Conference Boar d o f the Mathematica l Science s

CBMS Regional Conference Series in Mathematics

Number 10 0

Calderon-Zygmund Capacities an d Operator s

on Nonhomogeneou s Space s

Alexander Volberg

Published fo r th e Conference Boar d o f the Mathematica l Science s

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by th e American Mathematica l Societ y

Providence, Rhod e Islan d with suppor t fro m th e

National Scienc e Foundatio n

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http://dx.doi.org/10.1090/cbms/100

CBMS Regiona l Researc h Conferenc e

Nonhomogeneous Harmoni c Analysis , Weights , an d

Applications t o Problem s i n Comple x Analysi s an d Operato r Theor y

University o f Nort h Carolin a

May 13-17 , 200 2

Partially supporte d b y th e Nationa l Scienc e Foundatio n

2000 Mathematics Subject Classification. Primar y 42B20 ; Secondary 32A55 , 31A15 , 31C05 .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages /cbms-100

Library o f Congres s Cataloging-in-Publicatio n D a t a

Volberg, Alexander , 1956-Calderon-Zygmund capacitie s an d operators o n nonhomogeneous space s / Alexande r Volberg . p. cm . — (Regiona l conferenc e serie s i n mathematics, ISS N 0160-764 2 ; no. 100) . Includes bibliographica l references . ISBN 0-8218-3252- 2 (alk . paper) 1. Calderon-Zygmun d operator . I . Title . II . Series .

QA1.R33 329 no . 100 [QA329.2] 510 s—dc22 200306299 0 [515/.7246]

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Contents

Chapter 1 . Introductio n 1

Chapter 2 . Preliminarie s o n Capacitie s 7

Chapter 3 . Localizatio n o f Newton an d Ries z Potential s 1 1

3.1. Localizatio n lemma s 1 1 3.2. A building bloc k fo r th e constructio n o f specia l measure s 1 3 3.3. Localizatio n o n specia l cube s 1 3 3.4. Modificatio n o f distribution S. Constructio n o f auxiliar y measure s 1 4 3.5. Ahlfor s ball s 1 5 3.6. Th e principa l estimat e fo r auxiliar y measure s 1 6

Chapter 4 . Fro m Distributio n t o Measure . Carleso n Propert y 2 1

Chapter 5 . Potentia l Neighborhoo d tha t ha s Propertie s (3.13)-(3.14 ) 2 5 5.1. Capacitie s wit h Calderon-Zygmun d (CZ ) kernel s 2 6 5.2. Variationa l capacit y an d extrema l measure s 3 3 5.3. L p theor y o f nonhomogeneous C Z operators . Measur e o f orde r m 4 2 5.4. Ries z an d Cauch y kernels : 7 + x j o p 4 5 5.5. Cauch y kerne l an d analyti c capacit y 4 7

Chapter 6 . Th e Tre e o f the Proo f 5 1

Chapter 7 . Th e Firs t Reductio n t o Nonhomogeneou s Tb Theore m 5 5

Chapter 8 . Th e Secon d Reductio n 6 1

8.1. Suppresse d kernel s 6 1 8.2. Fro m real-value d kerne l t o vecto r value d kerne l 6 7 8.3. Fro m on e lattic e t o tw o lattice s 6 8 8.4. Cor e suppressio n 6 9

Chapter 9 . Th e Thir d Reductio n 7 1

Chapter 10 . Th e Fourt h Reductio n 7 3 10.1. // , 6, D, 77 decompositio n 7 3 10.2. Goo d function s an d ba d function s 7 4 10.3. Estimate s o f nonhomogeneous Calderon-Zygmun d operator s o n goo d

functions 7 6 10.4. Th e reductio n o f Theore m 9. 1 t o estimate s o f nonhomogeneou s

Calderon-Zygmund operator , namel y t o Theore m 10. 6 7 8

iv C O N T E N T S

Chapter 11 . Th e Proo f o f Nonhomogeneou s Cotlar' s Lemma . Arbitrar y Measure 8 3

Chapter 12 . Startin g the Proo f of Nonhomogeneous Nonaccretive Tb Theorem 9 3 12.1. Termina l an d transi t cube s 9 4 12.2. Projection s A and AQ 9 6

Chapter 13 . Nex t Ste p i n Theore m 10.6 . Goo d an d Ba d Function s 10 1 13.1. Goo d function s an d ba d function s agai n 10 1 13.2. Reductio n t o estimate s o n goo d function s 10 2 13.3. Splittin g (T(p good,ipgood) to thre e sum s 10 3 13.4. Thre e type s o f estimates o f f fc(x, y)f{x)g{y) d/j,(x) dfj,(y) 10 4 13.5. Estimat e o f long rang e interactio n su m oi 10 6 13.6. Shor t rang e interactio n su m 0-3 . Nonhomogeneous paraproduct s 10 9

Chapter 14 . Estimat e o f the Diagona l Sum . Remainde r i n Theore m 3. 3 12 1 14.1. Estimat e o f S i e r m 12 3 14.2. Estimat e o f E £r 12 6

Chapter 15 . Tw o Weigh t Estimat e fo r th e Hilber t Transform . Preliminarie s 12 7

Chapter 16 . Necessit y i n th e Mai n Theore m 13 3

Chapter 17 . Tw o Weigh t Hilber t Transform . Toward s th e Mai n Theore m 13 5 17.1. Ba d an d goo d part s o f / an d g 13 6

17.2. Estimate s o n goo d function s 13 7

Chapter 18 . Lon g Rang e Interactio n 13 9

Chapter 19 . Th e Res t o f the Lon g Rang e Interactio n 14 3

Chapter 20 . Th e Shor t Rang e Interactio n 14 5 20.1. Th e estimat e o f neighbor-terms 14 5 20.2. Th e estimat e o f stoppin g term s 14 5 20.3. Th e choic e o f stoppin g interval s 14 7

Chapter 21 . Difficul t Term s an d Severa l Paraproduct s 15 3 21.1. Firs t paraproduc t 15 4 21.2. Tw o mor e paraproduct s 15 6 21.3. Secon d paraproduct : miraculou s improvemen t o f th e Carleso n

property 15 9

Chapter 22 . Two-Weigh t Hilber t Transfor m an d Maxima l Operato r 16 1 22.1. Doublin g 16 1 22.2. N o doubling 16 3

Bibliography 165

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Titles i n Thi s Serie s

100 Alexande r Volberg , Calderon-Zygmun d capacitie s an d operator s o n nonhomogeneou s

spaces, 200 3

99 Alai n Lascoux , Symmetri c function s an d combinatoria l operator s o n polynomials , 200 3

98 Alexande r Varchenko , Specia l functions , K Z typ e equations , an d representatio n theory ,

2003

97 Bern d Sturmfels , Solvin g system s o f polynomia l equations , 200 2

96 Nik y Kamran , Selecte d topic s i n th e geometrica l stud y o f differentia l equations , 200 2

95 Benjami n Weiss , Singl e orbi t dynamics , 200 0

94 Davi d J . Sal t man, Lecture s o n divisio n algebras , 199 9

93 Gor o Shimura , Eule r product s an d Eisenstei n series , 199 7

92 Fa n R . K . Chung , Spectra l grap h theory , 199 7 91 J . P . Ma y e t a l . , Equivarian t homotop y an d cohomolog y theory , dedicate d t o th e

memory o f Rober t J . Piacenza , 199 6

90 Joh n R o e , Inde x theory , coars e geometry , an d topolog y o f manifolds , 199 6

89 Cliffor d Henr y Taubes , Metrics , connection s an d gluin g theorems , 199 6

88 Crai g Huneke , Tigh t closur e an d it s applications , 199 6

87 Joh n Eri k Fornaess , Dynamic s i n severa l comple x variables , 199 6

86 Sori n Popa , Classificatio n o f subfactor s an d thei r endomorphisms , 199 5

85 Michi o J imb o an d Tetsuj i Miwa , Algebrai c analysi s o f solvabl e lattic e models , 199 4

84 Hug h L . Montgomery , Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y an d harmonic analysis , 199 4

83 Carlo s E . Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y valu e

problems, 199 4

82 Susa n Montgomery , Hop f algebra s an d thei r action s o n rings , 199 3

81 Steve n G . Krantz , Geometri c analysi s an d functio n spaces , 199 3

80 Vaugha n F . R . Jones , Subfactor s an d knots , 199 1

79 Michae l Frazier , Bjor n Jawerth , an d Guid o Weiss , Littlewood-Pale y theor y an d th e

study o f functio n spaces , 199 1

78 Edwar d Formanek , Th e polynomia l identitie s an d variant s o f n x n matrices , 199 1

77 Michae l Christ , Lecture s o n singula r integra l operators , 199 0

76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 199 0

75 F . Thoma s Farrel l an d L . Edwi n Jones , Classica l aspherica l manifolds , 199 0

74 Lawrenc e C . Evans , Wea k convergenc e method s fo r nonlinea r partia l differentia l

equations, 199 0

73 Walte r A . Strauss , Nonlinea r wav e equations , 198 9

72 Pete r Orlik , Introductio n t o arrangements , 198 9 71 Harr y D y m , J contractiv e matri x functions , reproducin g kerne l Hilber t space s an d

interpolation, 198 9 70 Richar d F . Gundy , Som e topic s i n probabilit y an d analysis , 198 9

69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Stein , Invarian t theor y an d

superalgebras, 198 7

68 J . Wil l ia m Hel ton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer ,

Operator theory , analyti c functions , matrices , an d electrica l engineering , 198 7

67 Haral d Upmeier , Jorda n algebra s i n analysis , operato r theory , an d quantu m mechanics ,

1987

66 G . Andrews , g-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory , combinatorics, physic s an d compute r algebra , 198 6

TITLES I N THI S SERIE S

65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o differential equations , 198 6

64 Donal d S . Passman , Grou p rings , crosse d product s an d Galoi s theory , 198 6

63 Walte r Rudin , Ne w construction s o f function s holomorphi c i n th e uni t bal l o f C n , 198 6

62 Bel a Bollobas , Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 198 6

61 Mogen s Flensted-Jensen , Analysi s o n non-Riemannia n symmetri c spaces , 198 6

60 Gille s Pisier , Factorizatio n o f linea r operator s an d geometr y o f Banac h spaces , 198 6

59 Roge r How e an d Alle n Moy , Harish-Chandr a homomorphism s fo r p-adi c groups , 198 5

58 H . Blain e Lawson , Jr. , Th e theor y o f gaug e field s i n fou r dimensions , 198 5

57 Jerr y L . Kazdan , Prescribin g th e curvatur e o f a Riemannia n manifold , 198 5

56 Har i Bercovici , Cipria n Foia§ , an d Car l Pearcy , Dua l algebra s wit h application s t o

invariant subspace s an d dilatio n theory , 198 5

55 Wil l ia m Arveson , Te n lecture s o n operato r algebras , 198 4

54 Wil l ia m Fulton , Introductio n t o intersectio n theor y i n algebrai c geometry , 198 4

53 Wi lhe l m Klingenberg , Close d geodesie s o n Riemannia n manifolds , 198 3

52 Ts i t -Yue n Lam , Orderings , valuation s an d quadrati c forms , 198 3

51 Masamich i Takesaki , Structur e o f factor s an d automorphis m groups , 198 3

50 Jame s Eell s an d Lu c Lemaire , Selecte d topic s i n harmoni c maps , 198 3

49 Joh n M . Franks , Homolog y an d dynamica l systems , 198 2

48 W . Stephe n Wilson , Brown-Peterso n homology : a n introductio n an d sampler , 198 2

47 Jac k K . Hale , Topic s i n dynami c bifurcatio n theory , 198 1

46 Edwar d G . Effros , Dimension s an d C*-algebras , 198 1

45 Ronal d L . Graham , Rudiment s o f Ramse y theory , 198 1

44 Phil l i p A . Griffiths , A n introductio n t o th e theor y o f specia l divisor s o n algebrai c curves ,

1980

43 Wil l ia m Jaco , Lecture s o n three-manifol d topology , 198 0

42 Jea n Dieudonne , Specia l function s an d linea r representation s o f Li e groups , 198 0

41 D . J . N e w m a n , Approximatio n wit h rationa l functions , 197 9

40 Jea n Mawhin , Topologica l degre e method s i n nonlinea r boundar y valu e problems , 197 9

39 Georg e Lusztig , Representation s o f finit e Chevalle y groups , 197 8

38 Charle s Conley , Isolate d invarian t set s an d th e Mors e index , 197 8

37 Masayosh i Nagata , Polynomia l ring s an d affln e spaces , 197 8

36 Car l M . Pearcy , Som e recen t development s i n operato r theory , 197 8

35 R . Bowen , O n Axio m A diffeomorphisms , 197 8

34 L . Auslander , Lectur e note s o n nil-thet a functions , 197 7

33 G . Glauberman , Factorization s i n loca l subgroup s o f finit e groups , 197 7

32 W . M . Schmidt , Smal l fractiona l part s o f polynomials , 197 7

31 R . R . Coifma n an d G . Weiss , Transferenc e method s i n analysis , 197 7

30 A . Pelczyriski , Banac h space s o f analyti c function s an d absolutel y summin g operators ,

1977

29 A . Weinste in , Lecture s o n symplecti c manifolds , 197 7

28 T . A . Chapman , Lecture s o n Hilber t cub e manifolds , 197 6

27 H . Blain e Lawson , Jr. , Th e quantitativ e theor y o f foliations , 197 7

For a complet e lis t o f t i t le s i n thi s series , visi t t h e AMS Bookstor e a t www.ams.org/bookstore/ .