Hedenmalm H., Korenblum B., Zhu K. — Theory of Bergman spaces :: Электронная библиотека попечительского совета мехмата МГУ

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Hedenmalm H., Korenblum B., Zhu K. — Theory of Bergman spaces

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Название: Theory of Bergman spaces

Авторы: Hedenmalm H., Korenblum B., Zhu K.

Аннотация:

Preliminary Text. Do not use. 15 years ago the function theory and operator theory connected with the Hardy spaces was well understood (zeros; factorization; interpolation; invariant subspaces; Toeplitz and Hankel operators, etc.). None of the techniques that led to all the information about Hardy spaces worked on their close relatives the Bergman spaces. Most mathematicians who worked in the intersection of function theory and operator theory thought that progress on the Bergman spaces was unlikely. Now the situation has completely changed. Today there are rich theories describing the Bergman spaces and their operators. Research interest and research activity in the area has been high for several years. A book is badly needed on Bergman spaces and the three authors are the right people to write it.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2000

Количество страниц: 289

Добавлена в каталог: 03.06.2005

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Предметный указатель

$A^0_\alpha$ , the Bergman — Nevanlinna class      98 131
-subinner function      52
$A^2(\omega)$ , weighted Bergman space      245
$A^2(\omega)$ -inner function      270
-inner function      52
$A^p_\alpha$ -inner function      52
$A^p_\alpha$ -subinner function      87
$A_\alpha^{p+}$ , limit of Bergman spaces      130
$A_\alpha^{p-}$ , limit of Bergman spaces      130
      42
$BMO_{\partial}$       45
$B^2(\omega)$ , large weighted Bergman space      245
, the Besov space      23
$C(\overline{\mathbb D})$       13
      60
$C_0(\mathbb D)$       13
$C_0^{-\alpha}(\Gamma)$       137
, hyperbolic disk      38
, the normalized area element      1
$dA_\alpha$ , the weighted area element      2
, normalized arc length measure      59 104
$d\Lambda_E$ , push-out measure      114
, upper asymptotic $\kappa$ -density      105
$D^+_s(\Gamma)$ , upper Seip density      147
$D^+_u(\Gamma)$ , uniform separating upper asymptotic $\kappa$ -density of $\Gamma$       140
\kappa$-density      105
$D^-_s(\Gamma)$ , lower Seip density      153
$D^\alpha$ , “fractional differentiation”      18
$D_\alpha$ , “fractional integration”      19
$d_{\mathbb C}$ Euclidean metric on the plane      105
$d_{\mathbb T}$ , circle metric      104
, Green function for the Laplacian      60
, Green potential      60
, zero divisor for A      57
, the extremal function for I      57
$H(\mathbb D)$ , the space of all holomorphic functions      18
$HP^2(\omega)$       251
, the Hardy space      54
$H^\infty$       2
$H_\omega(z, w)$ , the harmonic compensator      266
$H_{\omega, D}$ , harmonic compensator      254
, zero-based invariant subspace      55
, invariant subspace generated by f      56
, the Bergman kernel      6
$K_\alpha(z, w)$ , the Bergman kernel for standard weights      6
$K_\omega(z, w)$ , weighted Bergman kernel      245
, “mean oscillation”      43
, mean oscillation      42
, counting function      100
, common multiplicity of zero at the origin      56
, Poisson kernel      29
, the Poisson kernel      114 217
, the sweep operator      87
$Q_\omega(z, w)$ , weighted harmonic Bergman kernel      252
      46
$VMO_{\partial}$       46
$\beta$ , the hyperbolic metric      16
$\Delta$ , invariant Laplacian      32
$\Delta$ , the Laplacian      32
$\delta(\mathb T)$ , diagonal      253
$\Gamma(z, w)$ , biharmonic Green function      62
$\Gamma[f]$ , biharmonic Green potential      62
$\Gamma_a(z)$ , approximate Stolz angle      111
$\Gamma_\omega$ , the weighted biharmonic Green function      242
$\hat\kappa(F)$ , Beurling — Carleson characteristic      104
$\kappa$ -absolutely continuous      201
$\kappa$ -area      148
$\kappa$ -bound      194
$\kappa$ -bounded above      194
$\kappa$ -density      105
$\kappa$ -smooth      201
$\kappa$ -variation      194
$\kappa\, B+$       194
$\Lambda(A, E)$ , partial logarithmic Blaschke sum on E      105
$\mathbb C$ , the complex plane      1
$\mathbb D$ , the open unit disk      1
$\mathbb R$ , the real line      1
$\mathbb T$ , the unit circle      1
$\mathbf B$ , the Berezin transform      29
$\mathbf B_\alpha, the weighted Berezin transform      29
$\mathbf P$ , Bergman projection      6
$\mathbf P_\alpha$ , weighted Bergman projection      6
$\mathcal A^p$ , the Bergman spaces      6
$\mathcal A^p$ -outer function      190 191
$\mathcal A^p_\alpha$ , the standard weighted Bergman space      2
$\mathcal A^{-\alpha}$ , standard growth space      110
$\mathcal A^{-\alpha}_+$ , limit of growth spaces      112
$\mathcal A^{-\alpha}_-$ , limit of growth spaces      112
$\mathcal A^{-\alpha}_0$ , , small growth space      110 137
$\mathcal A^{-\infty}$ , the big growth space      110
$\mathcal B$ , the Bloch space      13
$\mathcal B_0$ , the little Bloch space      13
$\mathcal D$ , the Dirichlet space      25
$\mathfrak s_F$ , Stolz star      105
$\mathfrak s_z$ , Stolz angle      104
$\mathfrak s_{z, \alpha}$ , Stolz angle with aperture $\alpha$       106
$\nabla$ , the gradient      51
$\omega$ , harmonic measure      218
$\omega$ , weight      242 244
$\omega$ -mean value disk      255
$\rho$ , the pseudohyperbolic metric      16
$\sigma(A, E)$ , partial Blaschke sum on E      105
$\sim$       2
$\varphi_z$ , the Moebius involution      6
$\varpi_r$ , harmonic measure      255
Absolutely continuous measure      201
Admissible      122
Analytic projection      12
Aperture      104 106
Arc length      104
Arc length, normalized      104
Arithmetic-geometric mean inequality      101
Asymptotic $\kappa$ -density, lower      105
Asymptotic $\kappa$ -density, upper      105
Atomic singularity      59
Balayage-type estimate      112
Berezin transform      28
Bergman kernel      6
Bergman metric      16
Bergman projection      6
Bergman space      2
Bergman — Nevanlinna class      98 131
Besov space      24 50
Beurling — Carleson characteristic      104
Beurling — Carleson set      104 200
Beurling's theorem      176
Beurling-type theorem      181
Bi-harmonic Green function      62
Bi-harmonic Green potential      62
Biharmonic Green function      59 242
Blaschke product      26
Bloch space      13
BMO      42
BMO in the Bergman metric      42
BMOA      13
Bounded $\kappa$ -variation      194
building blocks      219
Caratheodory-Schur theorem      87
Carleson measure      38
Carleson square      106 240
Cayley transform      113 165 167
Complementary arcs      104
Concave operator      182
Concave sequence      183
Concavity-type property of the biharmonic Green function      273
Conjugate exponent      18 160
Contractive divisibility property      59
Contractive multiplier      54
Contractive zero divisors      71
Covering      125 204

Cyclic function      190
Cyclic invariant subspace      56
Cyclic vector      78 85 190
density      104
Dirichlet space      25
Division property      189
Domination      190 191
Duality theorem of Linear Programming      124 204
Edge point in polyhedron      124
Eigenfunctions      36
Eigenvalue      37
Elementary factors      131
entropy      104
Exact type      111
Expansive multiplier property      59 66 70
Extraneous zero      72
Extremal function      57
Extremal problem      26 55 56 195
Extremal problem for I      56
Factorization      78
Fejer kernel      87
Fractional differentiation      18
Fractional integration      18
Functions of bounded variation      195
Gap sequence      15
Garsia's lemma      42
Generation      233
Geometric progression      107
Green function      59
Green potential      60
Green's formula      59
Growth space      110
Hadamard's variational formula      255 257
Hankel operator      49
Hardy space      12 54 78
Harmonic compensator      254 266
Harmonic conjugate      26
Harmonic majorant      55
Harmonic measure      218 219 225 232
Helly selection theorem      195
Herglotz measure      201
Herglotz transform      259
Homogeneous decomposition      178
Hurwitz's theorem      77
Hyperbolic center      42
Hyperbolic disk      38
Hyperbolic exponential type      111
Hyperbolic metric      16
Hyperbolic radius      42
INDEX      176
Index n      176
Index of an invariant subspace      176
Inner function      52 78
Inner function for Bergman spaces      53
Inner space      180
Inner-outer factorization      78
Interpolating sequence      50 136
Interpolation problem      137
Invariant Laplacian      32
Invariant subspace      55
Invariant subspace problem      187
Iteration scheme      222
Jensen's formula      98
Jensen-type inequality      116
Koebe function      260
Lacunary series      15
Laplace equation      62
Laplace — Beltrami operator      33
Laplacian      32 Laplacian invariant
Linear programming      119
Linear Programming, duality theorem      124 204
Little Bloch space      13
Localization trick      230
Logarithmic entropy      194
Logarithmically subharmonic      94 242 244
Lower asymptotic $\kappa$ -density      105
Lower Seip density      153
Lunula      227
Maximal inner space      187
Mean oscillation      42
Min-max equation      123
Minimal type      111
Moebius group      6
Moebius map      6
Moebius transformation      16
Mushroom      231
Mushroom forest      230
Mushroom hat      232
Mushroom stem      231
Nonoverlapping arcs      211
Normalized arc length      59 104
Oblique projection      119
Optimization problem      123
Outer function      190
Perturbation      139
Poincare metric      16
Point-evaluation      2
Poisson extension      114
Poisson formula      29
Poisson kernel      29 114
Poisson solver      273
Poisson transform      28 29 114 273
Positive function      2
Positive measure      2
Premeasure      190 193
Pseudohyperbolic metric      16
Push-out measure      114
Quasi-Banach space      2
Quasi-similar operators      188
Regular sequence      177
Reproducing for the origin      242
Reproducing kernel      57 242
Residue theorem      76
Restriction operator      155
Reverse triangle inequality      177
Sampling sequence      136
Schur's test      9 11
Seip density, lower      153
Seip density, upper      147
Separated sequence      39 50 138
Sequence of interpolation      136
Sequence of sampling      136
Sesquiholomorphic      252
Similar operators      188
Simple covering      125 204
Singly generated invariant subspace      56
Singular inner function      59
Singular measure      201
Spectral mapping theorem      37
Stieltjes integral      195
Stolz angle      104
Stolzstar      105
Strictly positive      2
Subinner function      86
Suppressive weight      273
Sweep of a function      88 264
Toeplitz operator      49
Truncation      202
Type, hyperbolic exponential      111
Uniform separating upper asymptotic $\kappa$ -density      140
Uniform upper asymptotic $\kappa-*$ -density      146
Uniformly discrete      39
Upper asymptotic $\kappa$ -density      105
Upper Seip density      147
Variational argument      56
Variational formula      255
VMO      42
Weakly cyclic      201
Weierstrass factorization      131
Weighted Bergman kernel      6

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