Harmonic Measur e Geometric an d Analyti c Point s o f View

http://dx.doi.org/10.1090/ulect/035

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University

LECTURE Series

Volume 3 5

Harmonic Measur e Geometric an d Analyti c Point s o f View

Luca Capogn a Carlos E . Keni g

Loredana Lanzan i

American Mathematica l Societ y Providence, Rhod e Islan d

E D I T O R I A L C O M M I T T E E

Jerry L . Bon a (Chair ) Eri c M . Friedlande r Adriano Garsi a Nige l J . Higso n

Peter Landwebe r

2000 Mathematics Subject Classification. Primar y 35-02 , 31-XX , 34A26, 35R35 , 28A75 .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/ulect-35

Library o f Congres s Cataloging-in-Publicatio n Dat a

Capogna, Luca , 1966 -Harmonic measur e : geometri c an d analyti c point s o f vie w / Luc a Capogna , Carlo s E . Kenig ,

Loredana Lanzani . p. cm . - (Universit y lectur e series , ISS N 1047-399 8 ; v. 35 )

Includes bibliographica l references . ISBN 0-8218-2728- 6 (alk . paper ) 1. Potential theor y (Mathematics) . 2 . Differential equations , Partial . 3 . Geometry, Differential .

I. Kenig , Carlo s E. , 1953 - . II . Lanzani , Loredana , 1965 - . III . Title . IV . Universit y lectur e series (Providence , R.I. ) ; 35.

QA404.7.C37 200 5 515/.96-dc22 200504409 5

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Contents

Introduction i x

Chapter 1 . Motivatio n an d statemen t o f th e mai n result s 1 1. Characterizatio n (l) a: Approximatio n wit h plane s 2 2. Characterizatio n (2) a: Introducin g BM O an d VM O 3 3. Multiplicativ e vs . additiv e formulation : Introducin g th e doublin g

condition 3 4. Characterizatio n (l) a an d flatnes s 4 5. Doublin g an d asymptoticall y optimall y doublin g measure s 7 6. Regularit y o f a domai n an d doublin g characte r o f it s harmoni c measur e 8 7. Regularit y o f a domai n an d smoothnes s o f it s Poisso n kerne l 1 0

Chapter 2 . Th e relatio n betwee n potentia l theor y an d geometr y fo r planar domain s 1 3

1. Smoot h domain s 1 4 2. No n smoot h domain s 1 5 3. Preliminarie s t o th e proof s o f Theorems 2. 7 an d 2. 8 2 0 4. Proo f o f Theore m 2. 7 2 5 5. Proo f o f Theorem 2. 8 2 9 6. Note s 3 7

Chapter 3 . Preliminar y result s i n potentia l theor y 3 9 1. Potentia l theor y i n NT A domain s 3 9 2. A brief revie w o f the rea l variabl e theor y o f weight s 4 6 3. Th e space s BM O an d VM O 4 8 4. Potentia l theor y i n C 1 domain s 5 2 5. Note s 5 3

Chapter 4 . Reifenber g flat an d chor d ar c domain s 5 5 1. Geometr y o f Reifenberg fla t domain s 5 5 2. Smal l constan t chor d ar c domain s 6 1 3. Note s 7 1

Chapter 5 . Furthe r result s o n Reifenber g fla t an d chor d ar c domain s 7 3 1. Improve d boundar y regularit y fo r 6— Reifenberg fla t domain s 7 4 2. Approximatio n an d Rellic h identit y 7 7 3. Note s 8 0

Chapter 6 . Fro m th e geometr y o f a domai n t o it s potentia l theor y 8 1 1. Potentia l theor y fo r Reifenber g domain s wit h vanishin g constan t 8 1 2. Potentia l theor y fo r vanishin g chor d ar c domain s 10 0

viii CONTENT S

3. Note s 11 2

Chapter 7 . Fro m potentia l theor y t o th e geometr y o f a domai n 11 3 1. Asymptoticall y optimall y doublin g implie s Reifenber g vanishin g 11 3 2. Bac k t o chor d ar c domains 12 4 3. lo g k G VMO implie s vanishing chor d arc ; Ste p I 12 6 4. lo g k G VMO implie s vanishin g chor d arc ; Ste p I I 13 9 5. Note s 14 6

Chapter 8 . Highe r codimensio n an d furthe r regularit y result s 14 7 1. Note s 15 1

Bibliography 153

Introduction

This book i s based o n a series of five lectures that Carlo s Kenig gave during th e 25th Arkansas Sprin g Lectures Serie s in March 2000 , at th e Universit y o f Arkansas.

In these lectures , Keni g describe d hi s joint wor k wit h Tatian a Tor o concernin g end-point analogue s o f the well-know n potentia l theoreti c resul t o f Kellogg , whic h says that th e density k of the harmonic measur e o f a C1'" domain , ha s logarithm i n Ca; an d of the 'converse' o f this result , the free boundary regularit y theorem of Alt-Caffarelli [2] , which says that unde r (necessary ) mil d hypothesis , i f log k is Ca , the n the domai n mus t b e of class C 1,a. Th e potentia l theoreti c result s ar e extension s o f the classica l functio n theoreti c wor k o f Lavrentiev [53 ] an d Pommerenk e [61] , an d the highe r dimensiona l result s o f Dahlber g [16 ] an d Jerison-Keni g [34] .

The free boundar y results , on the one hand, giv e a geometric measur e theoreti c characterization o f the support set s of measures which are " asymptotically optimally doubling" i n term s o f "flatness" condition s o n th e support , an d exten d th e Alt -Caffarelli highe r dimensiona l versio n [2 ] of th e "converse" resul t o f Pommerenke' s [61], to the end-point VM O case. Thi s type of end-point versio n of the Alt-Caffarell i result wa s first introduce d b y Davi d Jeriso n [32] .

The boo k follow s closel y th e forma t o f th e lectures . I n particular , fo r eac h o f the mai n Theorem s i n Chapte r 6 and i n th e first sectio n o f Chapte r 7 , we presen t a shor t "sketc h o f th e proo f whic h i s a n almos t verbati m cop y o f th e argumen t described i n th e lectures . Thes e brie f sketche s ar e followe d b y detaile d proofs . I n this wa y w e hop e t o communicat e th e mai n idea s an d conve y th e enthusias m an d the intuitiv e insigh t whic h mad e th e lecture s s o lively an d exciting .

We break thi s patter n i n th e proo f o f the las t tw o theorems (Section s tw o an d three i n Chapte r 7) , fo r whic h th e sketc h o f th e proo f alon e i s alread y quit e lon g and technicall y involved . Th e intereste d reade r wil l find detail s fo r thes e theorem s in [45 ] an d [46] . W e hop e tha t ou r presentatio n wil l provid e a "readin g key " t o help navigat e throug h thes e papers .

In orde r t o mak e th e presentatio n mor e self-containe d an d comprehensive , a review o f th e classica l result s fo r plana r domain s ha s bee n adde d i n Chapte r 2 , where conforma l mappin g i s the mai n too l t o approac h th e problems .

ix

x INTRODUCTIO N

Kenig woul d lik e t o than k T . Tor o fo r he r fundamenta l contributio n t o thei r joint wor k an d D . Jeriso n fo r man y conversation s o n th e subjec t throughou t th e years. Keni g woul d als o lik e t o than k Lui s Caffarell i an d Gu y Davi d fo r usefu l discussions, an d G . David fo r hi s role in their joint wor k in the highe r co-dimensio n case o f the geometri c measur e theor y results .

We ar e indebte d t o Joa n Carmona , Christia n Pommerenke , an d Joa n Verder a for discussing with us many of the two-dimensional results . I t i s a pleasure to than k Chaim Goodman-Straus s fo r producin g th e picture s i n th e book , an d Christin e Thiverge at the American Mathematica l Societ y for her assistance with this project .

Last bu t no t least , th e author s wis h to than k th e Nationa l Scienc e Foundatio n and th e Universit y o f Arkansa s fo r sponsorin g th e 200 0 Arkansa s Sprin g Lecture s Series.

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Titles i n Thi s Serie s

35 Luc a Capogna , Carlo s E . Kenig , an d Loredan a Lanzani , Harmoni c measure :

Geometric an d analyti c point s o f view , 200 5

34 E . B . Dynkin , Superdiffusion s an d positiv e solution s o f nonlinea r partia l differentia l

equations, 200 4

33 Kristia n Seip , Interpolatio n an d samplin g i n space s o f analyti c functions , 200 4

32 Pau l B . Larson , Th e stationar y tower : Note s o n a cours e b y W . Hug h Woodin , 200 4

31 Joh n Roe , Lecture s o n coars e geometry , 200 3

30 Anatol e Katok , Combinatoria l construction s i n ergodi c theor y an d dynamics , 200 3

29 Thoma s H . Wolf f (Izabell a Lab a an d Caro l Shubin , editors) , Lecture s o n harmoni c analysis, 200 3

28 Ski p Garibaldi , Alexande r Merkurjev , an d Jean-Pierr e Serre , Cohomologica l

invariants i n Galoi s cohomology , 200 3

27 Sun-Yun g A . Chang , Pau l C . Yang , Karste n Grove , an d Jo n G . Wolfson ,

Conformal, Riemannia n an d Lagrangia n geometry , Th e 200 0 Barret t Lectures , 200 2

26 Susum u Ariki , Representation s o f quantu m algebra s an d combinatoric s o f Youn g

tableaux, 200 2

25 Wil l ia m T . Ros s an d Harol d S . Shapiro , Generalize d analyti c continuation , 200 2

24 Victo r M . Buchstabe r an d Tara s E . Panov , Toru s action s an d thei r application s i n

topology an d combinatorics , 200 2

23 Lui s Barreir a an d Yako v B . Pesin , Lyapuno v exponent s an d smoot h ergodi c theory ,

2002

22 Yve s Meyer , Oscillatin g pattern s i n imag e processin g an d nonlinea r evolutio n equations ,

2001

21 Bojk o Bakalo v an d Alexande r Kirillov , Jr. , Lecture s o n tenso r categorie s an d

modular functors , 200 1

20 Aliso n M . Etheridge , A n introductio n t o superprocesses , 200 0

19 R . A . Minlos , Introductio n t o mathematica l statistica l physics , 200 0

18 Hirak u Nakajima , Lecture s o n Hilber t scheme s o f point s o n surfaces , 199 9

17 Marce l Berger , Riemannia n geometr y durin g th e secon d hal f o f th e twentiet h century ,

2000

16 Harish-Chandra , Admissibl e invarian t distribution s o n reductiv e p-adi c group s (wit h

notes b y Stephe n DeBacke r an d Pau l J . Sally , Jr.) , 199 9

15 Andre w Mathas , Iwahori-Heck e algebra s an d Schu r algebra s o f the symmetri c group , 199 9

14 Lar s Kadison , Ne w example s o f Frobeniu s extensions , 199 9

13 Yako v M . Eliashber g an d Wil l ia m P . Thurston , Confoliations , 199 8

12 I . G . Macdonald , Symmetri c function s an d orthogona l polynomials , 199 8

11 Lar s Garding , Som e point s o f analysi s an d thei r history , 199 7

10 Victo r Kac , Verte x algebra s fo r beginners , Secon d Edition , 199 8

9 S tephe n Gelbart , Lecture s o n th e Arthur-Selber g trac e formula , 199 6

8 Bern d Sturmfels , Grobne r base s an d conve x polytopes , 199 6

7 And y R . Magid , Lecture s o n differentia l Galoi s theory , 199 4

6 Dus a McDuf F an d Dietma r Salamon , J-holomorphi c curve s an d quantu m cohomology ,

1994

5 V . I . Arnold , Topologica l invariant s o f plan e curve s an d caustics , 199 4

4 Davi d M . Goldschmidt , Grou p characters , symmetri c functions , an d th e Heck e algebra ,

1993

3 A . N . Varchenk o an d P . I . Etingof , Wh y th e boundar y o f a roun d dro p become s a curve o f orde r four , 199 2

TITLES I N THI S SERIE S

2 Frit z John , Nonlinea r wav e equations , formatio n o f singularities , 199 0 1 Michae l H . Freedma n an d Fen g Luo , Selecte d application s o f geometr y t o

low-dimensional topology , 198 9