , 2002 . K. T . Sturm , O n th e geometr y define d b y Dirichle t forms , Semina r o n Stochasti c Processes , Random Field s an d Application s (Ascona , 1 993) , E . Bolthause n e t al. , eds. , Birkhauser , Progress i n Probabilit y 36 , 1 995 , 231 -242 . K. T . Sturm , Analysi s o n loca l Dirichle t space s III . Th e paraboli c Harnac k inequality , J . Math. Pure s Appl . 7 5 (1 996) , 273-29 7 K. T . Sturm , I s a Diffusio n Proces s Determine d b y It s Intrinsi c Metric? , Chao s Soliton s Fractals 8 (1 997) , 1 855-1 890 . T. Yamaguchi , Th e developmen t o f convergence theor y fo r Riemannia n manifold s (Japanese) , Sugaku 4 7 (1 995) , 46-6 1 T. Yamaguchi , Collapsin g Riemannia n 4-manifold s (Japanese) , Sugak u 5 2 (2000) , 1 72-1 86 . K. Yoshikawa , Degeneratio n o f algebrai c manifold s an d th e continuit y o f th e spectru m o f th e Laplacian, Nagoy a Math . J . 1 4 6 (1 997) , 83-1 29 . DEPARTMENT O F MATHEMATICS , KANAZAW A UNIVERSITY , KANAZAW A 920-1 1 92 , JAPA N

E-mail address: [email protected] p Translated b y ATSUSH I KASU E

Amer. Math . Soc . Transl . (2) Vol . 21 5 , 200 5

http://dx.doi.org/10.1090/trans2/215/06

Subfactor theor y an d it s applications : Operator algebra s an d quantu m field theor y Yasuyuki Kawahigash i ABSTRACT. W e revie w subfacto r theor y an d it s application s t o algebrai c quan tum field theor y wit h emphasi s o n classificatio n theory .

1. Brie f introductio n My specialit y i s subfacto r theor y withi n th e theor y o f operato r algebras , an d more specifically , I stud y mathematica l structure s appearin g i n operato r algebrai c approaches t o algebrai c quantu m field theory . Everyon e know s tha t th e field o f operator algebra s exists , bu t my experienc e suggest s tha t rathe r fe w kno w abou t concrete result s an d method s i n th e theor y o f operato r algebras , beyon d th e ap pearance o f it s nam e i n connectio n wit h noncommutativ e geometr y an d quantu m invariants i n lo w dimensional topology . Thus , takin g thi s opportunity , I would lik e to star t wit h explanation s o n ou r aim s an d method s i n operato r algebr a theor y fo r non-experts. Afte r that , I wil l revie w m y recen t researc h results . I refrai n fro m stating precis e definition s an d exac t statement s o f theorem s here , an d refe r th e reader t o th e reference s fo r al l suc h things . 2. Aim s o f th e theor y o f operato r algebra s The noncommutativ e geometr y o f Conne s an d quantu m invariant s i n 3-dimen sional topology , startin g wit h th e discover y o f th e Jone s polynomial , ar e partic ularly famou s amon g interaction s o f operato r algebr a theor y an d othe r fields o f mathematics. M y researc h i s closel y relate d t o th e latter , s o I hav e writte n an d talked abou t the m o n many occasions , bu t thes e ar e rather application s o f operato r algebra theor y an d ar e no t exactl y withi n operato r algebr a theor y i n th e narro w sense. S o befor e goin g int o suc h applications , I woul d lik e t o star t wit h interna l aims an d technique s withi n operato r algebr a theory . I refe r th e reade r t o [24 ] a s general textbook s o n operato r algebr a theory . We stud y algebra s o f bounde d linea r operator s o n a Hilber t space , whic h i s usually assume d t o b e separable an d infinit e dimensional , i n the theor y o f operato r This articl e originall y appeare d i n Japanes e i n Sugak u 5 4 (4 ) (2002) , 337-347 . 2000 Mathematics Subject Classification. Primar y 46L37 , 81 T05 ; Secondar y 46L60 , 81 T40 . Key words and phrases. Algebrai c quantu m field theory , subfactor , tenso r category , Virasor o algebra. ©2005 America n Mathematica l Societ y

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algebras. W e furthe r requir e tha t a n algebr a b e close d unde r th e *-operatio n an d an appropriat e topology . Dependin g o n th e choic e betwee n th e nor m topolog y and th e stron g operato r topology , w e obtai n a C*-algebr a an d a vo n Neuman n algebra. (Th e wea k operato r topolog y als o give s a vo n Neuman n algebra , an d so doe s th e stron g operato r topology. ) Th e stron g an d wea k operato r topologie s are weake r tha n th e nor m topology , an d closednes s unde r a weake r topolog y i s a stronger condition , needles s t o say , s o a vo n Neuman n algebr a i s automaticall y a C*-algebra, bu t th e method s use d t o stud y C*-algebra s an d vo n Neumann algebra s are ofte n technicall y different , s o man y operato r algebraist s thin k tha t th e theor y of operato r algebra s consist s o f tw o different , bu t no t entirel y disjoint , theories , (7*-algebras an d vo n Neuman n algebras . O f cours e ther e ar e severa l similaritie s and interaction s betwee n thes e two , s o i t i s importan t t o kno w wha t i s happenin g in th e other , an d som e peopl e wor k reall y i n both , bu t man y peopl e wor k i n onl y one o f them , an d I wor k o n vo n Neuman n algebras . (Man y peopl e outsid e th e theory o f operato r algebra s thin k tha t operato r algebr a theor y an d C*-algebr a theory ar e synonyms , an d thi s i s no t logicall y wron g i n th e abov e sens e tha t a vo n Neumann algebr a i s automaticall y a C*-algebra , bu t i t run s agains t th e feeling s o f many operato r algebraists. ) Here , not e tha t i t i s very importan t t o assum e tha t a n algebra i s close d unde r a n appropriat e topology . Fo r example , conside r a discret e countable grou p G an d it s lef t regula r representatio n A . Linea r combination s o f the unitar y operator s X g, g G G , mak e a n infinit e dimensiona l algebra , bu t w e need t o mak e a closur e i n a n appropriat e topolog y t o obtai n a G*-algebr a o r a von Neuman n algebra , i n orde r t o us e a genera l theor y o f operato r algebras . Som e infinite dimensiona l algebra s withou t completeness , o r eve n a topology , hav e ofte n been studie d i n othe r field s o f mathematics , bu t I fee l that , i n general , a trul y interesting structur e emerge s onl y afte r a completion . (I t i s tru e i n man y case s that w e se e interestin g structure s read y befor e a completion , an d the n w e d o no t need t o mak e a completion , bu t I fee l tha t thi s i s just becaus e w e happen t o b e i n a luck y situation. ) The mai n ai m i n th e theor y o f operato r algebra s i s t o understan d th e struc tures o f G*-algebra s an d vo n Neuman n algebras , an d thu s classificatio n theor y i s naturally a central topic . Tha t is , we would lik e to obtai n a (computable ) complet e invariant o r a complet e enumerativ e lis t fo r operato r algebra s themselves , thei r au tomorphisms, grou p action s o n them , o r thei r subalgebras . I n thi s sense , i t woul d be a n ultimat e goa l t o mak e a complet e classificatio n lis t o f al l th e operato r alge bras; bu t suc h a classificatio n seem s t o b e hopelessl y difficult , unfortunately , lik e the classificatio n problem s fo r al l topological space s o r al l Banac h spaces . Needles s to say , we have man y othe r importan t problem s beside s th e classificatio n problems , such a s investigatin g a property tha t hold s fo r al l o r mos t operato r algebras , deter mining the class of operator algebra s for which a certain conditio n holds , computin g (not necessaril y complete , bu t interesting ) invariants , an d constructin g intriguin g examples. Voiculescu' s fre e probabilit y theor y an d th e wor k o n exac t G*-algebra s by Kirchber g ar e particularl y importan t i n connectio n wit h suc h problems , bu t I limit thi s revie w t o problem s directl y relate d t o classifications , partl y du e t o m y limited tast e an d knowledge . An importan t ide a in classificatio n theorie s o f operator algebra s i s that thos e i n a nic e clas s ca n b e completel y classifie d b y a n algebrai c invariant . Anothe r equall y important ide a i s tha t w e ca n obtai n a simpl e characterizatio n o n whic h algebrai c

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invariants ca n indee d arise . Thes e idea s originate d i n a serie s o f stunnin g result s by Alai n Conne s i n th e 1 970 s an d hav e bee n source s o f man y studies . Mor e de tailed description s o f thes e ar e give n a s follows . Th e condition s fo r "nice " classe s are generall y calle d "amenability" . Thi s notio n originall y appear s fo r a discret e group G , an d i s defined a s existenc e o f a nontrivia l positiv e an d translatio n invari ant linea r functiona l o n £°°(G). (Her e positivit y o f a linea r functiona l mean s tha t it map s an y positiv e element s i n £°°(G) t o a positiv e number. ) Abelia n group s and finite group s ar e amenable . (However , eve n fo r th e intege r grou p Z , suc h a linear functiona l i s constructed onl y with a transcendental metho d arisin g fro m th e axiom o f choice . I t ofte n happen s i n operato r algebr a theor y tha t suc h a transcen dental ma p carrie s importan t piece s o f information. ) Fo r genera l discret e groups , amenability i s a conditio n "close " t o commutativit y i n a sense . Fo r example , fre e groups havin g a t leas t tw o generators ar e "th e mos t noncommutative " i n th e sens e that element s neve r commut e excep t fo r trivia l cases , an d the y ar e typica l exam ples o f non-amenabl e groups . A notio n o f amenabilit y i s extende d t o non-discret e groups easily . W e hav e a notio n o f amenabilit y fo r operato r algebra s a s a n ana logue o f thi s amenabilit y fo r groups , an d thi s conditio n fo r operato r algebra s ha s many equivalen t formulations . Connes ' greates t achievemen t withi n operato r alge bra theor y i s a characterization o f amenabilit y fo r vo n Neuman n algebras . Tha t is , he ha s prove d tha t amenabilit y o f a vo n Neuman n algebr a i s equivalen t t o hyper finiteness, whic h mean s tha t th e algebr a ca n b e approximate d internall y b y finit e dimensional algebras . (Not e tha t i t i s ofte n a usefu l an d importan t ide a t o pursu e an analogu e fo r operato r algebra s o f a notion fo r group s o r fields , base d o n th e ide a that groups , rings , an d field s ar e al l "similar". ) Thu s w e have importan t problems , such a s classification o f amenable operato r algebras , classificatio n o f actions o f (no t necessarily discrete ) amenabl e group s o n amenabl e operato r algebras , an d classifi cation o f amenabl e operato r subalgebra s (afte r a suitabl e definitio n o f amenabilit y for inclusion s o f operato r algebras) . I n th e las t nearl y thirt y years , w e hav e see n much grea t progres s o n suc h problems , bu t stil l w e ar e fa r fro m th e completion . The notio n o f amenabilit y fo r group s ma y no t b e ubiquitou s i n al l mathematics , but i t i s a decisivel y importan t conditio n i n operato r algebr a theory . W e hav e several reason s t o believ e tha t unifie d classificatio n wit h a simpl e invarian t woul d immediately fail , i f we proceed beyon d th e amenabl e class . Also , fro m a n operato r algebraic viewpoint , a n amenabl e discret e grou p i s "almos t lik e Z" , bu t th e othe r discrete group s ar e "entirel y different" . (Discret e group s o f a Li e group , suc h a s 5L(n,Z), ar e no w importan t object s i n th e theor y o f operato r algebras , bu t the y are ver y fa r fro m bein g amenable , an d fe w operato r algebrai c result s ar e know n for them . Fo r completel y genera l operato r algebras , i t i s believe d tha t whateve r strange conditio n w e may think, anything, ca n happen, a s long as it i s not obviousl y prohibited. Ther e i s a so-calle d "Gromo v principle " tha t an y non-trivia l statemen t for genera l discret e countabl e group s ha s a counterexample . Man y statement s o n discrete group s ca n b e translate d int o thos e operato r algebras , b y passin g t o grou p algebras, s o a statemen t analogou s t o thi s principl e i s believed t o hol d fo r operato r algebras.) Next I explain wha t a n "algebrai c invariant " is , in general . B y th e wor d "alge braic" , I mean tha t th e sam e or a similar invarian t ca n b e defined withou t topology . We usuall y conside r som e maps , suc h a s linea r functional s o n operato r algebras , * homomorphisms fro m on e operato r algebr a t o another , o r mor e complicate d maps ,

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and stud y th e se t o f al l suc h map s wit h som e equivalenc e relations . I t i s ofte n th e case tha t th e se t o f equivalenc e classe s i s "small " an d ha s som e algebrai c struc ture. A typica l suc h exampl e i s if-theor y fo r C*-algebras . "Som e equivalenc e relations" ofte n aris e fro m "trivialization " o f inne r automorphism s o f a n algebra , which i s o f th e for m x — i > Ad{u){x) = uxu* fo r som e unitar y u i n th e operato r algebra. Example s o f "som e algebrai c structure " ar e (finitely generated ) abelia n groups, tenso r categories , an d ergodi c flows . (I t ma y no t b e appropriat e t o cal l the las t on e "algebraic". ) Al l the classificatio n theorem s o n von Neuman n algebra s (and thei r subalgebras , grou p action s o n them) , initiate d b y Conne s an d develope d by Haagerup , Jones , Ocneanu , Popa , an d othe r people , clai m tha t suc h algebrai c invariants ar e complet e unde r assumption s o f som e amenability . I n th e classifica tion theor y o f simpl e separabl e C*-algebras , th e relevan t algebrai c invariant s ar e if-theoretic, an d man y classificatio n theorie s hav e bee n successfu l alon g th e lin e of th e Elliot t conjectur e statin g tha t amenabl e simpl e separabl e (7*-algebra s ar e completely classifie d b y the if-theoreti c invariants . (W e are , however, stil l far fro m a complet e solutio n t o thi s conjecture. ) Here , th e if-theoreti c invariant s ar e essen tially KQ- an d i f i-groups, bu t w e need some extra related dat a fo r the classification . I als o explai n basi c idea s tha t ar e commonl y share d i n thi s typ e o f classifica tion theories . A commutativ e vo n Neuman n algebr a i s o f a for m L°°(X, fi) an d a commutative (unital ) C*-algebr a i s of a form C(X), wher e (X , /JL) i n the forme r i s a measure spac e an d X i n the latte r i s a compact Hausdorf f space . (I f a commutativ e C*-algebra i s not unital , w e need to conside r continuou s function s o n a locally com pact Hausdorf f spac e vanishin g a t infinity. ) Thus , i t i s a n ol d ide a tha t theorie s o f von Neuman n algebra s an d C*-algebra s ar e "noncommutativ e integration/ergodi c theory" an d "noncommutativ e topology" , respectively . Th e noncommutativ e ge ometry o f Connes is also a recent far-reachin g extensio n o f this idea. On e importan t and rathe r surprisin g ide a in classification theorie s mentione d abov e i s that passin g to noncommutativ e algebra s make s classificatio n statement s simple r (althoug h i t does no t mak e technica l aspect s easie r a t all) . Fo r example , conside r classificatio n of automorphism s o f th e hyperfinit e H i factor , whic h ha s bee n on e o f th e mos t fundamental example s o f vo n Neuman n algebra s sinc e th e initiatio n o f th e theor y by Murra y an d vo n Neumann , an d is , o f course , amenable . I f w e hav e tw o au tomorphisms ce , /3 whose non-trivia l power s ar e neve r inner , the n w e hav e anothe r automorphism 6 and a unitary u in the algebr a so that w e have Ad(u)-a = 6-P'O" 1 . The inne r automorphis m Ad(u) i s regarded a s a "trivia l automorphism" , s o that a "generic" automorphism , i n th e sens e tha t n o non-trivia l power s becom e trivial , i s essentially uniqu e fo r thi s operato r algebra . Thi s i s a grea t achievemen t o f Connes . A commutativ e counterpar t o f a n automorphis m o f th e hyperfinit e H i facto r i n ergodic theor y i s a measure-preservin g ergodi c transformatio n o f a Lebesgu e spac e with a probabilit y measure , bu t a statemen t tha t al l suc h transformation s ar e con jugate i s obviousl y false . (Actuall y suc h transformation s ar e al l orbi t equivalent , and thi s fac t i s importan t fo r operato r algebr a theory , bu t orbi t equivalenc e lose s information abou t whic h grou p act s o n a Lebesgu e space , an d thi s i s a proble m from th e viewpoin t o f classificatio n o f grou p actions. ) I t i s important tha t w e hav e Ad(u) i n the abov e classification theore m o f Connes, an d th e fact tha t a noncommu tative algebr a ca n hav e a n inne r automorphis m no t equa l t o the identit y ma p play s a rol e i n uniquenes s here . Furthermore , sinc e commutativ e C*-algebra s shoul d b e amenable whicheve r definitio n w e ma y use , a counterpar t o f th e Elliot t conjectur e

S U B F A C T O R T H E O R Y AN D IT 1 S APPLICATION S 0 1

on if-theoreti c classificatio n o f simpl e amenabl e C*-algebra s i n th e commutativ e setting woul d b e a statement tha t (locally ) compac t Hausdorf f space s ar e classifie d up to homeomorphism wit h X-theory , whic h is , of course, entirely false . Recal l tha t commutative C*-algebra s hav e "many " ideal s (unles s i t i s th e scala r field C) , s o they ar e fa r fro m bein g simple . Thu s th e Elliot t conjectur e predict s tha t w e have a simple an d plai n classificatio n statemen t unde r a n assumptio n tha t C*-algebra s ar e close to being commutative i n terms of amenability, bu t fa r fro m bein g commutativ e in term s o f simplicity . I t ha s bee n recentl y show n tha t eve n fo r amenabl e simpl e C*-algebras, w e have various rathe r strang e phenomena , bu t still , i n compariso n t o topology, w e hav e a simple r pictur e afte r passin g t o th e noncommutativ e world . Finally, not e tha t a proo f o f suc h a classificatio n theore m i s alway s highl y technical an d relie s o n improvin g approximation s ste p b y ste p whil e controllin g errors. 3. Subfacto r theor y o f Jone s Next, I explai n ho w th e subfacto r theor y initiate d b y V . F . R . Jone s [1 3 ] fits int o a genera l framework a s abov e an d wh y i t i s relate d t o severa l "quantu m somethings". I have written an d talke d o n thes e topic s o n man y occasions , s o I wil l be rathe r brie f an d refe r th e reade r t o ou r boo k [9] . The definitio n o f a simpl e algebra , o f course , require s trivialit y o f two-side d ideals (close d unde r a n appropriat e topology) . Fo r vo n Neuman n algebras , i t ha s been known that thi s conditio n i s equivalent t o the propert y tha t th e center i s equal to C , an d fo r a historic reason , th e terminology "factor " ha s been used , rathe r tha n "simple von Neumann algebra" . Sinc e the days of Murray an d von Neumann, study ing factor s ha s bee n a centra l them e i n vo n Neuman n algebras . The n i n subfacto r theory, w e stud y th e situatio n wher e a facto rT V is include d i n anothe r facto r M . Usually, th e large r facto r M i s fixed, s o w e ofte n writ e " a subfacto rT V C M" , bu t we also use the terminology " a subfactor" t o mea n th e inclusio n o fTV i n M i n man y cases. Agai n b y th e ide a tha t groups , rings , an d fields ar e al l similar , w e conside r an analogu e o f a n inde x o f a subgrou p an d a degre e o f a n extensio n o f a field, an d call i t th e Jone s inde x o f a subfactor. Thi s Jone s inde x take s a value i n the interva l [1, oo] an d ca n b e a non-integer. I n thi s setting , w e are intereste d i n a n analogu e of the Galoi s theor y o f extension s o f fields an d it s "quantization" . (Usually , on e con siders a field and it s extensio n i n th e theor y o f fields, an d w e can similarl y conside r here a facto r an d it s extende d factor , bu t i n man y cases , i t i s more natura l t o fix a larger facto r an d stud y it s subfactors. ) A direct analogu e o f classica l Galoi s theor y has been studie d i n the theory o f operator algebra s fo r man y years , a s follows. Con sider a facto r M an d a n actio n o f a finite grou p G o n M suc h tha t an y non-trivia l element o f G act s a s a non-inner automorphis m o f M. (Her e the ide a tha t a n inne r automorphism i s "trivial " agai n appears. ) W e write M G fo r th e fixed point algebr a of thi s action , an d the n i t i s automaticall y a facto r an d w e ca n recove r th e grou p G jus t fro m th e inclusio n M G C G . Mor e concretely , w e conside r automorphism s of M tha t fix al l th e element s o f th e subfacto r M G , an d the n th e origina l actio n of G appears . W e als o hav e a Galoi s correspondenc e betwee n subgroup s o f G an d intermediate algebra s of M an d M G. Th e Jone s inde x o f this subfacto r M G C M i s the orde r o f G. I t i s an importan t ide a i n subfactor theor y t o exten d thi s viewpoin t to a genera l subfacto r N C M an d regar d N a s a "fixe d poin t algebra " unde r a n action o f somethin g lik e a group . Suc h a n ide a wa s materialize d i n a ver y stron g

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form o f paragrou p theor y b y Ocnean u [20] . Here , I explai n thi s theor y i n a for m due t o Long o an d Izumi . Recal l tha t i f a subfacto r indee d come s fro m a n actio n o f a finit e grou p a s a fixed poin t algebra , the n th e finite grou p ca n b e recovere d fro m the subfacto r a s a certai n automorphis m grou p o f th e large r algebra . Fo r a genera l subfactor, w e conside r endomorphisms rathe r tha n automorphism s an d obtai n a certain algebrai c system . (B y simplicit y o f M , an y endomorphis m i s automaticall y injective. I t i s mor e interestin g t o conside r a non-surjectiv e endomorphism . Fo r such a n endomorphism , th e imag e i s a prope r subalgebr a o f M tha t i s isomorphi c to M. Suc h a cas e doe s no t occu r fo r finite dimensiona l algebras , o f course , bu t this easil y happen s fo r infinit e dimensiona l algebras. ) W e have a product operatio n here, define d a s a compositio n o f endomorphisms . W e also hav e a direc t su m oper ation a s follows . (Not e tha t i f w e just ad d tw o endomorphism s a s maps , the n th e resulting ma p i s not a n endomorphis m i n general. ) Fo r tw o endomorphism s p , a o f M, w e define a homomorphis m x — i »I'

f

\ ) fr°

m

M t o M ® M 2 ( C ) . Th e

image i s in M(g)M2(C ) an d no t i n M , bu t i f M fall s i n a class of type II I factor tha t naturally arise s i n th e settin g o f algebrai c quantu m field theor y tha t I wil l explai n below, the n w e hav e a n eas y isomorphis m fro m M (g ) M2(C) ont o M , an d usin g this, w e obtain a n endomorphis m o f M tha t i s the direc t su m o f p, a. Furthermore , we ca n defin e a notio n o f a dimensio n fo r endomorphisms , an d i t take s a valu e i n [l,oo]. Thi s i s essentially th e sam e a s the squar e roo t o f the Jone s inde x mentione d above. W e als o hav e a notio n o f irreducibilit y fo r endomorphisms , an d i f a dimen sion i s finite, w e ca n mak e a n irreducibl e decompositio n o f a n endomorphis m i n a uniqu e wa y i n a n appropriat e sense . Th e dimensio n i s additiv e an d multiplica tive wit h respec t t o a direc t su m an d a composition , respectively , an d a n algebrai c system o f endomorphism s behave s similarl y t o a n algebrai c syste m o f unitar y rep resentations o f a compact group , wher e w e have a direct su m an d a tensor product . For a n endomorphism , w e hav e a notio n o f a conjugat e endomorphis m whic h i s a n analogue o f a contragredien t representation , an d w e hav e a direc t analogu e o f th e Frobenius reciprocit y fo r conjugat e endomorphisms . W e hav e tw o difference s be tween ou r endomorphism s her e an d unitar y representation s o f compac t groups , a s follows. First , ou r dimension s ar e no w no t necessaril y integers . Second , ou r com positions o f endomorphism s ar e no t necessaril y commutative , thoug h th e tenso r product operation , whic h i s th e correspondin g operatio n i n representatio n theory , is commutativ e i n a n obviou s sense . (W e hav e composition s o f endomorphism s o f an infinit e dimensiona l algebra . Ther e i s n o reaso n fo r suc h a n operatio n t o b e commutative.) Thi s algebrai c syste m i s a certai n tenso r categor y havin g endomor phisms a s objects. W e have a notion o f amenability fo r such a category, an d a tensor category havin g onl y finitely man y irreducibl e object s i s amenable , a s a n analogu e of a finite group . I n suc h a case , w e sa y tha t a categor y i s rational . Th e rule s o f irreducible decomposition s o f compositions o f two irreducible object s ar e calle d th e fusion rules . B y comparin g tw o kind s o f irreducibl e decomposition s o f a produc t of thre e objects , w e obtai n 6j-symbols , a s i n th e representatio n theor y o f compac t groups. W e simpl y cal l the m 6 j-symbols,o r quantum 6 j-symbols. A paragrou p is a n abstrac t algebrai c syste m an d axiomatize d wit h severa l propertie s o f suc h categories o f endomorphisms , lik e thos e o n fusio n rule s an d quantu m 6j-symbols . Popa's classificatio n theore m say s tha t i f w e have amenabilit y fo r bot h factor s an d

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paragroups, the n subfactor s ar e completel y classifie d b y paragroups . W e omi t de tails on these categories, quantu m 6j-symbols , an d paragroups , an d refe r th e reade r to the boo k [9 ] for precis e definitions . Her e we only mak e brie f remarks , i n compar ison t o simila r structure s appearin g i n conforma l field theor y an d quantu m groups , that th e fusio n rule s her e ar e no t necessaril y commutativ e an d tha t th e (quantum ) dimensions ar e alway s positiv e now . Her e w e als o remar k tha t i f a tenso r categor y arising fro m a subfacto r a s abov e ha s onl y finitely man y irreducibl e objects , the n we ca n construc t a complex-number-value d topologica l invarian t o f 3-dimensiona l oriented close d manifolds , an d mor e generally , 3-dimensiona l topologica l quantu m field theory, fro m suc h a category . Thi s i s a constructio n o f Ocneanu , generalizin g the on e b y Turae v an d Viro . (Not e tha t a categor y her e doe s no t hav e a braidin g in general . W e als o hav e operato r algebrai c method s t o produc e a braiding. ) Now Popa' s classificatio n theore m mentione d abov e fall s unde r th e genera l principle, explaine d above , tha t a n algebrai c invarian t become s complet e unde r amenability, an d i n thi s sense , th e resul t itsel f i s similar t o othe r classificatio n the orems, bu t w e now hav e a ne w feature , a s follows . A very nic e abstrac t characteri zation o f paragroup s ha s bee n obtaine d b y Ocneanu , bu t i t i s no t clea r a t al l wha t examples o f paragroup s w e indee d have , an d thi s proble m ca n b e studie d withi n a framewor k o f operato r algebras . O n on e hand , finitely generate d abelia n group s and ergodi c flows, whic h appea r a s invariant s o f operato r algebrai c objects , hav e been studie d wel l fo r a lon g time . O n th e othe r hand , th e histor y o f the studie s o f tensor categorie s i s relatively short , an d no t muc h i s known beyon d representatio n theories o f group s an d quantu m group s havin g deformatio n parameters . We , how ever, think , fro m th e viewpoin t o f subfacto r theory , tha t tenso r categorie s arisin g from quantu m group s wit h a deformatio n paramete r q ar e special ; w e hav e man y other tenso r categories , an d operato r algebra s ar e usefu l fo r studyin g them . (Th e biggest evidenc e fo r suc h a n ide a i s a combinatoria l stud y o f Haageru p [1 2 , 1 ] fo r subfactors wit h smal l Jone s index . H e mad e a n exhaustiv e searc h fo r paragroup s in a ver y limite d inde x range , foun d a larg e lis t o f candidate s tha t d o no t see m to b e relate d t o usua l quantu m grou p theory , an d prove d tha t tw o o f the m ar e indeed realized , afte r ver y har d computations. ) So , eve n i f on e i s intereste d onl y in tenso r categorie s an d topologica l invariant s arisin g fro m them , no t i n operato r algebras themselves , on e ca n stil l hop e t o obtai n ne w result s no t know n i n othe r fields b y operato r algebrai c ma t hods. On e informa l reaso n fo r thi s possibilit y i s that infinit e dimensiona l operato r algebra s ar e s o "large " tha t on e ca n pu t man y mathematical object s int o them . I n th e cas e o f operato r algebrai c invariant s suc h as finitely generate d abelia n group s an d ergodi c flows, we do no t obtai n reall y ne w examples throug h operato r algebras , s o th e ne w situatio n fo r tenso r categorie s i s different fro m thes e predecessor s an d interestin g fo r thi s reason . Thi s i s a basi s fo r a serie s o f ne w relation s betwee n subfacto r theor y an d "quantu m something" . 4. Algebrai c quantu m fiel d theor y There hav e bee n differen t mathematica l approache s t o quantu m field theory , and her e I explai n th e on e base d o n operato r algebras . Thi s approac h relie s o n algebras o f bounde d linea r operators , rathe r tha n operato r value d distributions , and i s calle d "algebraic " quantu m field theor y i n thi s sense . A standar d textboo k is [1 1 ] , an d on e ca n loo k a t th e proceeding s volum e [1 8 ] fo r recen t developments . Here I presen t a basi c settin g below .

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We fix a "space-time " an d conside r appropriat e bounde d region s i n it , thos e called doubl e cone s fo r example . T o each suc h regio n 0 w e assign a von Neuman n algebra A(0) generate d b y physical observable s i n 0. (W e are now in the realm of quantum mechanics , so physical quantities are represented a s operators on a Hilber t space.) W e have a fixed Hilber t spac e fo r al l 0's . Mathematicall y speaking , thi s gives a family o f operator algebra s parameterize d b y 0's. W e require a set of physically natura l axiom s t o hol d fo r suc h a famil y o f operator algebras , an d we stud y such familie s mathematically . Her e I explain onl y th e fundamental axioms , thoug h we often assum e variou s set s o f extra axiom s i n concrete studies . First , w e assume that w e have a larger operator algebr a for a larger space-time region 0, whic h mean s that w e have mor e observabl e quantitie s i n a large r region . Second , i f two region s 0 i , 0 2 ar e spacelike an d separated, the n th e two operator algebra s A{0\), A(02) commute, whic h i s called locality. Thi s mean s a relativistic requiremen t tha t i f one cannot reac h 0\ fro m 0 2 eve n a t th e spee d o f light, the n w e have n o interaction s between them , an d thus physica l quantitie s i n these two regions commute . Nex t we assume tha t w e have a projectiv e unitar y representatio n u g o f a grou p o f "space time symmetry " o n the Hilbert space , an d the equality u gA(0)u* = A(gO) holds , which i s called covariance. A physically natura l choic e of the space-time symmetr y is the inhomogeneous Lorent z grou p o f the 4-dimensional Minkowsk i space , bu t we will als o conside r othe r choice s o f the symmetr y below . W e also hav e axiom s o n a vacuum vecto r an d positivity o f energy, bu t I omit explanation s o f them here . W e often cal l a famil y {^4(0) } of operator algebra s a net o f operator algebras , becaus e the inde x se t o f double cone s 0 i s directed wit h respec t t o inclusions . Now w e have th e mathematica l proble m o f studying structure s o f such net s of operator algebra s unde r a suitabl e se t o f axioms , an d eve n classifyin g the m wit h a certai n algebrai c invariant . Ou r algebrai c invarian t i s the representatio n theor y of a net o f operator algebras . Suppos e w e have a net {A(0) } o f operator algebras . These operato r algebra s ac t o n a Hilber t spac e fro m th e beginnin g b y definition , but w e als o conside r thei r representation s o n othe r Hilber t spaces . Th e origina l Hilbert spac e ha s a special vecto r calle d a vacuum vector b y axiom, s o the origina l representation o n thi s Hilber t spac e i s calle d a vacuu m representation. W e the n consider a compatibl e famil y o f representation s o f th e algebra s {^4(0) } wit h th e covariance propert y o n anothe r Hilber t space , withou t requirin g a vacuu m vecto r there, an d stud y unitar y equivalenc e classe s o f suc h representations . W e furthe r have notion s o f a direc t sum , irreducibl e decomposition , a tenso r product , an d a dimension fo r suc h representations . Not e tha t w e now hav e representation s o f a family o f algebras , s o the notio n o f tenso r produc t i s not obviou s a t al l here . (I n quantum grou p theory, they hav e a coproduct, bu t we do not have such an operation here.) Also , considerin g a dimensio n i n th e ordinar y sens e i s meaningles s sinc e i t is always infinite . Thes e difficultie s ar e overcome wit h th e Doplicher-Haag-Robert s (DHR) theor y [8] . Tha t is , one first show s tha t an y representation, u p to unitar y equivalence, i s realized a s an endomorphism o f a net of operator algebras , base d on the Haa g duality, whic h i s a stronger for m o f locality. (Not e that a n endomorphis m of a net o f operator algebras , appropriatel y defined , give s a new representation o n the origina l Hilber t space . Th e DH R theor y say s tha t an y representatio n ca n be put i n thi s for m afte r a (possible ) chang e withi n th e unitar y equivalenc e class. ) Then th e operatio n calle d a "tenso r product " i s define d a s a compositio n o f two endomorphisms, an d a dimensio n i s als o define d nicely . I t i s obviou s tha t thi s

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situation i s simila r t o t h a t i n subfacto r theor y explaine d i n th e previou s section . Actually, th e D H R theor y i s muc h olde r t h a n Jones' s subfacto r theory , an d th e relations betwee n th e tw o wer e clarifie d b y Long o [1 7] . I n th e previou s section , we sa w t h a t th e compositio n operatio n o f endomorphism s i s no t commutative , bu t here, du e t o localit y i n th e axioms , w e hav e commutativit y o f composition , u p t o unitary equivalence . I n thi s way , w e obtai n a tenso r categor y o f representation s o f a ne t o f operato r algebras . As I mentione d above , th e mos t natura l choic e fo r th e space-tim e woul d b e th e 4-dimensional Minkowsk i space . Man y dee p result s hav e bee n obtaine d fo r thi s set ting, bu t i n connectio n t o "quantu m something" , w e hav e interestin g mathematica l structures o n space-tim e o f lowe r dimensions , suc h a s 1 , a s i n [1 0] , fo r example . The one-dimensiona l cas e i s no t a "space-time " an y more , b u t an y dimensio n i s allowed i n axiomati c mathematica l studies , an d fo r th e cas e o f on e spac e dimen sion an d on e tim e dimension , whic h i s quit e interestin g b o t h mathematicall y an d physically i n th e contex t o f conforma l field theory , a ne t o f operato r algebra s o n the 2-dimensiona l Minkowsk i spac e i s decompose d int o a tenso r produc t o f tw o net s of operato r algebra s o n one-dimensiona l space s i n a n appropriat e sense , an d suc h nets ar e calle d chiral nets an d hav e bee n wel l studied . On e migh t suspec t t h a t problems i n lowe r dimensions , particularl y i n dimensio n one , ar e mathematicall y easier, bu t thi s i s no t th e case . Fo r net s o f operato r algebra s o n a one-dimensiona l space, on e take s non-empt y bounde d ope n interval s a s O i n th e above , bu t the n a subtle an d interestin g problem , whic h doe s no t occu r i n dimensio n thre e o r higher , arises fro m th e fac t t h a t it s complemen t i s no t connected . Fo r example , a com position o f tw o endomorphism s arisin g fro m representation s i s commutativ e u p t o unitary equivalence , a s mentione d above , an d thi s unitar y operato r producin g th e unitary equivalenc e i n th e one-dimensiona l settin g bring s a braiding . I n thi s way , the representatio n categor y become s braided . I n dimensio n 3 o r higher , thi s braid ing automaticall y become s trivial , bu t man y non-trivia l example s o f interestin g braiding happe n i n th e one-dimensiona l setting , an d braide d tenso r categorie s cor responding t o th e Wess-Zumino-Witte n model s ar e realized . Also , th e dimension s of representation s ar e automaticall y integer s i n a highe r dimensiona l space-time , but w e hav e non-intege r dimension s o f representation s fo r net s o f operato r algebra s on a one-dimensiona l space , an d the y ar e counterpart s o f q u a n t u m dimension s i n quantum grou p theory . 5. T e n s o r c a t e g o r y , m o d u l a r invariant , a - i n d u c t i o n I hav e finally com e t o m y ow n researc h topics . T h e content s o f thi s sectio n describe wha t I hav e bee n studyin g afte r writin g th e boo k [9] . I refe r th e reade r t o the reference s fo r al l technica l m a t t e r s an d briefl y mentio n t h e mai n results . 5.1. C o m p l e t e l y r a t i o n a l A Q F T . A s explaine d above , a ne t o f operato r algebras o n one-dimensiona l "space-time " produce s a braide d tenso r category , bu t in connectio n wit h conforma l field theor y an d topologica l q u a n t u m field theory , we ar e intereste d i n tenso r categorie s whic h hav e onl y finitely man y irreducibl e objects an d a non-degenerat e braiding . T h e finiteness conditio n i s usuall y calle d rationality. Th e non-degenerac y o f a braidin g i s equivalen t t o invertibilit y o f th e 5-matrix, an d a braide d tenso r categor y wit h thi s non-degenerac y an d rationalit y is calle d a modular tensor category. Se e [25 ] fo r detail s o n thi s notion . Ou r mai n results i n [1 6 ] ar e t h a t w e hav e a simpl e operator-algebrai c sufficien t conditio n fo r

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the rationality o f the representation category , an d that th e braiding is automaticall y non-degenerate unde r thi s condition . W e cal l thi s operator-algebrai c conditio n complete rationality. B y result s i n [26] , [28] , w e kno w tha t th e net s o f operato r algebras arisin g fro m loo p groups of SU(n) [22 ] are completely rational . X u furthe r studied completel y rationa l net s o f operato r algebra s i n detai l i n [29 , 30 , 31 ] . Roughly speaking , complet e rationalit y i s preserve d unde r variou s operation s fo r nets o f operato r algebras . 5.2. a-inductio n an d modula r invariant . A s mentioned above , th e repre sentation theor y o f net s o f operato r algebra s i s quit e simila r t o tha t o f compac t groups. Th e metho d o f a-inductio n give s a machiner y i n th e representatio n theor y of net s o f operato r algebras , analogou s t o inductio n an d restrictio n i n th e ordinar y representation theory . A s an analogy o f a group an d it s subgroup, w e consider a net of operator algebra s an d a net o f subalgebras, an d w e produce a representation o f a larger ne t fro m on e of a smaller net . W e use the nam e "induction " her e becaus e we pass fro m smalle r algebra s t o large r algebras , bu t i n th e settin g o f nets o f operato r algebras, a large r ne t ha s a "smalle r symmetry" , s o thi s i s mor e lik e a restrictio n operation i n some sense. Thi s construction o f a-induction wa s first define d i n Longo and Rehre n [1 9] , and man y interestin g propertie s an d example s wer e foun d b y X u [27]. Bockenhaue r an d Evan s [2 ] furthe r studie d thi s construction . Aroun d th e same tim e a s Long o an d Rehren , Ocnean u [21 ] mad e a combinatoria l stud y o n th e A-D-E Dynki n diagram s fro m a quit e differen t motivation . I n [3] , we have prove d that bot h definitions , Longo-Rehre n an d Ocneanu , ca n be generalized t o a situatio n where w e have a braidin g i n the sens e o f Rehre n [23] , and w e have a unifie d defini tion there . B y combinin g bot h ideas , w e have obtained severa l result s i n [4 , 5 , 1 4] . We would lik e to construct a representation o f a larger ne t o f operator algebra s fro m a representatio n o f a smalle r net , an d us e a braiding i n the a-inductio n method . A braiding alway s come s i n a pai r o f a n ove r crossing an d a n unde r crossing, an d de pending o n these , w e have tw o kind s o f a-induction . W e distinguish the m b y usin g the ± symbol . Actually , w e canno t mak e a genuin e representatio n o f a large r ne t from on e o f a smalle r ne t i n general , an d w e obtai n onl y a "quasi"-representatio n in som e sense . I f w e tak e th e intersection s o f th e "quasi"-representation s arisin g from a-inductio n wit h a positive braidin g an d thos e wit h a negativ e braiding , the n it exactl y give s the representation s o f the large r net . I f th e representatio n categor y of the smalle r ne t i s modular (a s in the completel y rationa l case , for example) , the n we hav e a finite dimensiona l unitar y representatio n o f SX(2,Z) , an d w e ca n pro duce a matri x wit h non-negativ e intege r entrie s i n th e commutan t o f th e rang e o f this unitar y representatio n b y usin g positiv e an d negativ e a-inductions . Thi s wa s proved i n [3 ] unde r a ver y genera l assumption , base d o n wor k o f Ocnean u an d o f Bockenhauer an d Evan s [2] . Suc h matrice s hav e bee n studie d wel l in genera l unde r the nam e of "modula r invariants" , an d severa l classification theorems , startin g wit h [6], hav e bee n obtained . (Se e [7 ] for example. ) Ou r result s enabl e on e t o interpre t and realiz e suc h classificatio n result s wit h operator-algebrai c methods . Ther e ha s also bee n muc h stud y o f tenso r categorie s arisin g fro m a-inductio n whic h d o no t have a braidin g i n general . 5.3. A centra l charg e an d classificatio n o f net s o f factors . Recal l tha t a basi c ide a fo r classificatio n i s tha t w e shoul d hav e simpl e complet e algebrai c invariants fo r th e amenabl e case . Fo r net s o f operator algebras , eac h vo n Neuman n

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algebra A(O) assigne d t o a space-tim e regio n O i s alway s isomorphi c t o th e uniqu e hyperfinite typ e III i factor , a n amenabl e factor , unde r th e standar d se t o f axioms , s o the isomorphis m clas s o f eac h algebr a i s useles s fo r classificatio n o f nets . W e expec t t h a t representatio n theor y fo r completel y rationa l net s o n th e one-dimensiona l spac e would giv e a complet e invariant , bu t n o result s o f thi s typ e hav e bee n obtained . It i s als o totall y unclea r whic h modula r tenso r categor y arise s a s a representatio n category o f a ne t o f operato r algebras . Al l th e construction s s o fa r hav e bee n obtained dependin g o n th e case . Thu s th e genera l classificatio n an d constructio n problems ar e ver y fa r fro m a satisfactor y solution , bu t w e hav e recentl y obtaine d a first classificatio n theore m i n thi s directio n i n [1 5 ] a s follows . W e conside r net s of operato r algebra s o n th e one-dimensiona l space , an d w e compactif y th e spac e to S 1 . (Thi s i s jus t fo r simplicit y o f variou s descriptions . W e coul d wor k o n R withou t compactificatio n i f w e want . W e als o tak e th e orientation-preservin g diffeomorphism grou p o f S 1 a s th e grou p o f "space-time " symmetries . (I t i s no t mathematically clea r ho w stron g thi s assumptio n is , bu t w e hav e man y interestin g examples satisfyin g thi s condition. ) The n w e ca n defin e a rea l number , calle d th e "central charge" , fo r a ne t o f operato r algebra s throug h th e representatio n theor y of th e Virasor o algebra . I t i s well-know n t h a t i f thi s centra l charg e c i s les s t h a n one, the n i t ca n tak e onl y value s i n a discret e series , an d ou r mai n result s i n [1 5 ] say t h a t i n suc h a cas e w e hav e a complet e classificatio n an d listin g o f net s o f operator algebras . W e full y us e th e theorie s o f a-inductio n an d modula r invariant s mentioned above . Th e classificatio n resul t i s describe d wit h pair s o f th e A-D-E Dynkin diagrams . 6. F u t u r e d e v e l o p m e n t s Classification o f net s o f operato r algebra s stil l ha s man y ope n problems , suc h as finding a complet e invariant , characterizin g possibl e invariants , an d listin g al l the possibl e nets . W e als o hav e variou s problem s o n relation s amon g extr a axiom s in additio n t o th e standar d se t o f axioms . Furthermore , w e hav e limite d (o r no ) knowledge o n it s relation s t o q u a n t u m groups , verte x operato r algebras , th e monste r group, th e moonshine , an d topologica l q u a n t u m field theory . I finish thi s surve y hoping fo r furthe r development s i n thes e an d othe r yet-unknow n directions . References 1. M . Asaed a & ; U. Haagerup , Exotic subfactors of finite depth with Jones indices ( 5 + V l 3 ) / 2 and ( 5 + vT7)/2 , Comm . Math . Phys . 20 2 (1 999 ) 1 -63 . 2. J . Bockenhaue r & D . E . Evans , Modular invariants, graphs and a-induction for nets of subfactors I, Comm . Math . Phys . 1 9 7 (1 998 ) 361 -386 . I I 20 0 (1 999 ) 57-1 03 . Il l 20 5 (1 999 ) 183-228. 3. J . Bockenhauer , D . E . Evan s & Y . Kawahigashi , On a-induction, chiral projectors and modular invariants for subfactors, Comm . Math . Phys . 20 8 (1 999 ) 429-487 . 4. J . Bockenhauer , D . E . Evan s & Y . Kawahigashi , Chiral structure of modular invariants for subfactors, Comm . Math . Phys . 21 0 (2000 ) 733-784 . 5. J . Bockenhauer , D . E . Evan s & Y . Kawahigashi , Longo-Rehren subfactors arising from ainduction, Publ . Res . Inst . Math . Sci . 3 7 (2001 ) 1 -35 . 6. A . Cappelli , C . Itzykso n & J.-B . Zuber , The A-D-E classification of minimal and A\ conformal invariant theories, Comm . Math . Phys . 1 1 3 (1 987 ) 1 -26 . 7. P . D i Francesco , P . Mathie u & D . Senechal , i; Conformal Fiel d Theory" , Springer-Verlag , Berlin-Heidelberg-New York , 1 996 . 8. S . Doplicher , R . Haa g & J . E . Roberts , Local observables and particle statistics. I . Comm . Math. Phys . 23 , 1 99-23 0 (1 971 ) ; II . 35 . 49-8 5 (1 974) .

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9. D . E . Evan s & ; Y. Kawahigashi , "Quantu m Symmetrie s o n Operato r Algebras" , Oxfor d Uni versity Press , Oxford , 1 998 . 10. K . Fredenhagen , K.-H . Rehre n & ; B. Schroer , Super selection sectors with braid group statistics and exchange algebras, I . Comm . Math . Phys . 1 2 5 (1 989 ) 201 -226 ; II . Rev . Math . Phys . Special issu e (1 992 ) 1 1 3-1 57 . 11. R . Haa g "Loca l Quantu m Physics" , Springer-Verlag , Berlin-Heidelberg-Ne w York , 1 996 . 12. U . Haagerup , Principal graphs of sub factorsin the index range 4 < 3 + \ / 2 , i n "Subfactor s — Proceeding s o f th e Taniguch i Symposium , Katata " (ed . H . Araki , e t al.) , Worl d Scientific , 1994, 1 -38 . 13. V . F . R . Jones , Index for subfactors, Invent . Math . 7 2 (1 983 ) 1 -25 . 14. Y . Kawahigashi , Generalized Longo-Rehren subfactors and a-induction, Comm . Math . Phys . 226 (2002 ) 269-287 . 15. Y . Kawahigash i & ; R. Longo , Classification of Local Conformal Nets. Case c < 1 , t o appea r in Ann . Math. , math-ph/0201 01 5 . 16. Y . Kawahigashi , R . Long o & M . Miiger , Multi-interval subfactors and modularity of representations in conformal field theory, Comm . Math . Phys . 21 9 (2001 ) 631 -669 . 17. R . Longo , Index of subfactors and statistics of quantum fields, I . Comm . Math . Phys . 1 2 6 (1989); II . 21 7-24 7 & 1 3 0 (1 990 ) 285-309 . 18. R . Long o (ed.) , "Mathematica l Physic s i n Mathematic s an d Physics" , Field s Inst . Commun . 30, Amer . Math . S o c , Providence , RI , 2001 . 19. R . Long o & K.-H . Rehren , Nets of subfactors, Rev . Math . Phys . 7 (1 995 ) 567-597 . 20. A . Ocneanu , Quantized group, string algebras and Galois theory for algebras, i n Operator algebras and applications, Vol. 2 (Warwick, 1 987), (ed . D . E . Evan s an d M . Takesaki), Londo n Mathematical Societ y Lectur e Not e Serie s 36 , Cambridg e Universit y Press , Cambridge , 1 988 , 119-172. 21. A . Ocneanu , Paths on Coxeter diagrams: From Platonic solids and singularities to minimal models and subfactors (Note s recorde d b y S . Goto) , Par t 5 i n Lectures on operator theory, (ed. B . V . Rajaram a Bha t e t al.) , Th e Field s Institut e Monographs , 1 3 , Amer . Math . Soc , Providence, RI , 1 999 , pp . 243-323 . 22. A . Pressle y & G . Segal , "Loo p Groups" , Oxfor d Universit y Press , Oxford , 1 986 . 23. K.-H . Rehren , Braid group statistics and their supers election rules, in : "Th e Algebrai c Theor y of Superselectio n Sectors" , D . Kastle r ed. , Worl d Scientific , Singapore , 1 990 , 333-355 . 24. M . Takesak i ,"Theor y o f Operato r Algebras . I , II , III" , Springer-Verlag , Berlin-Heidelberg New York , 2002 . 25. V . G . Turaev , "Quantu m Invariant s o f Knot s an d 3-Manifolds" , Walte r d e Gruyter , Berlin New York , 1 994 . 26. A . Wassermann , Operator algebras and conformal field theory III: Fusion of positive energy representations of SU(N) using bounded operators, Invent . Math . 1 3 3 (1 998 ) 467-538 . 27. F . Xu , New braided endomorphisms from conformal inclusions, Comm . Math . Phys . 1 9 2 (1998) 347-403 . 28. F . Xu , Jones-Wassermann subfactors for disconnected intervals, Comm . Contemp . Math . 2 (2000) 307-347 . 29. F . Xu , Algebraic coset conformal field theories I, Comm . Math . Phys . 21 1 (2000 ) 1 -44 . 30. F . Xu , 3-manifold invariants from cosets, preprin t 1 999 , math.GT/9907077 . 31. F . Xu , Algebraic orbifold conformal field theories, Proc . Nat . Acad . Sci . U.S.A . 9 7 (2000 ) 14069-14073. DEPARTMENT O F MATHEMATICA L SCIENCES , UNIVERSIT Y O F T O K Y O , T O K Y O , 1 53-891 4 , JAPA N

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Brownian motio n an d valu e distributio n theor y o f holomorphic map s an d harmoni c map s At sushi Atsuj i

1. Picard' s theore m an d Brownia n motio n Let Z b e comple x Brownia n motio n o n C , namel y Z t = X t -f - \f^lY t, wher e X and Y ar e independent rea l valued Brownia n motion s an d / a holomorphic functio n on C . I t i s well-known tha t fo r som e comple x Brownia n motio n Z

f(Zt) = Z Pt, Pt = I

\f'(Z

2 t)\ ds.

Jo This beautifu l relationshi p betwee n Brownia n motio n an d holomorphi c function s i s called conforma l invarianc e o f Brownia n motion , an d i t wa s discovere d b y P . Levy , who was a pioneer o f the theor y o f Brownian motion . Thi s ensure s som e possibilit y of investigatin g comple x analysi s b y Brownia n motion . B. Davi s gav e a n elegan t proo f o f Picard's littl e theore m usin g this relationshi p ([16], [1 7]). 1 Let u s loo k a t th e trajector y o f Z i n th e figure below . (Not e tha t Brownia n motion o n R 2 doe s no t hi t a point . 0 stands fo r identificatio n o f th e origi n wit h a neighborhood o f th e origin. )

Davis prove d THEOREM 1 . Brownian motion on C\ { —1,1} starting from 0 makes a path homotopic to the one in the figure almost surely. Here C is the complex plane which identifies the origin with 0 . This articl e originall y appeare d i n Japanes e i n Sugak u 5 4 (3 ) (2002) , 235-248 . 2000 Mathematics Subject Classification. Primar y 60J65 ; Secondar y 30D35 , 58J65 . ^-See als o [1 9 ] an d [9] . Thes e book s ar e goo d guide s t o th e relationshi p betwee n classica l function theor y an d probabilit y theory . ©2005 America n Mathematica l Societ y 109

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It i s well-know n t h a t comple x Brownia n motio n i s recurrent . Namely , fo r an y open subse t [ f c C , limsup ljj(Z t) = 1 , a.s. 2 t—*oo

Hence Brownia n motio n startin g fro m th e origi n make s a p a t h comin g bac k t o a neighborhood o f th e origi n (6 ) wit h probabilit y 1 . Moreover , th e stron g Marko v property o f Brownia n motio n implie s t h a t i t come s bac k t o th e neighborhoo d in finitely often . O n th e othe r hand , conforma l invarianc e implie s t h a t a nonconstan t holomorphic functio n / map s Brownia n motio n t o a time-change d Brownia n mo tion. Assum e t h a t / omit s tw o point s { — 1,1}. Theore m 1 implie s t h a t th e imag e of Brownia n motio n b y th e holomorphi c functio n make s a loo p lik e th e figure, whil e the Brownia n motio n o n th e domai n o f th e functio n finitely ofte n come s bac k t o the neighborhood . Sinc e C i s simpl y connecte d an d / i s continuous , thi s lead s u s to a contradiction . This proo f i s concerne d wit h probabilisti c an d geometri c propertie s o f th e do main an d th e targe t separately . Thi s ha s fo r u s th e meri t o f makin g eac h rol e i n the theore m clearer . T h e proo f i s a synthesi s o f them . As w e know , a universa l coverin g o f C \ { — 1,1} i s conformall y equivalen t t o the uppe r plane , whic h i s a typica l exampl e o f hyperboli c space . Equippe d wit h the Poincar e metric , i t become s a complet e spac e o f constan t negativ e curvature . By takin g accoun t o f liftin g i t i s natura l t o conside r th e cas e o f simpl y connecte d and negativel y curve d manifold s a s a targe t o f maps . I n fact , Davi s use d transienc e (i.e., non-recurrence ) o f a lifte d proces s o n th e universa l coverin g an d th e fac t i t converges t o infinit y a s tim e tend s t o infinit y i n hi s proof. 3 I f you exten d thes e cases , keeping t o th e probabilisti c natur e o f th e proof , yo u wil l conside r th e situatio n t h a t a proces s i n th e domai n o f map s ha s a propert y clos e t o recurrenc e an d th e imag e process i n th e targe t converge s t o infinit y a s tim e tend s t o infinity . I n thi s directio n a probabilisti c improvemen t o f th e Picar d theore m wa s don e b y W . S . Kendall , wh o gave anothe r versio n o f th e Goldberg-Ishihara-Petridi s theore m ([38]) . 2. P i c a r d t y p e t h e o r e m s fo r h a r m o n i c m a p s and convergenc e o f martingale s We exhibi t a relationshi p betwee n harmoni c map s an d martingales . Consider a manifol d wit h a connection . W e ca n conside r geodesie s correspond ing t o th e connection . W e sa y t h a t a functio n g o n th e manifol d i s geodesicall y convex i f t — > g(j(t)) i s conve x fo r an y geodesi c 7 . D E F I N I T I O N 2 (cf . [22]) . Le t (Q,^ 7 , {Ft}, P) b e a filtered probabilit y spac e and Y a continuou s stochasti c proces s o n (ft , T ', T t, P) takin g value s i n a manifol d N wit h a connection . W e sa y t h a t Y i s a T-martingal e i f g(Y) i s a submartingal e on (ft, T, {Tt}, P) fo r an y conve x functio n g define d locall y o n N. A P-martingal e o n A r i s ofte n calle d simpl y a martingal e o n N below . Yo u ca n see t h a t th e notio n o f martingale s i s associate d wit h a connection . ' T - " come s fro m this. 2

Probabilists hav e th e habi t o f sayin g tha t a n even t occur s "a.s. " ( = "almos t surely" ) instea d of "wit h probabilit y one" . 3 Davis prove d Theore m 1 as a property o f Brownia n motio n o n R 2 . I n thi s directio n Goldber g and Muelle r [30 ] generalize d Davis ' proo f b y studyin g th e windin g propert y o f Brownia n motio n on Rieman n surface s t o obtai n a Picar d theore m fo r meromorphi c function s o n Rieman n surfaces .

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We no w giv e a definitio n o f finely harmoni c maps , whic h i s a wide r clas s tha n the usua l harmoni c map s an d i s relate d t o Brownia n motion , instea d o f th e usua l harmonic map s i n th e differential-geometri c sense . DEFINITION 3 . A Bore l measurabl e ma p / fro m a Riemannia n manifol d M to N i s calle d a finely harmoni c ma p i f h o / i s subharmoni c fo r an y geodesicall y convex functio n h define d locall y o n N.

In particular , a smoot h finely harmoni c ma p i s a usua l harmoni c ma p whic h i s defined b y a ma p o f vanishin g tensio n field (cf . [58]) . It woul d b e interestin g t o kno w whe n a finely harmoni c ma p become s har monic. Thi s proble m i s als o relate d t o th e Dirichle t proble m o f harmoni c maps . For probabilisti c approache s t o thes e problems , se e [43] , [3] , [52] , [15]. The followin g show s a fundamenta l relationshi p betwee n Brownia n motion , martingales an d harmoni c maps . 4 (Bismu t [49]) . A C 2-map f : M— > N is harmonic if and only if f maps any Brownian motion on M to martingales on N . THEOREM

We nex t giv e a Picar d typ e theore m fo r harmoni c maps , du e t o Goldberg , Ishihara, an d Petridis . THEOREM 5 ian manifold of simply connected less than —k N is a harmonic map of bounded dilatation, then f

We sa y tha t / : M - > N i s o f (K-)bounde d dilatatio n i f A ^ < K 2\&\ wher A(D > A (2) > .. . ar e eigenvalue s o f df(df)*. W. S . Kendal l gav e tw o type s o f proof s o f thi s theorem .

e

THEOREM 6 (Kendal l [38]) . Let M be a complete Riemannian manifold, N a Hadamard manifold whose sectional curvature satisfies —o?< Sect^ < —b 2 < 0. Let f : M — >TV be a harmonic map of bounded dilatation. If M admits no nonconstant bounded harmonic function, then f is constant. REMARK 7 . I f RICM > 0 , M admit s n o nonconstan t bounde d harmoni c func tion ([60]) . Henc e Theore m 6 require s a weake r assumptio n o n th e domain s o f maps tha n th e Goldberg-Ishihara-Petridi s theorem . Bu t i t need s a lowe r boun d o n the sectiona l curvatur e o f th e targe t manifold . Thu s w e canno t sa y whic h o f the m is the stronge r result .

We conside r probabilisti c propertie s o f M an d N a s i n Davis ' proof . The assumptio n tha t M admit s n o nonconstan t bounde d harmoni c functio n i s equivalent t o the triviality o f an invariant cr-fiel d o f Brownian motion . Le t Brownia n motion b e define d o n a probabilit y spac e (f£ , T, P x, x G M). An invarian t a-fiel d i s a collectio n o f A G T satisfyin g lA°0t =

U, t

< C, P

x -a.s., V

x G M,

where ( i s th e lifetim e o f th e Brownia n motio n an d 0 t : fi — > ft i s define d b y 0tUj(s) =

Ld(t + s).

Let u s exten d th e notio n o f "bounde d dilatation " t o martingales , i n orde r t o characterize martingale s o n N.

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Let Y b e a T-martingal e o n a d-dimensiona l Riemannia n manifol d N. There exis t a loca l martingal e Y o n H d an d a rando m orthonorma l fram e E(t) : H d— » Ty{t) satisfyin g a stochasti c differentia l equatio n dsY =

Zd

dsE = H

sY, EdsY,

where H^ : TyN— > T^O(N) (O(N) i s a n orthonorma l fram e bundle ) i s th e horizon tal lif t an d ds stand s fo r th e Stratonovic h integral. 4 H i s calle d stochasti c paralle l transport an d Y i s calle d stochasti c development . Set

(d(Y\Y*)t\ (assume it s existence) . W e sa y Y i s o f (K-)bounde d dilatatio n i f th e eigenvalue s \M(t) > \ {2\t) > . . . o f A , satisf y A ^ < K 2\^2\ It i s well-known t h a t a n n-dimensiona l H a d a m a r d manifol d wit h — a2 < Sect A/- < 2 —b oo .

From thi s w e immediatel y hav e a generalizatio n o f th e Casorati-Weierstras s theorem. COROLLARY 23 . If f is a nonconstant holomorphic map from a Kahler manifold which admits no nonconstant bounded harmonic function to complex projective space Pn ( C ) 7 then f can omit no sets of hyperplanes of positive measure in P n ( C ) . REMARK 24 . "Positiv e measure " i n th e abov e statemen t ca n b e replace d b y "positive projectiv e capacity" . Projectiv e capacit y wa s introduce d b y Molzo n [50] .

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Any algebrai c submanifol d i n C n admit s n o nonconstan t bounde d harmoni c function ([56]) . Henc e COROLLARY

25 . The Casorati-Weierstrass theorem holds on any algebraic sub-

manifold in C n. 5.3. Secon d mai n theorem . I n thi s sectio n w e exhibi t a probabilisti c ap proach t o th e secon d mai n theorem . Th e followin g tw o lemma s ar e essential . LEMMA 2 6 ([4]) . i ) Let Y a Brownian motion on a two-dimensional compact Riemannian manifold M, and let k G L l+t(M,dv) ( e > 0) , k > 0 . There exist constants c\,C2 > 0 such that, for any stopping time T, T

E[f k(Y t)dt) < ClE[T} + c 2. Jo ii) Let Y be a martingale on a d-dimensional compact Riemannian manifold M, and let k G Ld(M,dv), k > 0 . There exist constants 61 ,6 2 > 0 such that for any stopping time T T

E[ [ k(Y Jo

t)£tdt]

< dE[[Y, Y] T] + C 2 ,

^=-((^)f

where

and Y is the stochastic development ofY.

LEMMA 2 7 ([4]) . Let X be Brownian motion on H d and K(x) a nonnegative locally integrable function on H d. For S > 0 there exists E$ C [0 , oo) with \E$\ < oo such that for r ^ E$ E[K(XTr)] 0 such that, for any stopping time T,

Set K(z)dzdz = inequality, J2m(a3,r) +

E[f k(Y t)dt] < ClE[T}+c2. Jo k o f f*w. Then , b y th e first mai n theore m an d Jensen' s

N^r) = E[\ogK(X Tr)} +

^ S[log(log[/(X

Tr ),

a 3})2}

< S [ l o g i f ( X r r ) ] + 0(logT(r)) . The las t ter m ca n b e estimate d a s before . Henc e w e hav e q

Y^rn^j.r) +

N x(r) < 2T(r) + 0(logT(r) ) + logC(r ) + 0(1 )

for r outsid e a n exceptiona l se t o f finite length . It i s a proble m whethe r o r no t logO(r ) = o(T(r)). Bu t i t i s no t simpl e excep t in th e cas e o f M — C n . Fo r example , i n th e cas e o f Hadamar d manifold s wit h sectional curvatur e pinche d b y negativ e constant s w e hav e lo g C(r) = 0(r), an d there exist s a bounde d holomorphi c functio n whic h satisfie s T(r) = O(l) . I t ma y be interestin g t o conside r estimate s o n C(r) whe n a manifol d admit s n o constan t bounded harmoni c functions . I t ma y als o b e interestin g t o kno w an y probabilisti c meaning o f thi s 'remainde r term' . Using Lemm a 2 6 ii ) enable s u s t o trea t th e cas e o f highe r dimensiona l targets . But Lemm a 2 6 ii ) work s onl y i n th e equidimensiona l case . Instea d o f thi s demeri t we ca n rela x som e par t o f th e assumption s o f nonsingularit y an d norma l crossin g

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of divisor s o f th e targe t manifold s i n th e result s o f Carlso n an d Griffith s [1 1 ] b y Lemma 2 6 ii). I t woul d b e interestin g t o se e i f suc h a n inequalit y a s Lemm a 2 6 ii) holds fo r genera l martingales . Eremenko an d Sodi n ([23] , [24] ) obtaine d a secon d mai n theore m i n non-equi dimensional cases , i n particula r a generalizatio n o f th e secon d mai n theore m fo r holomorphic curve s in P n ( C ) du e t o H . Carta n (cf . [51 ]) , by usin g potentia l theor y on th e comple x plane . I t seem s closel y relate d t o th e probabilisti c natur e o f th e second mai n theorem , bu t th e autho r ha s n o ide a abou t i t yet . Nevanlinna's origina l proof o f his second mai n theore m wa s based o n his lemma on the logarithmic derivative. W e appl y ou r techniqu e t o this . W e ca n generaliz e this to an estimate for a class of £-subharmonic function s ([6]) . Suc h an estimate ha s applications outsid e comple x analysis . Fo r instance , w e ca n se e tha t i f a minima l surface o f finite genu s an d wit h finite numbe r o f end s ha s finite projectiv e volume , then i t ha s finite tota l curvature , eve n i f i t i s no t properl y immersed . Acknowledgement The autho r woul d lik e t o than k th e refere e fo r a carefu l readin g o f thi s articl e and helpfu l advice . References [1] A . Ancona , Theori e d u potentie l su r le s graphe s e t le s varietes . Ecol e d'et e d e Probabilitie s de Saint-Flou r XVIII—1 988 , 1 -1 1 2 , Lectur e Note s i n Math. , 1 427 , Springer , Berlin , 1 990 . [2] A . Ancona , R . Lyons , Y . Peres , Crossin g estimate s an d convergenc e o f Dirichle t function s along rando m wal k an d diffusio n paths . Ann . Probab . 2 7 (1 999) , no . 2 , 970-989 . [3] M . Arnaudon , X.-M . Li , A . Thalmaier , Manifold-value d martingales , change s o f probabilities , and smoothnes s o f finel y harmoni c maps . Ann . Inst . H . Poincar e Probab . Statist . 3 5 (1 999) , no. 6 , 765-791 . [4] A . Atsuji , Nevanlinn a theor y vi a stochasti c calculus . J . Funct . Anal . 1 3 2 (1 995) , no . 2 , 4 7 3 510. [5] A . Atsuji , A Casorati-Weierstras s theore m fo r holomorphi c map s an d invarian t cr-field s o f homomorphic diffusions . Bull . Sci . Math . 1 2 3 (1 999) , no . 5 , 371 -383 . [6] A . Atsuji , A lemm a o f logarithmi c derivativ e fo r som e (5-subharmoni c functions . Comple x Variables 4 6 (2001 ) , no . 3 , 1 95-206 . [7] A . Atsuji , Parabolicity , divergenc e theore m fo r 1 } b e i.i.d . wit h a p.d.f . £ >~1 f({x — 0)/£) > wher e th e unknow n parameters (0,£ ) G R x R^ . Here , /(• ) i s know n an d continuous , an d it s domai n space ma y depen d o n 0 alone . Let T n = T n(Xi,...,Xn) an d U n — U n{Xi, ...,X n) b e estimator s o f 9 an d £ respectively base d o n sample s (Xi,...,X n). Suppos e tha t T n an d U n satisf y th e following conditions : (a) Fo r an y fixed n ( > 2) , T n i s independen t o f (U2, • -, U n). (b) (i ) For some 7 > 0 , for a measurable function g : R^ —* i? + , the distributio n of ^(Tn — 9)/g{£) doe s no t depen d o n (n,0,£) ; (ii) Th e distributio n o f n 7(Tn — 9)/g(U n) doe s no t depen d o n (0 , £). Given preassigne d widt h 2d > 0 an d confidenc e coefficien t 1 — a > 0 , le t u s construct a confidenc e interva l fo r 0 by usin g Stein' s two-stag e procedure . Havin g recorded Xi, ...,X m o f a suitabl e siz e m ( > 2) , calculat e U m an d defin e (i)

N = raaxjm, \{b mg(Um)/d)1 ^} +

1},

STATISTICAL I N F E R E N C E I N T W O - S T A G1 E SAMPLIN G 2

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where b m > 0 is a desig n constan t and , her e an d throughout , [u] wil l stan d fo r th e greatest intege r les s tha n u. Next , tak e a n additiona l sampl e X m +i, ...,X/ v o f siz e N—m accordingl y and calculat e T/v based on all the samples. Then , the fixed-width confidence interva l for 6 is finally obtained a s [T/ v — d, T/v + d], keepin g in mind tha t the lowe r o r uppe r confidenc e limi t wil l b e modifie d suitabl y i f th e domai n spac e depends o n 0. Le t u s verif y satisfactio n o f the probabilit y requirement . Notin g th e stopping rul e (1 ) an d assumption s (a)-(b) , w e hav e P{\TN -0\ P{w?\T m - 9\/g(0 < b

mg(Um)/g(0)-

So, th e probabilit y requiremen t i s satisfied b y designin g b m a s a constan t suc h tha t P{m?\Tm - e\/g{U m) /i),

where /(• ) stand s fo r th e indicato r functio n o f (•) . Choos e 0 = /i , £ = a, an d fo r n > 4 le t T n = min(Xi , ...,X n ), U n = (n — 1 ) _ 1 Y^i=i(Xi — T n ). I n thi s case , se t 7 = 1 an d g(x) — x ( > 0) . Then , assumption s (a)-(b ) ar e satisfied . Exchang e d in th e stoppin g rul e (1 ) wit h 2d an d defin e th e fixed-width confidenc e interva l fo r \i b y [T/ v — 2c? , T/v] . Tha t interva l ha s confidenc e 1 — a whe n b m i s designe d a s the uppe r a poin t o f th e T-distributio n wit h (2 , 2(r a — 1 ) ) d.f.s , whil e th e ris k o f T/v i s bounded abov e b y Ad 2 when b 1 7l i s designed a s the positiv e squar e roo t o f th e second orde r momen t fo r th e sam e F-distribution . Several estimatio n problem s fo r parameter s suc h a s th e locatio n paramete r o f a negativ e exponentia l distribution , th e scal e paramete r o f a Paret o distribution , and th e mea n o f a n invers e Gaussia n distributio n hav e th e sam e natur e a s fo r the mea n o f a norma l distributio n i n th e sens e o f th e essenc e o f Stein' s two-stag e procedure. Therefore , i t i s possibl e t o appl y th e methodologie s i n thi s articl e t o these problems . Se e Ghury e [G58 ] an d Mukhopadhya y [M82 ] fo r th e details . I n

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M A K O T O AOSHIM A

addition, th e estimatio n proble m fo r regressio n parameter s o f a linea r regressio n model, an d th e rankin g an d selectio n proble m o f population s ca n b e considere d i n a simila r context . Even i f th e essenc e o f th e two-stag e procedur e i n th e one-dimensiona l cas e of a populatio n distributio n remain s i n th e multi-dimensiona l case , still , handlin g the statisti c tha t appear s i n th e multi-dimensiona l cas e become s complicated . Fo r problems o f processin g severa l population s simultaneously , i t become s eve n mor e complicated. Th e efficienc y o f inferenc e reall y depend s o n ho w yo u handl e proper ties o f th e underlyin g distributio n i n methodologie s fo r a n individua l problem . I n all problems , th e two-stag e procedur e aim s a t guaranteein g a requiremen t abou t a ris k i n authenticit y (instea d o f approximation) . Wit h tha t i n mind , w e pursu e an optima l wa y t o reduc e th e sampl e siz e require d i n inference . I n Section s 3-4 , a multivariate norma l distributio n i s take n a s a populatio n distribution , s o tha t w e are no t bothere d b y specifi c detail s peculia r t o distribution . W e shall solv e a n indi vidual inferenc e proble m b y usin g a two-stag e procedure . Ther e ar e severa l case s when assumption s (a)-(b ) ar e no t satisfied . I n Sectio n 5 , abou t inferenc e problem s for suc h cases , w e shal l discus s robustnes s o f a two-stag e procedure , alon g wit h several othe r technique s t o handl e suc h cases . 3. A multivariat e norma l populatio n Let {Xi\ i > 1 } be a sequenc e o f i.i.d . rando m vector s wit h p-variat e norma l distribution A p (/i, E), wher e 5 ] > 0 . Le t u s conside r th e followin g tw o inferenc e problems abou t th e mea n vecto r /i . (PI) Fo r a prespecifie d constan t e > 0 , fin d a n estimato r 6 n = 6 n(Xi, ...,X n ) for / i suc h tha t E(\\S n — /i|| 2 ) < e fo r an y (/i , E). Her e an d throughout , th e nor m is th e Euclidia n nor m an d ||X|| 2 = X'X, wher e X' stand s fo r th e transpos e o f a vector X. (P2) Fo r a valu e d > 0 o f th e radiu s an d a confidenc e coefficien t 1 — a > 0 , both give n beforehand, determin e th e sampl e siz e n suc h tha t th e confidenc e regio n CR(5n) — {/i| ||/ i — 8 n\\ < d} fo r / i satisfie s P(/ x G CR(S n)) > 1 — a fo r an y (/i,E). Having recorde d X i , . . . , X n o f size n, i t i s natural t o conside r th e sampl e mea n Xn a s a n estimato r o f /i . Whe n E i s known , w e hav e E(\\Xn-fi\\2) =

trC£)/n,

so that th e sampl e siz e n neede d t o solv e (PI ) i s determined a s the smalles t intege r such tha t n > £r(E)/e . A s fo r (P2) , le t a b e th e uppe r a poin t o f th e chi-square d distribution function , F p (-), withp d.f. , i.e. , F p(a) = 1 — a. Conside r th e confidenc e region CR(X n). Whe n E i s known , th e sampl e siz e n neede d t o solv e (P2 ) i s determined a s th e smalles t intege r suc h tha t n > a\/d 2, wher e A denotes th e maximum laten t roo t o f E . However , i f E i s unknown , th e sampl e siz e n shoul d be a functio n o f a sampl e throug h a n estimat e o f E . T o handl e thi s process , th e two-stage procedur e come s int o consideration . A s fo r (PI) , i t i s solve d a s follows . [Two-stage procedur e (PI) ] (Tl) First , tak e a pilo t sampl e X i , . . . , X m o f size m ( > 4 ) an d calculat e SV n = (m — 1 ) _ 1 Y^iL\{Xi — X rn)(Xi — X mY a s a n estimat e o f E . Defin e th e stoppin g rule (2) A

f = ma x (m, [c

rntr(Srn)/e]

+

1 } , wher e c m = (m — l)/(r a — 3) .

STATISTICAL I N F E R E N C E I N T W O - S T A G1 E SAMPLIN G 2

9

(T2) Next , tak e an additional sampl e X m + i , . . . , XN o f size N — m. B y combining the initial sample and the additional sample, estimate (j, by XN = N"1 X^= i -^iWhen p = 1 , Birnbau m an d Heal y [BH60 ] showe d tha t th e abov e two-stag e procedure solve s proble m (PI) , an d i t wa s extende d t o th e cas e whe n p > 2 by Kubokawa [K90] . (Refe r t o Ra o [Ra73 , pp . 486-487 ] a s well. ) A s for a choic e of the siz e ra, Cohe n an d Sackrowit z [CS84ab ] studie d th e Baye s decisio n rul e wit h respect t o suitable prio r distribution s o f the parameters . When w e assum e tha t w e hav e a know n rea l numbe r s* suc h tha t tr(Xl ) > s+(> 0) for any XI (> 0) , Aoshima [A00 ] showed that procedur e (PI ) is second-order efficient unde r th e condition tha t m = m(e) an d lim £ ^ 0 ^^ = s* > by developing the techniques give n b y Mukhopadhyay an d Dugga n [MD97 , 99] . This mean s tha t th e average sampl e siz e and the associated ris k ca n be expanded u p to the second orde r when e ^ 0. I t woul d b e interestin g i f a reductio n o f the sampl e size , require d i n (PI), i s considered wit h th e help o f improvements o n estimators togethe r wit h th e stopping rul e b y incorporating prio r informatio n abou t nuisanc e parameters . For th e specfic cas e T, = cr 2Ip, Ghos h an d Sen [GS83 ] showe d tha t th e James Stein typ e estimato r

6N=XN-^Zp^ 2AXN, a% iV l l J v | | i= 1

=J2l\X t-XN\\2/{p(N -I) + 2}

dominates XN whe n p > 3 . A s fo r a purel y sequentia l procedure , Ghos h e t al . [GNS87] showe d a simila r ris k dominanc e result . Nataraja n an d Strawderma n [NS85] an d Kubokaw a an d Sale h [KS94 ] studie d th e improvement s o f the stoppin g rule togethe r wit h th e estimator suc h tha t i n the shrinkage procedure s th e sampl e size is less than o r equal t o N an d the shrinkage estimato r i s asymptotically bette r than XJ V whe n 5 ] = 3. A s for th e cas e o f arbitrar y X , th e James Stein typ e estimato r ha s not bee n developed . When certai n prio r distribution s ar e assume d fo r th e parameters , th e Baye s sequential estimatio n i s pursued t o minimiz e th e Baye s ris k ove r al l stopping rule s and ove r al l estimators. Arro w e t al . [ABG49 ] showe d tha t i n the Bayes sequentia l estimation problem , th e optima l estimato r fo r square d los s i s give n b y th e Baye s solution for any stopping rule . I t can be shown tha t a n optimal stoppin g rul e exists; however it is given by the method o f backward inductio n an d it is often inaccessible . To overcom e thi s difficulty , Bicke l an d Yahav [BY68 ] devise d th e APO (asymptot ically pointwis e optimal ) rule , whic h derive s a n explicit stoppin g rul e whos e Baye s risk typically i s close to the Bayes risk of the optimal rule . Whe n a conjugate distri bution i s taken a s a prio r distribution , th e APO rule ha s a ris k o f the same exten t as the Bayes stoppin g rul e asymptoticall y a s the observation cos t approache s zero . When p = 1 , Woodroofe [W81 ] showed tha t certai n stoppin g rule s ar e asymptoti cally non-deficien t a s the observatio n cos t approache s zero : Tha t verificatio n wa s extended t o th e multivariat e cas e b y Naga o [N97ab] . Whe n th e prio r distributio n is no t completel y know n bu t auxiliar y dat a ar e availabl e fo r estimatin g unknow n parameters o f the prio r distribution , Martinse k [M87 ] considered a genera l empir ical Baye s approac h whic h approximate s th e optima l rule . O n the othe r hand , t o overcome the difficulty, Alv o [Alv77 , 78], Akahira an d Koike [AkK96] , Koike [Ko99 ] and other s considere d a heuristi c approac h givin g stoppin g rule s fo r whic h th e excess ris k incurre d ove r th e optimu m Baye s ris k i s bounded an d possibly evaluate d explicitly fro m th e prio r distribution . Furthe r evolutions , suc h a s improvement s

130

MAKOTO AOSHIM A

of th e inequalit y s o that a certai n distributio n famil y achievin g th e boun d exists , could b e anticipated i n this research . As fo r (P2) , it i s solved a s follows . [Two-stage procedur e (P2)] (Tl) First , tak e a pilo t sampl e X i , ...,Xm o f siz e m ( > p) an d calculat e th e maximum laten t roo t ^ m o f S m. Defin e th e stopping rul e (3) N

- ma x {m, [a

2 mem/d ]

+

l} ,

where a m = p(m — 1 ) F p ^ rn - V {a)/(m — p) wit h F r^s(a) th e upper a poin t o f the F distribution havin g d.f. s (r , s). (T2) Next , tak e a n additiona l sampl e X m + i , ...,XN o f size N — m. B y combining th e initia l sampl e an d th e additiona l sample , defin e th e confidenc e regio n CR(XN) wit h X N = N' 1 J2? =i -X"»Healy [He56 ] showe d tha t th e above two-stag e procedur e solve s proble m (P2). Aoshima [AOO ] suggeste d tha t th e stoppin g rul e (3 ) replacin g a m b y th e smalle r constant a ^ = pF P j m _i(a) stil l enables the procedure to solve problem (P2) . Whe n p is large and m is small, the reduction o f N b y this correction i s remarkable. Unde r the assumption s tha t A is simple and there exists a known and positive A * such tha t A > A * for any S ( > 0) , Mukhopadhyay [M99 ] and [AOO ] considered a sequenc e of m = m(d) suc h tha t l i m ^ o mc P — a ^* a n d studie d th e second-order efficienc y o f procedure (P2 ) as d—• 0 . Recal l that a denotes the upper a point of the chi-square d distribution wit h p d.f. A s for a purely sequentia l procedur e fo r proble m (P2) , se e Srivastava [Sriv67] , an d refe r t o Woodroof e [W77 ] for it s second-orde r efficienc y when p = 1 : Fo r a multivariat e case , it s second-orde r efficienc y ha s no t bee n resolved fully , bu t Dmitrienko an d Govindarajulu [DB00 ] proved ris k boundedness . It woul d b e interestin g t o conside r a James-Stei n typ e estimato r improvin g CR(XN) a t leas t whe n 5 ] = 0 i s give n a s u — dm suc h tha t /

(T2) Next , tak e a n additiona l sampl e X m + i , . . . , X^ o f size N — m. B y com bining th e initial sampl e an d the additional sample , calculat e a generalized sampl e mean vecto r X ^ = {tr{T\Y r),..., tr(T pYf))'', wher e Y = [ X i , . . . , X m , X m + i , . . . , Xft] an d Ti = [t^i,... , t^m, t2 m + i , . . . , tift\,1 < i < p, ar e p x N rando m matrice s satisfying th e following thre e conditions : VV tn —

... — tim'i

STATISTICAL I N F E R E N C E I N T W O - S T A G1 E SAMPLIN G 3 1

(ii) Til = e u wher e 1 : T V x 1 = (1 ,... , 1 ) ' an d e { : pxl = (0,... , 0,1, 0,..., 0)'; (hi) TT' = (d 2/dm)Ip ® Sm\ wher e T ^ p 2 x N =jT' ly ...,T fp)f. Then, defin e th e confidenc e regio n CR(X ^) wit h X ^. The conditiona l distributio n o f X^ give n S m i s N p(fjb,(Vd2/arn)Ip). Whe n 2 p = 1 , Stei n [S45 ] showe d tha t ^dm/{pd ){X fj — /i ) ha s Student' s t-distributio n with v — m — 1 d.f. Whe n p = 2 , Chatterje e [C59 ] gav e it s distribution . Whe n p > 3 , th e distributio n o f X ^ i s complicate d t o handl e exactly . I n thi s case , a large sampl e approximatio n coul d b e considered . Hyakutak e an d Siotan i [HS87 ] obtained a n asymptoti c expansio n fo r th e distributio n o f y /drn/(pd2)(X^ — JJL) u p to th e orde r 0(v~ 2) b y usin g th e differentia l operato r method . Then , it s limitin g distribution i s N P(Q,IP). Chatterjee [C60 ] showe d tha t th e abov e two-stag e procedur e als o solve s prob lem (P2 ) wit h coverag e probabilit y exactl y equa l t o 1 — a. Th e stoppin g rul e (4 ) is the leas t intege r meetin g th e necessar y an d sufficien t conditio n t o choos e T sat isfying thre e condition s (i)-(iii ) simultaneously . (Severa l method s o f generatin g T wer e give n b y Hyakutak e [H86 ] an d Dudewic z an d Tanej a [DT87]. ) Chatter jee [C60 ] showe d tha t procedur e (P2) ' i s les s efficien t tha n (P2 ) i n term s o f bot h the sampl e siz e an d th e coverag e probability . Furthermore , fo r an y (/i , X), w e have CR(XJV) C CR{X fj) wit h probability 1 whe n placed a t th e same center. (Aoshim a [A94a] gav e assessment s abou t bot h mea n an d varianc e o f th e sampl e siz e i n pro cedure (P2 ) an d compare d the m numericall y wit h thos e give n b y Hyakutak e an d Siotani [HS89 ] i n (P2)'. ) However , o n th e othe r hand , procedur e (P2) ' ha s th e in teresting propert y tha t U = d rn{pd2)~1 \\Xj^\\2 ha s distributio n dependin g onl y o n ||/x||2 bu t completel y fre e fro m nuisanc e parameter s X) . Chatterje e [C59 ] success fully showe d tha t procedur e (P2) ' yield s a tes t fo r H o : JJL = 0 wit h powe r functio n completely fre e fro m E . (A s for th e distributio n o f U', a n asymptoti c expansio n wa s studied b y Mukaihat a an d Fujikosh i [MF93 ] togethe r wit h it s erro r bound. ) Alber s [Alb92] investigated t o wha t exten t thi s nic e feature o f having a power independen t of certain aspect s o f the unknow n underlyin g distributio n ca n b e generalize d t o th e case o f ran k tests . It i s no t know n whethe r th e tes t give n b y procedur e (P2 ) i s uniforml y mor e powerful tha n th e tes t give n b y (P2) r . Chatterje e [C60 ] constructe d a tes t whic h is uniforml y mor e powerfu l an d slightl y mor e economica l tha n (P2)' , b y replacin g m + p 2 wit h m + p 2 — 1 i n th e stoppin g rul e (4 ) an d b y replacin g (d 2/dm) wit h tr(Sm)/N i n condition (iii ) abou t T . (Not e that fo r A f given by those replacement s such a T ca n alway s b e found. ) I t i s eviden t tha t procedur e (P2) ; wit h th e abov e modification ca n als o yield a confidence regio n slightl y bette r tha n CR(X^). How ever, a s suggeste d i n [C60] , this sligh t improvemen t require s th e sacrific e tha t th e associated powe r alread y depend s o n X) . Se e als o Chatteje e [C91 ] . A s fo r a purel y sequential procedure , refe r t o Li u [Li97c] , who studie d a tes t proble m whe n p = 1 . When som e structur e o f X I is suppose d b y prio r information , severa l improve ments o f the stoppin g rul e ca n b e considered . Fo r instance , le t u s conside r th e cas e that £ ha s a n intraclas s correlatio n mode l such a s T, = a 2{(\ — p)IpJr pll'}, wher e 1 = (1 ,...,1 ) / . Then , a s for th e maximu m laten t roo t o f E, w e have A = max(ri , T2), where T\ = & 2{l + (p — l)p } i s simpl e whil e T 2 = a 2(l — p) i s p — 1 multiple. Not e that rim = P _ 1 l / ^ m l an d f 2m = (p— l ) ~1( t r ( 5 m ) — fim ) ar e unbiased estimate s of T\ an d T 2 respectively. A natua l estimatio n o f A could b e max(fi,T2) . Hyakutak e

132

M A K O T O AOSHIM A

et al . [HTA95 ] considere d a n improvemen t o f procedur e (P2 ) b y modifyin g th e stopping rul e (3 ) a s (5) i

V = max{ra , [a

m

max(fi m ,f 2rn )/ w n n e th e oVs are unknow n an d positiv e scalars . Unde r tha t genera l structur e o f S , Aoshim a and Mukhopadhya y [AM99 ] gav e asymptoti c assessment s u p t o th e secon d orde r when d — • 0 abou t th e averag e sampl e siz e an d th e coverag e probabilit y fo r th e modified procedur e (P2) . Fo r a purel y sequentia l procedure , se e Naga o [N96] . 4. Severa l multivariat e norma l population s Let u s consider estimatin g a linear function o f mean vectors coming from severa l multivariate norma l populations . Fo r eac h i ( 1 < i < fc), le t {Xij\ j > 1 } b e a sequence o f i.i.d . rando m vector s wit h p-variat e norma l distributio n A ^ / x ^ S ^ ) , where S $ > 0 . Fo r (61 , ...,&&) give n i n advanc e fo r th e ai m o f inference , w e defin e £ = X^2= i ^ /V Le t u s conside r th e followin g tw o inferenc e problem s abou t £ . (P3) Fo r a prespecifie d constan t e > 0 , find n = (ni , ...,n^), th e sampl e size s from eac h population , an d a n estimato r S n o f £ suc h tha t i£(|| 0 o f th e radiu s an d a confidenc e coefficien t 1 — a > 0 , both give n beforehand , determin e n = (rii , ...,nfc) suc h tha t th e confidenc e regio n CR(Sn) = {£ | || £ - 1 - a fo r an y (Mt>Ei), l 4 ) an d calculat e Sim = (m — 1 ) _ 1 Yl'jLii-X-ij — Xim)(Xij — X irnY a s an estimate o f S^ fo r eac h i. Define th e stopping rul e o f each populatio n b y k

(6) Ni

\biWtr{S irn)/e \ n\bi\y/tr{Sim) J2

= ma x -

+1

i=l

where c m = (m — l ) / ( m — 3), which i s the same on e as in (2). (T2) Next , tak e a n additiona l sampl e X irn+i,..., Xi^. o f size Ni — m fo r eac h i. Le t N = (JVi,... , Nk). B y combinin g th e initia l sampl e an d th e additiona l sample, calculat e XiN i — N~x Ylj=i -%-ij fo r eac h i. Then , estimat e £ b y T N = Yli=lbiXiNtAoshima an d Takada [AT02 ] showed tha t th e above two-stage procedur e solve s problem (P3 ) and improves the predecessor researc h give n in Ghosh e t al. [GMS97 , Chap. 6 ] in terms o f the sample size . As for (P4) , it i s solved a s follows . [Two-stage procedur e (P4) ] (Tl) First , tak e a pilo t sampl e Xn^.^Xim o f siz e m ( > p) an d calculat e the maximu m laten t roo t £i m o f Sim fc>r each i. Defin e th e stoppin g rul e o f eac h population b y (7)

Nt

+1

m a x < 77i,

where u m > 0 is determined b y solving th e equation s

(k-l)Fp

kR p , m —1

Umm

p(m — 1 )

= 1 - a

when p — 1, 2, and UnrnX

(1 - Fm-i(x)) k-ldFm-i{x) = 1 -a 1 m when p > 3. Here , F r^s(-) an d F r(-) denot e th e c.d.f. s o f th e F-distributio n wit h (r, s) d.f.s . an d chi-squared distributio n wit h r d.f. , respectively . (T2) Next , tak e an additional sampl e X ^ m + i , . . . , Xnsr z o f size Ni — m fo r each i. Let N = (iVi , ...,iVfc). B y combining th e initial sampl e an d the additional sample , calculate Xi = N-'E Nt 1 ^ ^ for eac h i. Then , defin e th e confidenc e regio n wit h T N = E t i biX iNi. Aoshima e t al . [ATS02 ] showe d tha t th e abov e two-stag e procedur e solve s problem (P4) . Proble m (P4 ) had been studie d b y Chapman [Ch50] , Ghosh [Gh75] , Mukhopadhyay an d Liberman [ML89 ] an d many othe r researchers , bu t a n optima l solution ha d not bee n obtaine d eve n asymptotically . A s for specifi c case s o f (P4), when p = 1 and k = 2 , Banerje e [B67 ] gave a n asymptoti c optima l solutio n an d Schwabe [Sc93 ] gav e a n improvemen t o f [B67 ] i n terms o f the sample size . Takad a and Aoshim a [TA96 ] develope d th e Banerjee-Schwabe resul t fo r a multivariate case CR{TK)

134

M A K O T O AOSHIM A

but wit h Ti l = erf Hi (Hi i s a know n an d positiv e definit e matrix) , i = 1 ,2 , an d this wa s followe d b y Takad a an d Aoshim a [TA97 ] fo r th e cas e whe n k > 3 . Fol lowing thes e papers , [ATS02 ] gav e a n asymptoti c optima l solutio n t o proble m (P4 ) in whic h 5 ^ ( 1 < i < k) i s completel y unknown . Fo r second-orde r asymptoti c properties i n thi s context , se e Aoshim a an d Mukhopadhya y [AM02] . Assumin g fo r each i tha t Yti = p) fo r thei r pilo t samples . Sample size s fo r tha t testin g ar e determine d s o tha t a tes t rul e wit h significan t level a > 0 agains t Hi : ^ ^ / x 2 guarantee s powe r (5 whe n \\fi 1 — /JL 2\\ = # o fo r given So > 0 an d / ? > 0 . Fo r constructin g simultaneou s confidenc e intervals , se e Hyakutake an d Siotan i [HS87] , Siotan i [Sio87] , Siotan i e t al . [SHF85 ; Sees . 5.6.3 , 6.4.3] an d Aoshim a [A94b] , and th e reference s give n i n them . When p > 2 , we consider multipl e compariso n experiment s fo r correlate d com ponents (£1 ,... , £p) o f £ = ^2 i=ibifi{. Whe n w e suppos e tha t ther e ar e severa l remedies t o b e compare d wit h eac h other , sa y k = 2 , an d thos e effect s ar e observe d at p point s o f a tim e series , th e use r woul d b e typicall y intereste d i n th e directio n and th e magnitud e o f differences—whic h point s ar e mor e significan t differences , and b y ho w much—wit h respec t t o p correlate d tim e component s (£i,...,£ p ) o f Mi ~ M 2 (fr i = 1 ?^ 2 — ~1)- Fo r multipl e compariso n methods , Tukey' s (1 953 ) method o f al l pairwis e multipl e comparison s (MCA) , Hsu' s (1 984 ) metho d o f mul tiple comparison s wit h th e bes t (MCB) , an d Dunnett' s (1 955 ) metho d o f multipl e comparisons wit h a control (MCC ) ar e well known an d referre d t o by many authors . (See Hochber g an d Tamhan e [HT87] , Hirots u [Hi92] , Hsu [Hs96] , an d Nagat a an d Yoshida [NY97 ] a s relate d textbooks. ) Havin g recorde d Xn, ... 1 Xini fro m eac h i ( 1 < i < &) > we hav e T n = J2i=i biX iriz wit h n = (ni , ...,nfc). Then , th e com ponents (£i,...,£ p ) ar e estimate d b y T n = (T i n , ...,T p n ). Tukey' s MC A method , Hsu's MC B metho d an d Dunnett' s MC C metho d giv e th e followin g simultaneou s confidence interval s whe n d i s specifie d (suitabl y narrow) . (MCA) Fo r th e p(p — l)/ 2 difference s o f componen t effects , SCI(Tn) =

{£ | £ r - & G [Trn - T

sn

- d,T

rn

- T

sn

+ d] , 1 < r < s < p}.

(MCB) Fo r comparin g eac h componen t wit h th e bes t o f the othe r component s when a large r componen t effec t i s suppose d t o b e better ,

SCI(Tn) = {£ | £ r - max£ s e [-(T r n - maxT

sn

- d) _, +(T

r = l,...,p} , where -f-x + = max{0 , x} an d — x~ = min{0,x} .

rn

- maxT

sn

+ d) + ],

STATISTICAL I N F E R E N C E I N T W O - S T A G E S A M P L I N G

135

(MCC) Fo r comparing eac h componen t wit h a control component , SCI(Tn) =

- T m - d , T rn - T

{£\ Zr-Z Pe [Tm

pn

+ d] , r = 1 , ...,p - 1 } ,

where th e component p is the control . Then, w e conside r th e followin g proble m fo r eac h o f (MCA) , (MCB ) an d (MCC). (P5) Fo r a value d > 0 and a confidence coefficien t 1 — a > 0 both give n before hand, determin e n = (ni,...,rifc ) suc h tha t th e simultaneou s confidenc e interval s SCI(Tn) satisf y P( £ e SCI(T n)) > 1 - a fo r any (/i 2, E2 ), 1 < i < k. The coverag e probabilit y o f simultaneous confidenc e interval s depend s o n the variance-covariance structur e onl y throug h th e variance o f the pairwise difference s of th e componen t mea n estimates . W e assume tha t 5] ^ = (cr^ rs) ha s a spherica l structure suc h a s 2 had not been tackle d fully , considerin g interactio n amon g populations . When Ti (1 < i < k) i s known, th e sample size s n neede d t o solv e (P5 ) woul d be determine d a s th e smalles t intege r suc h tha t n $ > z 2\bi\riY2i=i \^i\ ri/d2 fo r each i i f one considers minimizin g th e total sampl e siz e X^= i ni- Here , fo r eac h of (MCA), (MCB ) an d (MCC), z > 0 is determined a s a solution t o a certain equatio n depending o n (p , a). (Se e Aoshima [A01 ] for th e details.) Whe n r ^ ( 1 < i < k) i s unknown, th e sampl e size s n shoul d b e determine d throug h estimate s o f r^ , 1 < i < k. Fo r each o f (MCA) , (MCB ) and (MCC) , proble m (P5 ) is solved a s follows . [Two-stage procedur e (P5)] (Tl) First , tak e a pilo t sampl e Xij — (Xiji, ...,Xij PY, j = l,...,m , o f siz e m ( > 2 ) and calculat e pm

Sim =

v

2i^

Z^(X ljr -

X %j. - X

Lr

+ Xi.)

r=lj = l

with v — (p — l)(r a — 1 ) as an estimate o f rf fo r each i. Here , pm

r = l j=l

and pm

r=l j =l

Define th e stopping rul e o f each populatio n b y (8) Ni

k

2 2 = ma x < m , t n\bt\Sl7n^2\b2\Sirn/d 2=1

+1

136

M A K O T O AOSHIM A

where t m > 0 i s determine d a s a solutio n t o th e followin g equatio n fo r eac h o f (MCA), (MCB ) an d (MCC) : (MCA) kpl

I

\*(x)-$(x-t

myfc/V)l

( 1 - F„{y)) k-1 d$(x)dF

v (y)

1 — a: (MCB) A ^ y

{(*

(MCC) kj j

|$(x+t

+ £ m y ^ ) } {l-F

v{y))

k l

- d{x)dFv{y) =

l-a-

mV /^)-$(^-t mV ^A;)}

x ( 1 - i^fo))*" 1 d*{x)dF v{y) =

1 -a ,

where $(• ) denote s th e standar d norma l distributio n functio n an d F r{-) denote s th e chi-squared distributio n functio n wit h r d.f . (T2) Next , tak e a n additiona l sampl e X ^ m + i , . . . , X^ o f size Ni — m fo r eac h i. Let A T== (iVi, ...,7Vjfc). B y combinin g th e initia l sampl e an d th e additiona l sample , calculate Xi^ % = N~ l ^7= 1 ^ij f° r e a c n * • Then , construc t th e simultaneou s confidence interval s SCI(T^) base d o n th e component s ( T I N , . . . , T P N ) ° f T N = Aoshima [A01 ] showed that th e above two-stage procedure solves problem (P5) , and als o discusse d th e cas e whe n th e XI^' s d o no t hav e an y specifi c structure . Le t tma, tmb an d t mc denot e th e t m valu e fo r eac h o f (MCA) , (MCB ) an d (MCC) . Then t m^ < t m c < t m a , an d henc e th e sampl e shoul d b e require d fo r eac h multipl e comparisons metho d i n suc h order . Th e reaso n wh y (MCB ) require s th e smalles t sample siz e i s because fewes t confidenc e interval s ar e require d t o b e simultaneousl y correct. Not e tha t (MCB ) implie s th e inferenc e o f bot h th e indifference-zon e an d the subse t selectio n methodologie s fo r rankin g an d selectio n o f the bes t componen t with a specifie d probabilit y o f correc t selection . Se e Hs u [Hs96 ] fo r th e details . A special case of (MCB ) i n this context wa s given by Hyakutake [H00 ] under intraclas s correlation model s whe n k = 2 . Fo r th e second-orde r analysi s relate d t o procedur e (P5), se e Aoshim a an d Takad a [ATOO ] and Aoshim a an d Miyajim a [AMiOl] . For other relate d topic s to which two-stage procedure s ar e applicable, Aoshim a and Mukhopadhya y [AM98 ] considered Scheffe-typ e simultaneou s confidenc e inter vals fo r a doubl e linea r combinatio n o f M = [/LX 1 , ..., jik\. Hyakutak e [H92 ] an d Aoshima, Aok i an d Ka i [AAK04 ] considere d a n applicatio n t o selec t a mos t prefer able component o f the Marshall-Olki n [M067 ] multivariat e exponentia l distributio n in lif e testing . Aoshim a an d Che n [AC99 ] an d Aoshima , Che n an d Panchapakesa n [ACP03] considere d a n applicatio n i n a vote r preferenc e proble m whe n ther e i s a nuisance cel l i n whic h Dirichle t integrals , ofte n use d i n invers e sampling , d o no t work. W e note tha t th e two-stag e procedur e i s simpler an d faste r tha n a sequentia l procedure i n makin g a surve y o f voter s i n whic h som e voter s chang e thei r mind s quickly afte r the y se e new s o n T V an d th e Interne t an d i t i s essentia l t o mak e th e survey a t on e tim e poin t t o a s man y voter s a s possible . 5. Beyon d th e condition s o f th e two—stag e procedur e It i s natura l t o enquir e whethe r departur e fro m normalit y ha s an y advers e ef fect o n th e performanc e o f th e two-stag e procedure . However , th e robustnes s o f the two-stag e procedur e ha d no t bee n studie d full y s o far . Whe n condition s (a )

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7

and (b) , give n i n Section 2 , are not satisfied, th e distribution o f a statistic i s quite complicated an d even a n asymptoti c expansio n o f the distribution i s difficul t t o derive. Som e robustnes s studie s ha d been mad e b y simulation fo r various type s of departure fro m normalit y (se e Ramkaran [Ra83 ] an d its references) o r had bee n done o n some certai n specifi c model s (se e Blumentha l an d Govindarajulu [BG77 ] and Aoshim a an d Kano [AK97 ] fo r instance). T o start thi s section , w e shall theo retically attemp t t o show th e robustnes s o f the two-stag e procedur e alon g th e line s of Aoshim a an d Wakaki [AW01 ] . Let X , {Xi] i > 1 } be a sequence o f i.i.d. rando m vector s wit h value s i n R p. Let / x = E(X), S = Cov(X) ( > 0) , and let A > A 2 > .. . > A p be the laten t roots o f S. Le t Xj denot e th e j - th elemen t o f X, an d [iiY...ir th e moment o f X defined b y /x^...^ = E(Xi 1 • • • Xir). Similarly , th e corresponding cumulan t o f X is denote d b y K^...^. Le t Kr b e the r-th order tenso r whos e (ii,... , ir)-element i s K,i1...ir. Havin g recorde d X i , . . . , X n o f size n , /x is estimated b y the sample mea n Xn. Le t us consider th e following inferenc e proble m abou t \i. (P6) Fo r a valu e d > 0 an d a confidenc e coefficien t 1 — a > 0 bot h give n beforehand, determin e th e sampl e siz e n such tha t th e ellipsoida l confidenc e regio n ECR(Xn) — {/x| n(/x — X nyT, (/ x — X n) < c] for some c > 0 satisfies P(/ x G ECR(Xn)) > 1 — a'fo r an y (^x, S) wit h th e maximum diamete r < 2d. When the population distribution i s -/Vp(/x, £ ), w e may use procedure (P2 ) give n in Section 3 and defin e the region by ECR{X^) = {/x | N(IJL — XN)'S^ (ii — XN) < a m } , wher e S m, a m an d N ar e given i n (Tl) of (P2) an d XN i s given i n (T2) o f (P2). Then , Heal y [He56 ] showe d tha t procedur e (P2 ) give s a solution t o problem (P6) wit h P(/L X G ECR(X N)) = 1 - a fo r an y (/x,D) . I t i s eviden t tha t th e maximum diamete r o f ECR(XN) < 2d in view of the stoppin g rul e (3) . Not e tha t the distributio n o f Tjy- = N(X^ — ii) fS^{Xjsf — /x ) i s coincident wit h Hotelling' s T 2 distribution , an d hence th e constant c is given b y a m , whic h i s the upper a point o f Hotelling's T 2 distribution . It i s interesting to investigate how robust th e above solution is against departur e from normality , tha t is , whethe r th e regio n ECR(XN) give n by am guarantee s its confidence coefficien t t o be the require d 1 — a. I t is difficult t o give the distributio n of Tjy- without th e assumptio n o f normality i n a form tha t i s easy to handle theoret ically. W e note tha t th e distribution o f Hotelling's T 2 statisti c unde r nonnormalit y is quit e difficul t t o deriv e eve n i n a single-stag e samplin g scheme , fo r whic h it s asymptotic expansio n for m wa s give n by Kano [Ka95 ] and Fujikoshi [Fu97] . A s for the two-stag e samplin g scheme , a differen t approac h fro m thos e forme r studie s is required fo r derivatio n o f an asymptoti c expansio n o f the distribution. Aoshim a and Wakak i [AW01 ] gave an asymptotic expansio n fo r the distributio n o f Tjy unde r nonnormality whe n m — > o o and investigated th e robustness o f the solutio n give n by procedur e (P2) . W e conside r th e following assumptions : (AO) Cramer' s conditio n fo r the join t distributio n o f ( X, XX') holds . (Al) Th e maximu m laten t roo t A of E i s simple. (A2) £ ( | | X | | 8 + r ) < ( X ) , r > 0 . (A3) li m md 2 = C o for some constant CQ G (0, aA), where a denotes th e uppe r ra—•oo

a poin t o f the chi-square d distributio n wit h p d.f.

138

M A K O T O AOSHIM A

Then, th e distributio n o f Tjy - is given i n a n asymptoti c expansio n a s follows : 3

P (T 2N

V

2>

= X^X!* 1 1 ^'**' = l fe=l V lj = l

and rjf^ — Y2^=i Kiijj- (See Aoshim a an d Wakak i [AW01 ] fo r th e details. ) When th e populatio n distributio n i s symmetric , (3% = 0 . Whe n th e populatio n distribution i s a norma l distribution , th e abov e formul a i s coinciden t wit h a n as ymptotic expansio n o f Hotelling' s T 2 distribution . Th e averag e o f Tj y depend s on bot h th e third - an d fourth-orde r cumulant s i n th e orde r o f m~ 1 , whil e i n a single-stage samplin g schem e i t depend s onl y o n th e third-orde r cumulant . Thi s reflects o n variou s simulatio n studie s conducte d b y Blumentha l an d Govindarajul u [BG77], Ramkara n [Ra83 ] an d other s whe n p = 1 . Puttin g x — am i n th e abov e formula, on e ca n asymptoticall y evaluat e th e coverag e probabilit y o f th e regio n ECR{XN) give n b y procedur e (P2) . W e observ e tha t i f th e populatio n distri bution i s symmetri c an d longer-taile d tha n th e norma l distribution , th e coverag e probability exceed s 1 — a whe n p < p* fo r som e give n p* ( G (1 ,2) ) dependin g on p. Further , i f on e applie s Cornish-Fishe r expansio n t o th e distributio n suc h a s _1 P{TJ!J < a + 6m ) = 1 — a - f o(ra _ 1 ), w e coul d obtai n th e constan t a + bm~ l as a n optio n t o modif y procedur e (P2 ) wit h a m agains t nonnormality . Afte r tak ing thi s modification , w e observe i n th e situatio n describe d abov e tha t th e averag e sample size required i n procedure (P2 ) become s less than tha t require d i n construct ing th e regio n ECR(X n) = {/x | n(/ x — Xn ) / S ~ 1( / i — Xn) < a} wit h confidenc e _1 1 — a-f - o(?n ) fo r know n X) . I t woul d b e interestin g t o conside r a n improvemen t of the estimat e wit h highe r orde r moment s a s seen i n Uno an d Isoga i [UI00] . A s fo r an improvemen t o f th e stoppin g rule , a monoton e Bartlett-typ e correctio n fo r th e T2 statisti c migh t b e considere d unde r nonnormalit y b y extendin g th e technique s of Fujikosh i [FuOO ] and others . Let u s conside r th e estimatio n o f parameter s othe r tha n th e mean . Fo r in stance, whe n considerin g a bounde d ris k proble m abou t th e varianc e o f a norma l distribution, th e condition s o n independenc e o f estimate s require d fo r th e two stage procedur e ar e no t satisfied . Birnbau m an d Heal y [BH60 ] propose d a differen t two-stage procedur e i n whic h th e ultimat e estimato r i s define d b y usin g onl y a n additional sample . Thi s procedur e yield s a bounde d ris k poin t estimat e no t onl y for th e varianc e o f a norma l distributio n bu t als o fo r th e parameter s o f a Poisso n distribution, a binomia l distributio n an d a hypergeometri c distribution , an d als o for a scale parameter o f the location-scal e famil y o f distributions: I f followed b y th e

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9

use of Tchebychev's inequality , a fixed-width confidenc e interva l of given confidenc e is naturall y produced . Blu m an d Rosenblat t [BR69b ] applie d a simila r techniqu e to obtai n a fixed-width confidenc e interva l o f give n confidenc e fo r th e moment s o f a distributio n wit h increasin g failur e rate . However , th e procedur e give n b y [BH60 ] necessarily cause s inefficiency , s o i t wa s necesssar y t o develo p technique s s o a s t o overcome thi s inconvenience . Fo r estimatio n fo r th e varianc e o f a norma l distribu tion an d fo r th e paramete r o f a unifor m distributio n [7(0 , 6>), Graybill an d Connel l [GC64ab] gav e a techniqu e t o reduc e th e sampl e siz e b y usin g a n inequalit y inher ent t o thos e distribution s instea d o f Tchebychev' s inequality . (Tha t techniqu e wa s applied b y Takad a [T86] , Aoshim a an d Govindarajul u [AG02] , an d others. ) Fo r point estimatio n wit h bounde d ris k fo r a paramete r o f the scal e famil y o f distribu tions, Kubokaw a [K89 ] propose d a techniqu e t o improv e th e ultimat e estimato r b y combining wit h a n initia l sample . Later , Kubokaw a [K90 ] applie d thi s techniqu e t o point estimatio n wit h bounde d ris k fo r th e generalize d varianc e |E | o f N p(fi, £ ) . I n addition, especiall y wit h fixed-width confidenc e interval s fo r th e variance , Sproul e [Sp74] applie d a n appropriat e larg e sampl e theor y o f /7-statistic s t o th e sampl e variance i n a purel y sequentia l samplin g scheme . Se e Ghos h e t al . [GMS97 ] fo r related references . In thi s article , severa l topic s hav e bee n omitte d fo r brevity , especiall y abou t other los s function s suc h a s thos e base d o n a n asymmetri c los s o r a square d los s modified a t th e boundar y o f th e paramete r space . Th e los s functio n use d i n se quential analysi s doe s no t hav e th e invarian t propert y o f scal e transformation s a s seen i n thi s article . Thi s i s becaus e th e sampl e siz e i s determine d b y pickin g u p information abou t th e populatio n distributio n throug h estimatio n o f th e scal e pa rameter. Ther e i s ver y littl e wor k o n estimatio n fo r nonlinea r function s o f mean s using two-stag e procedures . Recently , Zhen g e t al . [ZSS98ab ] considere d a two stage procedur e fo r th e proble m o f estimatin g th e produc t o f means , whic h arise s in situations o f determining are a base d o n measurements o f length an d widt h i n environmental applications . Ther e ar e othe r problems , suc h a s sequentia l tim e serie s (Sriram [Srir87,01 ] , Le e an d Srira m [LS99] , Galtchou k an d Kone v [GK01 ] , Shio hama an d Taniguch i [ST01 ] , an d others) , sequentia l change-poin t detectio n (La i [La95], Siegmun d an d Venkatrama n [SV95] , Yakir [Y98] , and others) , an d sequen tial densit y estimatio n (Isoga i [1 93 , 99] , Martinse k [Ma92 , 93] , X u an d Martinse k [XM95], Hond a [Hon98] , an d others) , whe n considerin g th e precisio n o f inferenc e beforehand. However , judging fro m th e characte r o f the researc h fields, thes e prob lems shoul d b e handle d wit h sequentia l samplin g scheme s rathe r tha n two-stag e sampling schemes . Lastly , i t woul d b e interestin g i f a necessar y an d sufficien t num ber o f stage s fo r samplin g coul d b e determine d i n som e sense , an d i t migh t b e applied t o compute r simulation s an d engineerin g i n th e nea r future . References [AkK96] Akahira , M . an d Koike , K. , O n som e propertie s o f statistical sequentia l decisio n rule s (i n Japanese), Sugak u 4 8 (1 996) , 1 84-1 95 ; Englis h t r a n s l , Sugak u Exposition s 1 1 (1 998) , 197-213. [Alb92] Albers , W. , Asymptoti c expansion s fo r two-stag e ran k tests , Ann . Inst . Statist . Math. , 44 (1 992) , 335-356 . [Alv77] Alvo , M. , Bayesia n sequentia l estimation , Ann . Statist. , 5 (1 977) , 955-968 . [Alv78] Alvo , M. , Sequentia l estimatio n o f a truncatio n parameter , J . Amer . Statist . Assoc , 7 3 (1978), 404-407 .

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[A94a] Aoshima , M. , Healy's sampl e siz e of two-stage procedur e i n heteroscedastic simultaneou s inference, Commun . Statist.—Theor y Meth. , 2 3 (1 994) , 1 297-1 31 0 . [A94b] Aoshima , M. , I n heteroscedasti c simultaneou s inference , th e Heteroscedasti c Metho d II i s mor e economica l tha n th e origina l HM , Amer . J . Math . Manage . Sci. , 1 4 (1 994) , 167-196. [A98] Aoshima , M. , A two-stag e procedur e fo r fixed-size confidenc e regio n whe n covarianc e matrices hav e som e structures , J . Japa n Statist . S o c , 2 8 (1 998) , 59-67 . [A00] Aoshima , M. , Second-orde r propertie s o f improved two-stag e procedur e fo r a multivari ate norma l distribution , Commun . Statist.—Theor y Meth. , 2 9 (2000) , 61 1 -622 . [A01] Aoshima , M. , Sampl e siz e determinatio n fo r multipl e comparison s wit h component s o f a linea r functio n o f mean vectors , Statistics : Reflection s o n the Pas t an d Vision s fo r th e Future: A n Internationa l Conferenc e i n Hono r o f Prof. C.R . Ra o on the Occasio n o f His 80th Birthday , R . Khattre e an d S.D . Peddada , eds. , Commun. Statist.—Theor y Meth. , 30 (2001 ) , 1 773-1 788 . [AAK04] Aoshima , M. , Aoki , M . an d Kai , M. , Two-stag e procedure s fo r selectin g th e bes t com ponent o f a multivariat e exponentia l distribution , Applie d Sequentia l Methodologies , N. Mukhopadhyay , S . Datt a an d S . Chattopadhyay , eds. , Marce l Dekker , Inc. , Ne w York, 2004,1 9-34 . [AC99] Aoshima , M . an d Chen , P. , A two-stag e procedur e fo r selectin g th e larges t multinomia l cell probabilit y whe n nuisanc e cel l i s present , Sequentia l Anal. , 1 8 (1 999) , 1 43-1 55 . [ACP03] Aoshima , M. , Chen , P . an d Panchapakesan , S. , Sequentia l procedure s fo r selectin g th e most probabl e multinomia l cel l whe n a nuisanc e cel l i s present , Commun . Statist. — Theory Meth. , 3 2 (2003) , 893-906 . [AG02] Aoshima , M . an d Govindarajulu , Z. , Fixed-widt h confidenc e interva l fo r a lognorma l mean, Internat . J . Math . Math . Sci. , 29 (2002) , 1 43-1 54 . [AK97] Aoshima , M . an d Kano , Y. , A not e o n robustnes s o f two-stag e procedur e fo r a multi variate compounde d norma l distribution , Sequentia l Anal. , 1 6 (1 997) , 1 75-1 87 . [AMiOl] Aoshima , M . an d Miyajima , H. , Sampl e siz e reductio n o f two-stag e procedure , Kokyuroku 1 224 , RIMS, Kyoto , 2001 , 61-72. [AM98] Aoshima , M . and Mukhopadhyay , N. , Fixed-width simultaneou s confidenc e interval s fo r multinormal mean s i n severa l intraclas s correlatio n models , J . Multiv . Anal. , 6 6 (1 998) , 46-63. [AM99] Aoshima , M . and Mukhopadhyay , N. , Second-order propertie s o f a two-stag e fixed-size confidence regio n whe n th e covarianc e matri x ha s a structure , Statistica l Inferenc e an d Data Analysis- A Meetin g i n Honou r o f Prof . N . Sugiura , Commun . Statist.—Theor y Meth., 2 8 (1 999) , 839-855 . [AM02] Aoshima , M . an d Mukhopadhyay , N. , Two-stage estimatio n o f a linea r functio n o f nor mal mean s wit h second-orde r approximations , Sequentia l Anal. , 2 1 (2002) , 1 09-1 44 . [AT00] Aoshima , M . an d Takada , Y. , Second-orde r propertie s o f a two-stag e procedur e fo r comparing severa l treatment s wit h a control , J . Japa n Statist . S o c , 3 0 (2000) , 27-41 . [AT02] Aoshima , M . an d Takada , Y. , Bounde d ris k poin t estimatio n o f a linea r functio n o f k multinormal mea n vector s whe n covarianc e matrice s ar e unknown , Advance s o n Theo retical an d Methodologica l Aspect s o f Probabilit y an d Statistics , N . Balakrishnan , ed. , Taylor & Francis , Ne w York, 2002 , 279-287 . [ATS02] Aoshima , M. , Takada , Y . an d Srivastava , M.S. , A two-stag e procedur e fo r estimatin g a linear functio n o f k multinorma l mea n vector s whe n covarianc e matrice s ar e unknown , J. Statist . Plan . Inference , 1 0 0 (2002) , 1 09-1 1 9 . [AW01] Aoshima , M . an d Wakaki , H. , Two-stag e procedur e unde r nonnormality , Bull . Interna tional Statist . Inst . 53r d Session , 1 (2001 ) , 99-1 00 . [ABG49] Arrow , K.J. , Blackwell , D . and Girshick , M.A. , Bayes an d minima x solution s o f sequen tial decisio n problems , Econometrica , 1 7 (1 949) , 21 3-244 . [BaS56] Bahadur , R.R . an d Savage , L.J. , Th e nonexistenc e o f certai n statistica l procedure s i n nonparametric problems , Ann . Math . Statist. , 2 7 (1 956) , 1 1 1 5-1 1 22 . [B67] Banerjee , S. , Confidenc e interva l o f preassigne d lengt h fo r th e Behrens-Fishe r problem , Ann. Math . Statist. , 3 8 (1 967) , 1 1 75-1 1 79 . [BhS73] Bhargava , R.P . an d Srivastava , M.S. , On Tukey' s confidenc e interval s fo r th e contrast s in th e mean s o f th e intraclas s correlatio n model , J . Roy . Statist . S o c , B 3 5 (1 973) , 147-152.

STATISTICAL I N F E R E N C E I N T W O - S T A G1 E SAMPLIN G 4 1

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3

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[ZSS98a] Zheng , S. , Seila , A.F . an d Sriram , T.N. , Sequentia l fixed-widt h confidenc e interva l fo r the produc t o f tw o means , Ann . Inst . Statist . Math. , 5 0 (1 998) , 1 1 9-1 45 . [ZSS98b] Zheng , S. , Seila , A.F . an d Sriram , T.N. , Asymptoticall y ris k efficient tw o stage procedur e for estimatin g th e produc t o f k ( > 2 ) means , Statist . Decisions , 1 6 (1 998) , 369-387 . INSTITUTE O F MATHEMATICS , UNIVERSIT Y O F TSUKUBA , IBARAK I 305-8571 , JAPA N

E-mail address: [email protected] p Translated b y MAKOT O AOSHIM A

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Titles i n Thi s Serie s 215 Selecte d Paper s o n Differentia l Equation s an d Analysi s 214 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volume X 213 Ernes t Vinberg , Editor , Li e Group s an d Invarian t Theor y 212 V . M . B u c h s t a b e r an d I . M . Krichever , Editors , Geometry , Topology , an d Mathematical Physic s 211 K . N o m i z u , Editor , Selecte d Paper s o n Analysi s an d Differentia l Equation s 210 S . G . Gindikin , Editor , Li e Group s an d Symmetri c Spaces . I n Memor y o f F . I . Karpelevich 209 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volume I X 208 M . V . Karasev , Editor , Asymptoti c Method s fo r Wav e an d Quantu m Problem s 207 Yu . M . Suhov , Editor , Analyti c Method s i n Applie d Probabilit y 206 M . A . Shubi n an d M . S . Agranovich , Editors , Partia l Differentia l Equation s (Mar k Viskik's Seminar ) 205 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volume VII I 204 K . N o m i z u , Editor , Selecte d Paper s o n Classica l Analysi s 203 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volume VI I 202 V . Turae v an d A . Vershik , Editors , Topology , Ergodi c Theory , Rea l Algebrai c Geometry. Rokhlin' s Memoria l 201 Michae l S e m e n o v - T i a n - S h a n s k y , Editor , L . D . Faddeev' s Semina r o n Mathematica l Physics 200 L . Lerman , G . Polotovskii , an d L . Shilnikov , Editors , Method s o f Qualitativ e Theory o f Differentia l Equation s an d Relate d Topic s 199 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volume V I 198 R . A . Minlos , Seny a S h l o s m a n , an d Yu . M . Suhov , Editors , O n Dobrushin' s Way . From Probabilit y Theor y t o Statistica l Physic s 197 Vladimi r Arnold , M a x i m K o n t s e v i c h , an d A n t o n Zorich , Editors , Pseudoperiodi c Topology 196 Ya . Eliashberg , D . Fuchs , T . R a t i u , an d A . Weinstein , Editors , Norther n California Symplecti c Geometr y Semina r 195 O . V . Belegrade k e t al. , Mode l Theor y an d Application s 194 A . A s t a s h k e v i c h an d S . Tabachnikov , Editors , Differentia l Topolog y Infinite-Dimensional Li e Algebras , an d Application s (D . B . Fuchs ' 60t h Anniversar y Collection) 193 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volume V 192 Le v B e k l e m i s h e v , M a t i P e n t u s , an d Nikola i Vereshchagin , Provability , Complexity, Grammar s 191 A . Yu . Morozo v an d M . A . Olshanetsky , Editors , Mosco w Semina r i n Mathematical Physic s 190 S . Tabachnikov , Editor , Differentia l an d Symplecti c Topolog y o f Knot s an d Curve s 189 V . B u s l a e v , M . Solomyak , an d D . Yafaev , Editors , Differentia l Operator s an d 188 OVolume Spectral . A . Ladyzhenskaya ITheor V y (M . Sh ., Birman' Editor ,s Proceeding 70t h Anniversar s o f th y eCollection St . Petersbur ) g Mathematica l Society .

TITLES I N THI S SERIE S 187 M . V . Karasev , Editor , Coheren t Transform , Quantization , an d Poisso n Geometr y 186 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Geometr y o f Differentia l Equations 185 B . Feigi n an d V . Vassiliev , Editors , Topic s i n Quantu m Group s an d Finite-Typ e Invariants (Mathematic s a t th e Independen t Universit y o f Moscow ) 184 P e t e r K u c h m e n t an d Vladimi r Lin , Editors , Voronez h Winte r Mathematica l School s (Dedicated t o Seli m Krein ) 183 K . N o m i z u , Editor , Selecte d Paper s o n Harmoni c Analysis , Groups , an d Invariant s 182 V . E . Zakharov , Editor , Nonlinea r Wave s an d Wea k Turbulenc e 181 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y 180 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y Theory 179 V . M . B u c h s t a b e r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology : On th e Crossroa d 178 V . Kreinovic h an d G . M i n t s , Editors , Problem s o f Reducin g th e Exhaustiv e Searc h 177 R . L . D o b r u s h i n , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume ) 176 E . V . Shikin , Editor , Som e Question s o f Differentia l Geometr y i n th e Larg e 175 R . L . D o b r u s h i n , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume ) 174 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . N e t s v e t a e v , Editors , Mathematic s in St . Petersbur g 173 V . K h a r l a m o v , A . Korchagin , G . Polotovskii , an d O . Viro , Editors , Topolog y o f Real Algebrai c Varietie s an d Relate d Topic s 172 K . N o m i z u , Editor , Selecte d Paper s o n Numbe r Theor y an d Algebrai c Geometr y 171 L . A . B u n i m o v i c h , B . M . Gurevich , an d Ya . B . P e s i n , Editors , Sinai' s Mosco w Seminar o n Dynamica l System s 170 S . P . N o v i k o v , Editor , Topic s i n Topolog y an d Mathematica l Physic s 169 S . G . Gindiki n an d E . B . V i n b e r g , Editors , Li e Group s an d Li e Algebras : E . B . Dynkin's Semina r 168 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s 167 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l Equations 166 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volume II I 165 Yu . Ilyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 1 6t h Proble m 164 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s 163 L . A . Bokut' , M . Hazewinkel , an d Yu . G . R e s h e t n y a k , Editors , Thir d Siberia n School "Algebr a an d Analysis " 162 S . G . Gindikin , Editor , Applie d Problem s o f Rado n Transfor m 161 K . N o m i z u , Editor , Selecte d Paper s o n Analysis , Probability , an d Statistic s 160 K . N o m i z u , Editor , Selecte d Paper s o n Numbe r Theory , Algebrai c Geometry , an d Differential Geometr y 159 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volume I I For a complet e lis t o f title s i n thi s series , visi t t h e AMS Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .