Gilbert J., Murray M. — Clifford Algebras and Dirac Operators in Harmonic Analysis :: Электронная библиотека попечительского совета мехмата МГУ

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Gilbert J., Murray M. — Clifford Algebras and Dirac Operators in Harmonic Analysis

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Название: Clifford Algebras and Dirac Operators in Harmonic Analysis

Авторы: Gilbert J., Murray M.

Аннотация:

The aim of this book is to unite the seemingly disparate topics of Clifford algebras, analysis on manifolds, and harmonic analysis. The authors show how algebra, geometry, and differential equations play a more fundamental role in Euclidean Fourier analysis. They then link their presentation of the Euclidean theory naturally to the representation theory of semi-simple Lie groups.

Язык:

Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 334

Добавлена в каталог: 03.06.2005

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Предметный указатель

weight      118
functions, analytic over-determinedness of      115 118
spaces of Clifford analytic functions, and critical index of subharmonicity      242—243
spaces of Clifford analytic functions, boundary regularity of      242—243
spaces, of Clifford analytic functions in $\mathbb R^n_+$       119—125
theory, classical      108—119
theory, for Clifford analytic functions in a Lipschitz domain in $\mathbb R^n$       135—141
theory, for Lipschitz domains in $\mathbb C$       115—119
$\eth_A$       204
$\eth_\tau$       220—232
$\mathcal C^*$ algebra      55
$\mathcal M_{\mathbb C}(G)$       145—146
$\mathcal M_{\mathbb C}(\mathrm{Spin}(n))$       220
$\mathfrak A_n$ tangential and normal components of, on $\partial\mathbb R^n$       125
$\mathfrak A_n$ universal Clifford algebra for $\mathbb R^n$       49—65
$\mathfrak A_n$ , Banach algebra structure of      53—65
$\mathfrak A_n$ -module      60
$\mathfrak A_n$ -module, structure of      62—63
$\mathfrak A_n, \mathcal C^*$ algebra structure of      55—65
$\mathfrak C_n$ : universal Clifford algebra for $\mathbb C^n$       56
$\mathfrak R H^p(\mathbb R)$       113
$\mathrm{Spin}_0(V,Q)$ : connected component of Spin(V,Q)      49
Adams, J.F.      206 244
Adjoint Dirac operator, $\overline{\mathcal D}$       207
Ahlfors, L.      318
Algebraically over-determined operator      206
Alvarez-Gaume, L.      318
Analytic over-determinedness      223
Arima, R.      319
Associated vector bundle      266—267
Atiyah — Singer index theorem      244
Atiyah — Singer index theorem, for graded Dirac operators      309—317
Atiyah — Singer index theorem, for spin manifolds      315—316
Atiyah, M.      244 318—319
Axial polynomial      164
Bargmann, V.      284 318
Beardon, A.      85
Bergman space      287—288
Berline, N.      318
Bianchi identities      259
Bismut, J.      318
Bleecker, D.      318
BMO      133
Bochner — Weitzenbock formula      249 258 317
Bochner, S.      199
Booss, B.      318
Boothby, W.      317
Borel — Pompieu theorem      101
Bourguignon, J.-P.      317
Brackx, F.      141
Brauer, R.      94 141
Brocker, T.      85 201
Bruhat decomposition      280
Bruhat decomposition, for $\mathrm{Spin}_0(n,1)$       278
Burkholder — Gundy — Silverstein theorem, classical      114
Burkholder — Gundy — Silverstein theorem, for $H^p(\mathbb R_+^n, \mathfrak H)$       122
Calderon — Zygmund type operators      289—290
Calderon, A.P.      126 135 141 239
Capelli      181
Capelli operator      181
Cartan composition      159 222
Cartan — Schur — Weyl theorem      163
Cartan, E.      143 150 163
Cauchy integral operator, for $\mathbb R^n_+$       121
Cauchy integral operator, for a Lipschitz domain in $\mathbb R^n$       242—243
Cauchy integral operator, on Tan $H^p(\mathbb R^{n-1})$       125
Cauchy integral theorem, for Clifford analytic functions      101
Cauchy — Binet identity      180 230
Cauchy — Riemann operators      87—93
Cauchy — Szego kernel      293
Cauchy — Szego transform      293 295
Cayley      69
Cayley mapping      70
Cayley operator      180—181
Cayley, A.      181
Chern, S.S.      311
Chevalley, C.      85 181
Class 1 property      164
Class 1 property, for harmonic polynomials of matrix argument      198
Class 1 property, of $\mathcal H_m(\mathbb R^n)$       171
Clifford algebra      8
Clifford algebra bundle, over a Riemaimian manifold      264 267
Clifford algebra norm      267—268
Clifford algebra, $\mathbb Z(2)$ -grading of      14
Clifford algebra, as a Lie algebra      67—68
Clifford algebra, center of      27—28
Clifford algebra, dimension of      10 11 31
Clifford algebra, even and odd subspaces of      13
Clifford algebra, ideal structure of      28—30 32
Clifford algebra, matrix realizations of      18—22
Clifford algebra, universal      12
Clifford analytic function, definition of      97
Clifford analytic functions, Cauchy integral theorem for      101
Clifford analytic functions, Cauchy theorem for      103
Clifford analytic functions, maximum modulus principle for      104
Clifford analytic functions, Mean-value theorem for      103
Clifford analytic functions, Morera's theorem for      103
Clifford analytic functions, subharmonicity properties of      105—108
Clifford analytic functions, Weierstrass theorem for      104
Clifford group      38 39 42—45
Clifford group, definition of      42
Clifford module      60
Clifford module, structure of      62—63
Clifford norm, on $\mathfrak A_n$       53 267—268
Clifford norm, on $\mathfrak C_n$       56
Clifford operator norm, in $\mathfrak C_n$       56 128
Clifford operator norm, on $\mathfrak A_n$       53
Clifford semigroup      39
Clifford semigroup, definition of      41
Clifford, W.K.      8 85
Coifman, R.R.      126 134 141 201
Complex Spinor space      151 206
Conformal group      36—38
Conformal map      36
Conformal sphere      37
Conformal transformations      277 281—282
Conjugate Poisson integral      110
Conjugate Poisson kernel      110 121
Conjugation      17 128
Connection      250
Connection, Euclidean      250—251
Connection, Riemannian      251
Covariant derivatives      253
Creation and annihilation operators      15
Critical index of subharmonicity      232—244
Critical index of subharmonicity, defined      233
Curvature operator      254 257
Dahlberg, B.      141
Delanghe, R.      141
Derived representation, on a complex G-module      146
Determined elliptic operator      100
Dilation, of $\mathbb R^n$       278
Dimension, of a Clifford algebra      10 11 31
Dirac $\mathcal D$ operator      207
Dirac operator, basis independence of      95
Dirac operator, standard Euclidean      96
Dirac operators      93—97
Dirac operators, on hyperbolic and spherical space      272—284
Dirac, P.A.M.      83 93 141
Dirichlet problem, for Dirac operator in a Lipschitz domain in $\mathbb R^n$       243
Dirichlet problem, for Laplace's equation in a Lipschitz domain in $\mathbb R^n$       135—137
Div-curl system      224
Divergence, on a Riemannian manifold      256
Duren, P.      141
Eigenspace representation, of $\mathrm{Spin}(n)\copyright\mathbb R^n$       219
Elementary symmetric polynomials      69 178
Elliptic operator      205
Equivalence, of first order differential operators      204—205
Equivalent G-modules      144
Equivariant transformation      144

Euclidean motion group      36
Euler characteristic      311
Euler operator      181
Fatou theorem      110
Fefferman, C.      141
Fiber norm      267
Fischer inner product      164
Flett, T.M.      109
Fourier transform, for $\mathbb R$       111
Fourier transform, for $\mathbb R^{n-1}$       121
Freed, D.      317
Friedman, A.      318
Fueter operators      89 92
Fueter, R.      89 140—141
Fundamental representations      147
Fundamental representations, of Spin(n)      159—160 231
G-invariant inner product      145
Garnett, J.      141
Gauss — Bonnet theorem      311 317
Gauss — Weierstrass kernel      297
Gegenbauer polynomials      168 173
Gegenbauer polynomials, of matrix argument      191
Gegenbauer, L.      199
Gelbart, S.      199 201
Generalized Cauchy — Riemann (GCR) operator      206 233
Generalized Cauchy — Riemann (GCR) operator, rotation-invariant      236—237
Generalized Cauchy — Riemann (GCR) system      233
Generalized Cauchy — Riemann (GCR) systems      140
Getzler, E.      248 305 309 317—319
Gilkey, P.      311 318
Goodearl, K.R.      85
Grad, on a Riemannian manifold      255
Graded Dirac operator      97 203 208 210
Graded Dirac operator, determined by an involution      209—213
Graded Dirac operators, rotation-invariance of      216—218
Grassmann      8
Greiner, P.      319
Gross, K.      190
Group of similarities      37
Group representation      144
Hadamard three-circles theorem      109
Hamilton      8
Hardy — Littlewood maximal function      112 122—123
Hardy, G.H.      108
Harish — Chandra      181
harmonic oscillator      297
Harmonic polynomials of matrix argument      193—200
Harvey, F. Reese      85
Heat kernel, existence and uniqueness of      298—301
Heat kernel, of a second order elliptic operator      296—301
Heat kernels, asymptotics for      296—309
Heighest weight, of a representation      150
Helgason, S.      85 201 318
Herz, C.      190 199 201
Higher gradients operators      225 228
Higher gradients operators, critical index of subharmonicity of      239—242
Highest weight submodule      159
Highest weight vector      150—151
Hilbert bundle      267
Hilbert transform for a Lipschitz domain in $\mathbb R^n$ , estimates for      126—135
Hilbert transform, for $\mathbb R$       111
Hilbert transform, for a Lipschitz domain in $\mathbb R^n$       126 242—243
Hilbert — Schmidt inner product      146 204
Hirzebruch signature theorem      317
Hodge deRham -system      99—100 217—218 222 224 243—244 260 311
Hodge laplacian      260 263 307—308 311—312
Hodge-deRham -system, critical index of      237—239
Hodge-deRham -system, subharmonicity of      237—239
Horizontal lift      269
Horizontal vectors      269
Howe, R.      201
Husemoller, D.      85
Hyperbolic metric      273
Hypergeometric function      168
Hypergeometric function, of matrix argument      190
Imaginary unit      50
Induced representation      286
Injectively elliptic operator      205
Integration by parts theorem      101
Intertwining operator      286—287 290—294
inversion      37 279
Irreducible unitary representation      145—146
Isometry      35
James, A.      201
k-multivectors      50 153 159 262
Kenig, C.E.      115 141 244
Knapp, A.      318
Kobayashi, S.      317
Koornwinder, T.      201
Koosis, P.      141
Koranyi, A.      141
Kostant, B.      201
Kunze, R.      284 318
Laplacian, factorization of      91—93
Laplacian, on a Riemannian manifold      254—255
Laplacian, on a Riemannian manifold, defined      256
Levi, E.E.      299
Lichnerowicz' formula      259
Limits of discrete series representations      288
Linear fractional transformations      275—284
Lipschitz domain, in $\mathbb C$       115
Lipschitz domain, in $\mathbb R^n$       126
Lipschitz, R.      72 85
Maas, H.      201 318
Macdonald, I.      201
Marcus, M.      85
Maxwell stress tensor      244
Maxwell's reciprocal theorem      245
Mcintosh, A.      126 141
McKean, H.P.      311 318
Merryfield, K.      290
Meyer, Y.      126 134 141
Minkowski space      6
Moebius group      37—38
Moebius transformation      37
Muckenhoupt, B.      118
Murray, M.      141
Neumann problem, for Laplace's equation in a Lipschitz domain in $\mathbb R^n$       136
Nomizu, K.      317
Non-tangential limit      120
Non-tangential limits      110
Non-tangential maximal function      114 122
Norm function      41
Norm function on $\mathfrak A_n$ and $\mathfrak C_n$       128
Normal coordinate system      261
Normalized basis, for a quadratic space      7
O'Meara, O.T.      85
O(n)-harmonic function      194
O(n)-harmonic functions      227
Operator of Dirac type      203
Operator of Dirac type, definition of      208—209
Orthogonal group O(V,Q)      33—38
Orthogonal transformations      32—38
Orthonormal frame bundle      265
Orthonormal frame bundle, oriented      265
Parasarathy, R.      318
Parker, T.      318
Patodi, V.      301 311 318
Pauli matrices      9 25 93
Periodicity theorem, for $\mathfrak A_n$       57
Pfaffian      72 308
Pfaffian, mapping properties of      72—77
Pim(V,Q)      47
Pin group      47
Pipher, J.      245
Planck's constant      296
Pochhammer symbol      168 190
Poincare group      36
Poisson integral      110
Poisson kernel      110 121

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