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Krantz S.K. — Partial Differential Equations and Complex Analysis
Krantz S.K. — Partial Differential Equations and Complex Analysis

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Название: Partial Differential Equations and Complex Analysis

Автор: Krantz S.K.

Аннотация:

Ever since the groundbreaking work of J.J. Kohn in the early 1960s, there has been a significant interaction between the theory of partial differential equations and the function theory of several complex variables. Partial Differential Equations and Complex Analysis explores the background and plumbs the depths of this symbiosis.

The book is an excellent introduction to a variety of topics and presents many of the basic elements of linear partial differential equations in the context of how they are applied to the study of complex analysis. The author treats the Dirichlet and Neumann problems for elliptic equations and the related Schauder regularity theory, and examines how those results apply to the boundary regularity of biholomorphic mappings. He studies the ?-Neumann problem, then considers applications to the complex function theory of several variables and to the Bergman projection.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1992

Количество страниц: 320

Добавлена в каталог: 04.07.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\bar{\partial}$-Neumann boundary conditions      210
$\bar{\partial}$-Neumann problem      184
A priori estimate      52
Action of a pseudodifferential operator on a Sobolev space      61
Algebraic properties of distributions      38
Asymptotic expansion      66—68
Basic estimate      206
Bell’s lemma      257
Bergman kernel      133
Bergman kernel for the ball      137
Bergman metric      144
Bergman metric on the ball      146
Bergman projection      252
Bergman projection and the Neumann operator      253
Bessel potential      114 212
Bigraded spherical harmonics      172
Biholomorphic self-maps of the ball      130
Boundary regularity for the Dirichlet problem for the invariant Laplacian      151ff 182
Boundary value problems reduced to pseudodifferential equations      116
Calderon — Zygmund operator      55
Cauchy principal value integral      55
Cauchy problem      2
Cauchy — Riemann equations      129
Cayley transform      10
Character of a group      26
Closed range of the $\partial$ operator      246
Closed range property for a boundary value problem      100
Coercive estimate      196 206
Commutator of operators      66 208
Compactly supported distributions      41
Complex Jacobian      135
Complexified tangent space      185
Condition r      252
Constant coefficient boundary value problems      95
Constant coefficient partial differential operators      52
Convolution      43
Convolution of distributions      43
Convolution of functions      29
Difference operators      7
Dini — Kummer test      176
Dirichlet problem      2 23
Dirichlet problem for the, Laplace — Beltrami operator      148 150
Dirichlet problem on the disc      4 5
Disc      1
Distributions      40
Domain with smooth boundary      20
Elliptic boundary value problem      109
Elliptic operator      23 79 191
Elliptic regularity on an a smooth domain      124
Elliptic regularization      184
Ellipticity      25
Euclidean volume element      131
Existence for a boundary value problem      100
Existence for a general elliptic boundary value problem      114
Existence for an elliptic operator      85
Extension theorem for Sobolev spaces      92
Faa de Bruno      260
Fefferman’s mapping theorem      256
Fefferman’s theorem, proof      263
Finite differences      210
Formal adjoint of a partial differential operator      193
Fourier inversion formula      34
Fourier transform      26
Fourier transform of a distribution      40
Fourier — Laplace transform      48
Fractional integration operator      55
Friedrichs extension lemma      196
Friedrichs mollifiers      44
Frobenius integrability condition      278
Frobenius integrability theorem      278
Gauss — Weierstrass kernel      33
Gegenbauer polynomial      169
Generalized schwarz inequality      215
Grushin operator      279
Hardy spaces      139
Harmonic projector      246
Harmonic space      276
Hilbert space adjoint of a partial differential operator      193
Hodge star operator      188
Hodge theory for the d operator      246
Holomorphic extension phenomenon      267
Holomorphic function      131
Hopf’s lemma      22
Hormander calculus      57
Hypergeometric equation      174
Hypoelliptic operator      78
Hypoellipticity      207
Inhomogeneous Cauchy — Riemann equations      248
Interior regularity for an elliptic operator      84
Interpolation of operators      19
Invariant Laplacian      129
Inverse Fourier transform      30
k times continuously differentiable      8
k times continuously differentiable function      2 9
Kahler manifold      144
Kerzman’s theorem      263
Key properties of a calculus      65
Kodaira vanishing theorem      248
Kohn — Nirenberg calculus      56
Kohn — Nirenberg formula      71
Kohn, J.J.      252
Laplace — Beltrami operator      147
Laplace — Beltrami operator for the Poincare — Bergman metric      129
Laplacian      1 2
Levi form      205
Levi polynomial      264
Levi problem      267
Lewy unsolvable operator      269 270
Lewy’s example of an unsolvable operator      85
Lipschitz spaces      7 125
Local operator      76
Local solvability and analytic continuation      276
Lopatinski condition      96 98 100 109
Lumer’s Hardy spaces      139
Main estimate      207 239
Method of freezing coefficients      54
Mityagin/Semenov theorem      9
Mobius transformation      128 130
Neumann boundary conditions      200
Neumann operator      246 247
Nonisotropic balls      150
Nonisotropic geometry      142
Norm estimate for solutions of the $\partial$ problem      237
Overdetermined system of partial differential equations      190
Paley — Wiener theorem      42 49
Parallels orthogonal to a vector      166
Parametrix      53 81
Parametrix for a boundary value problem      109
Parametrix for an elliptic boundary value problem      110
Peetre’s theorem      76
Plancherel’s formula      35
Pluriharmonic function      181
Pluriharmonic functions      142
Poincare metric on the disc      144 146
Poisson kernel on the disc      5
Poisson — Szego kernel      140 180
Poisson — Szego kernel on the ball      141
Poisson — Szego kernel on the disc      141
Principal symbol      80 88
Pseudoconvex      205
Pseudodifferential operator      23 55
Pseudodifferential operators under change of variable      88
Pseudolocal      77
Pseudolocal operator      76
Raabe’s test      176
Regularity for the Dirichlet problem      10
Regularity in the Lipschitz topology      125
Regularity of a boundary value problem      99
Rellich’s lemma      60
Riemann — Lebesgue lemma      27
Riemannian metric      143
Riesz potential      55
Riesz — Thorin theorem      36
Rotations and the Fourier transform      28
Schauder estimates      125
Schur’s lemma      212
Schwartz distribution      37
Schwartz function      36
Schwartz space      36
Schwartz’s theorem      47
Siegel upper half space      270
Singular function      266
Singular support of a distribution      77
Smooth boundary continuation of conformal mappings      21
Smoothing operator      81
Sobolev -1/2 norm      234
Sobolev imbedding theorem      58
Sobolev spaces      57
Solid spherical harmonics      157
Special boundary charts      209
Spherical harmonics      153 156
Strongly pseudoconvex      205
Structure theorem for distributions      41
Subelliptic estimate      184
Summability kernel      30
Support of a distribution      40
Symbol of an adjoint pseudodifferential operator      68
Symbol of an operator      54 56 190
Szego kernel      139
Szego kernel on the ball      141
Szego kernel on the disc      141
Szego projection and analytic continuation      272
Tangent space      185
Tangential Bessel potential      217
Tangential holomorphic vector field      271
Tangential Sobolev norm      217
Tangential Sobolev spaces      217
Trace theorem for Sobolev spaces      90
Transformation of the Bergman kernel under biholomorphic mappings      135
Volume form      187
Weierstrass nowhere-differentiable function      8
Well-posed boundary value problem      99 100
Zonal harmonic      161 170
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