Extension Theorems of Hahn-Banach Type for Nonlinear Disjointly ...

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352 CONNES AND I,‘.Yterlsiotls ?f approximatel\~ jinile groups bj, solvable groups are appro.rimate[v jhite.
The Theta Transform and the Heisenberg Group RI(.IIAIW TOLIMIEKI 1. INTRODUCTIVE In this paper we continue the program of [7] of establishing a “good” function theory for the Ileisenberg manifold. The point of view hcrc is seemingly very different and has been motivated by the aim of building a function theory which would reprove some of the known results involving automorphisms of the Heisenberg manifold. At the same time, our tools unify under this theory some ideas of Hardy in abelian Fourier analysis regarding the growth of an P-function and its Fourier transform. The main idea we consider is a variant of the complex Fourier transform which we call the theta transform and is more natural from the theory of the Heisenberg manifold. Surprisingly, perhaps, it also appears, as indicated by the Hardy theorem mentioned abol-c and the results on the Hermite functions, to be more natural for some problems in abelian harmonic analysis as well. J,et .Y be the three-dimensional Heisenberg group, and r the discrete subgroup of A. consisting of those matrices with .x, -\v, at Z. The homogeneous space of right cosets r’X is a compact manifold admitting a unique probability measure invariant under right translations rr ~+ CY~, y E 1V. This measure will be assumed throughout. Th c ,pr.c : c~ * I’ 1\T is called the Heisenberg manifo/d. Let LP(~‘I,-V) be the Hilbert space of square integrable functions on f’ -1.. ( qnngilt c 1477 b, .\cadenuc Press, Inc. .\II righrs of reproduction in my form rescrvcd. ISSS 0012-1236 353
Page 1 and 2: Extension Theorems of Hahn-Banach T
Page 3 and 4: EXTENSIOK THEOREMS FOR FUNCTIOK.4LS
Page 5 and 6: EXTENSION THEOREMS FOR FL'NCTIONALS
Page 7 and 8: EXTENSION THEOREMS FOR FLJNCTIONALS
Page 9 and 10: EXTEIiSION THEOREMS FOR FUSCTIONALS
Page 11 and 12: EXTENSION THEOREMS FOR FUNCTIONALS
Page 13 and 14: Hence, by Proposition 4.6, EXTENSIO
Page 17 and 18: ESTENSION THEOREMS FOR FIVVCTIONALS
Page 19 and 20: EXTENSION THEORERIS FOR FUNCTIONALS
Page 21 and 22: EXTENSION TfIEOREMS FOR FUNCTIONALS
Page 23 and 24: EXTENSION THEOREMS FOR FUNCTIONAW 3
Page 27 and 28: EXTENSION THEOREMS FOR FUNC’l’I
Page 29 and 30: l\jow, let f be a function in F,/ ,
Page 33 and 34: KXTEXSION THEOREMS FOR FLWCTIOXA1.S
Page 35 and 36: MEASURE SPACE AI-TOiVIORPHIBRIS 337
Page 37 and 38: YIEASURE SPACE AUTOMORPIIISMS 339 W
Page 39 and 40: C‘hoosc a representative (F,,),,N
Page 41 and 42: MEASURE SPACY AUTOMORPIIIS~I~ 343 (
Page 43 and 44: MEASURE SPACE AYI’O~IORPHIS~ld 34
Page 45 and 46: MEASURE SPACE AUTOMORPHISMS 347 (3.
Page 47 and 48: such that and Pvoqf. We denote On C
Page 49: 4. ~WPLICATIONS \\‘c want to desc
Page 53 and 54: THE THETA TRANSFORRZ 355 The origin
Page 55 and 56: THE THETA TRANSFORM 357 It is conve
Page 57 and 58: THE THETA TRAXSFORM 359 zohew x1 T
Page 59 and 60: TIIE TfIEI‘A ‘lxANSFOK~1 361 Fo
Page 61 and 62: THE THETA TRANSFORhI 363 where J u,
Page 63 and 64: A DUNFORD-PETTIS THEOREM FOR L1/Hal
Page 65 and 66: A DlJNFORD-PETTIS THEOREM FOR L1/Hu
Page 67 and 68: the mapping A DVNFORD-PETTIS THEORE
Page 69 and 70: A DCNFORD-PETTIS THEOREM FOR LIIHu
Page 71 and 72: .t DCNFORD-PETTIS THEOREM FOR L’,
Page 73 and 74: A DUNFORD-PETTIS THEOREM FOR Ll/H=-
Page 75 and 76: Lebesgue’s theorem implies that A
Page 77 and 78: ,omivnr. OF FLW~~IONAI. ANALYSIS 24
Page 79 and 80: REPRESENTATIONS OF NILPOTENT LIE GR
Page 81 and 82: IIEPRESENTATIOKS OF NILPOTENT LIE G
Page 83 and 84: I
Page 87 and 88: REPRESENTATIONS OF SII.PoTEST LIE C
Page 91 and 92: KWKESE~TATIOSS OF NILPOTENT LIE (:I
Page 95 and 96: JOURNAL OF FUNCTIONAL ANALYSIS 24,
Page 97 and 98: TOPOLOGI('AI. ASPECTS 399 maps i,,
Page 99 and 100: where TOPOLOGICAL ASPECTS 401 1,’
Page 101 and 102:
TOPOLOGICAL ASPECTS 403 PROPOSITION
Page 103 and 104:
TOPOLOGICAL ASPECTS 405 (iv) S is b
Page 105 and 106:
TOPOLOGICAL ASPECTS 407 ever!- o-co
Page 107 and 108:
TOPOLOGICAL ASPECTS 409 I,et a = (J
Page 109 and 110:
The polar decomposition of the map
Page 111 and 112:
TOPOLOGICAL ASPECTS 413 Proof. It i
Page 113 and 114:
TOPOLOGICAL ASPECTS 415 algebras a
Page 115 and 116:
TOPOLOGICAL ASPECTS 417 where M, <
Page 117 and 118:
‘I’OPOLOCICAL ASPECTS 419 RIore
Page 119 and 120:
By the same method: TOPOLOGICAL ASP
Page 121 and 122:
TOPOLOGICAL ASPECTS 423 Since f bel
Page 123 and 124:
TOPOLOGICAL ASPECTS 425 20. H. H. S
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Extension Theorems of Hahn-Banach Type for Nonlinear Disjointly ...

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