Bennett C., Sharpley R.C. — Interpolation of Operators :: Электронная библиотека попечительского совета мехмата МГУ

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Bennett C., Sharpley R.C. — Interpolation of Operators

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Название: Interpolation of Operators

Авторы: Bennett C., Sharpley R.C.

Аннотация:

This book presents interpolation theory from its classical roots beginning with Banach function spaces and equimeasurable rearrangements of functions, providing a thorough introduction to the theory of rearrangement-invariant Banach function spaces. At the same time, however, it clearly shows how the theory should be generalized in order to accommodate the more recent and powerful applications. Lebesgue, Lorentz, Zygmund, and Orlicz spaces receive detailed treatment, as do the classical interpolation theorems and their applications in harmonic analysis. The text includes a wide range of techniques and applications, and will serve as an amenable introduction and useful reference to the modern theory of interpolation of operators.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1988

Количество страниц: 483

Добавлена в каталог: 01.12.2013

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Предметный указатель

$L^{1}+L^{\infty}$ and $L^{1}\cap L^{\infty}$       §2.6 74
$\Delta_{2}$ condition      266
$\overline{\partial}$ operator      §5.9 §5.10 402
$\overline{\partial}$ operator, fundamental solution      403
$\sigma(X,Y)$ -topology      24
$\sigma(X,Y)$ -topology, boundedness      24
$\sigma(X,Y)$ -topology, completeness      25
Absolutely continuous norm      §1.3 14
Absolutely continuous norm and dominated convergence      16
Absolutely continuous norm and duality      23
Absolutely continuous norm and reflexivity      23
Absolutely continuous norm and separability      27 29
Admissible growth      209
Admissible norm      426
Admissible operator      99 115 301
Almost everywhere convergence      §4.5 120 138
Analytic atom      435
Analytic family of operators      §4.3 209
Approximate identity      178
Associate norm      8 12
Associate space      9
atom      369
Atomic decomposition      371 372
Average of a function      117 352
Averaging operator      57 61 150 202
B-spline      335
Banach function norm      2
Banach function norm, dominated convergence theorem      16
Banach function norm, Fatou property      4 6
Banach function norm, Fatou's lemma      5 6
Banach function norm, lattice property      6
Banach function space      3
Banach function space, $\sigma(X,Z)$ -completeness      25
Banach function space, compactness      31
Banach function space, completeness      5
Banach function space, duality      23
Banach function space, reflexivity      23 30
Banach function space, separability      29
Banach limit      110
Banach's principle      237
Besov space      §5.4 332 341
Bilinear form      188
Bilinear operator      255
Blaschke product      364 Appendix
BLO      see "Bounded lower oscillation"
BMO      see "Bounded mean oscillation"
Boundary function      364
Bounded lower oscillation      388
Bounded mean oscillation      §5.6 368 §5.7 380 392
Bounded mean oscillation, decomposition into BLO functions      390 400
Boyd index      see "Indices"
Boyd's theorem on the Hilbert transform      154
Boyd's theorem on weak-type operators      153
Brudnyi — Krugljak theory      326 429
Calderon couple      427
Calderon operator      133 142 153 163 176 222 256 262 313
Calderon operator and the Hilbert transform      138 387
Calderon operator and the maximal conjugate function      160
Calderon operator and the maximal Hilbert transform      134
Calderon operator and weak-type conditions      223 263 386
Calderon — Ryff theorem      114
Calderon's theorem on $L^{1}$ and $L^{\infty}$       116
Calderon's theorem on weak-type operators      144
Carleson measure      401
Carleson norm      401
Characterizations of interpolation spaces      329 116 419 425
Class $\theta$       310
Class $\theta$ and log convexity      316
Compact operator      202
Compact operator and interpolation      203
Compatible couple      97
Complementary Young's function      271
Conditional expectations      see "Averaging operator"
Conjugate couple      174
Conjugate, Poisson kernel      176 177
Conjugate-function operator      160 163 251
Convolution      155
Convolution operator      258 259
Covering lemma      121 377
Covering lemma, Vitali variant      118
Covering lemma, Whitney      348
Cwikel's formula for the K-functional      322
Decreasing rearrangement      39
Decreasing rearrangement of products      88
Decreasing rearrangement of sums      41
Decreasing rearrangement, properties      41
Decreasing rearrangement, radial      91
Decreasing rearrangement, signed symmetric      91
Decreasing rearrangement, symmetric      91
Density theorem      315
DeVore — Scherer theorem      360
Difference operator      331
Dilation operator      148
Distribution function      36
Distribution function, properties      37
Divisibility      325
Dominated Convergence Theorem      16
Duality      §1.4 174
Duality for $H^{1}$ and BMO      374
Duality for $H^{p}$ and Lip $\alpha$       433
Duality for $L^{p,q}$       221
Duality for Orlicz spaces      275
Dyadic cubes      121
Elementary maximal function      §2.3 52 219 377 382
Elementary maximal function and the Hardy — Littlewood maximal function      122
Elementary maximal function, properties      52 54 74
Embeddings, ( $\theta$ , q) interpolation spaces      300 301 316
Embeddings, Banach function spaces      7 13 72 77 78
Embeddings, Besov and Lebesgue spaces      346
Embeddings, Besov and Sobolev spaces      334 346
Embeddings, BLO      389
Embeddings, BMO and Lebesgue spaces      382
Embeddings, BMO and weak- $L^{\infty}$       383
Embeddings, Lorentz spaces      217 286
Equimeasurable functions      37 80
Equivalence theorem      314
Exact interpolation pair      104 105
Extension operator      431
F*      see "Decreasing rearrangement"
f**      see "Elementary maximal function"
Fatou's lemma      6
Fefferman's duality theorem      374
Fejer kernel      179
Finite rank operator      202
Fourier coefficient      156 228
Fourier multiplier      159 163 433
Fourier series      156 179 285
Fourier transform      200 282
Fractional integral operator      228 254
Function norm      see "Banach function norm"
Fundamental function      §2.5 65 276 286
Fundamental function, properties      66 67
Fundamental indices      see "Indices"
Gagliardo completion      296
Gagliardo couples      320
Hadamard three-lines theorem      195
Hardy space      §5.6 §5.9 §5.10 363
Hardy space of rearrangement-invariant spaces      425
Hardy space, atomic      369
Hardy space, characterizations      372 432
Hardy space, real      §5.6 365
Hardy — Littlewood inequality for rearrangements      §2.2 44
Hardy — Littlewood maximal function      see "Hardy — Littlewood maximal operator"
Hardy — Littlewood maximal operator      §3.3 117 122 125 154 177 250 352
Hardy — Littlewood maximal operator and BMO      389
Hardy — Littlewood maximal operator and weak-type (1,1)      119 125
Hardy — Littlewood maximal theorem      125
Hardy — Littlewood theorem for Hardy spaces      364
Hardy — Littlewood — Polya relation      56 105 108 114 166 169 172 305 418
Hardy — Littlewood — Sobolev theorem      229

Hardy's inequalities for averages      124
Hardy's inequalities for the Fourier transform      431
Hardy's Lemma      56
Hausdorff — Young theorem      200 283
Hilbert transform      §3.4 126 138 154 366 367 372
Hilbert transform, Kolmogorov's theorem      139
Hilbert transform, Riesz's theorem      139
Hoelder's inequality      9 60
Hoermander condition      433
Holmstedt's formula      307
Indices for $\Lambda_{a}(X)$       286
Indices for LlogL and $L_{exp}$       247
Indices for Lorentz spaces      218 220
Indices for Orlicz spaces      277
Indices, Boyd      149 165 177 178
Indices, fundamental      177
Intermediate space      99
Interpolating sequence      435
Interpolation inequality for derivatives      337
Interpolation method, maximal      429
Interpolation method, minimal      428
Interpolation of compact operators      203
Interpolation pair      102
Interpolation segment $\sigma$       141
Interpolation space      §3.1 105
Interpolation space, ( $\theta$ ,q) space      299 301 §5.2
Interpolation space, monotone      §5.3 319
Interpolation spaces between $H^{1}$ and $L^{\infty}$       374
Interpolation spaces between $H^{l}$ and $H^{\infty}$       §5.10 415
Interpolation spaces between $L^{1}$ and $L^{\infty}$       §3.2 116 305
Interpolation spaces between $L^{1}$ and BMO      §5.8 398
Interpolation spaces between $L^{p}$ and $W^{p}_{r}$       341
Interpolation spaces between ( $\theta$ ,q) spaces      see "Reiteration theorem"
Interpolation spaces between Besov spaces      343
Interpolation spaces between Sobolev spaces      §5.5 362
J-functional      293
J-method of interpolation      314 429
Jackson kernel      179
Jackson's theorem for rearrangement-invariant spaces      179
John — Nirenberg lemma for a cube      381
John — Nirenberg lemma for distribution functions      393
John — Nirenberg lemma for rearrangements      393
Joint weak type      §3.5 143 253
Joint weak type and weak type (p, q)      223
Jones construction      405
K-functional      302 384
K-functional for $H^{1}$ and $H^{\infty}$       §5.10 411 414
K-functional for $L^{1}$ and $L^{\infty}$       298
K-functional for $L^{1}$ and BMO      393
K-functional for $L^{p}$ and the Sobolev space $W^{p}_{k}$       339 431
K-functional for ( $\theta$ ,q) spaces      see "Holmstedt's formula"
K-functional for C and Lip 1      436
K-functional for the Hardy space $H^{1}$ and $L^{\infty}$       373
K-functional for the Sobolev spaces, $W^{1}_{k}$ and $W^{\infty}_{k}$       360
K-functional for the Sobolev spaces, $W^{p}_{k}$ and $W^{q}_{k}$       362 see
K-method      305 329
K-method of interpolation      §5.1
Kernels for $\overline{\partial}$ operator      404 405
Kernels, Dirichlet      179
Kernels, Fejer      179
Kernels, Jackson      179
Kolomogorov's theorem      139
Landau's resonance theorem      10
Laplacian operator      402
Lattice property      6
Least concave majorant      70
Lebesgue differentiation theorem, multivariate case      430
Lebesgue differentiation theorem, univariate case      120
Lebesgue spaces      3
log convex estimate for Sobolev spaces      338
log convex estimate for spaces of class $\theta$       316
Lorentz spaces, $L^{p,q}$       216 300 374 398 415
Lorentz spaces, $M_{\phi}$       69
Lorentz spaces, $\Lambda(X)$ and M(X)      71 261
Lorentz spaces, $\Lambda_{a}(X)$       286
Lorentz — Luxemburg theorem      10
Lorentz — Shimogaki theorems      89 154 §3.7 169 173 329
Lorentz — Zygmund space      253 285
Lorentz' lemma      80
Luxemburg norm      see "Orlicz spaces"
Luxemburg representation theorem      62
Marchaud's inequality      332 334
Marcinkiewicz interpolation theorem      §4.4
Marcinkiewicz interpolation theorem and weak- $L^{\infty}$       386
Marcinkiewicz interpolation theorem for Lebesgue spaces      226
Marcinkiewicz interpolation theorem for Lorentz spaces      225
Marcinkiewicz interpolation theorem, limiting case      248 253
Marcinkiewicz multiplier theorem      433
Maximal operator      237
Maximal operator, maximal conjugate function      160
Maximal operator, maximal Hilbert transform      127 130
Maximal operator, median      421
Maximal operator, nontangential      175 177 251 363 372 373 411
Maximal operator, radial      175
Maximal operator, signed      400
Measure-preserving transformations      §2.7 80
Minimally smooth boundary      430
Modulus of continuity      157 332 431
Monotone, interpolation (intermediate) space      §5.3 319
Monotone, Riesz — Fischer space      305 329
Moon's theorem      234
Multi-index      335
Multi-index differential operator      335
Multi-index length      335
Multilinear operator      201
Multilinear strong type interpolation      202
Multilinear weak type interpolation      255
Multiplier      see "Fourier multiplier"
Nontangential maximal operator      see "Maximal operator"
Norm-convergence of Fourier series      §3.6 156 163
Norm-fundamental      12
Order ideal      16
Orlicz class      266
Orlicz space      §4.8 270
Orlicz space, fundamental function      276
Orlicz space, indices      277
Orlicz space, Luxemburg norm      268
Orlicz space, Orlicz norm      273
Paley's theorem for Fourier series      228
Parseval's formula      200
Poisson integral      175
Poisson kernel      175 176 206 284 363
Positive integral operator      185
Q-maximal function      352
Quasiconcave      69
Quasilinear      143
Re $H^{1}$       §5.6 365
Rearrangement-invariant norm      59
Rearrangement-invariant space      59 116
Reduction Theorem      343
Reflexivity      see "Banach function space"
Reiteration theorem      311 430
Relative Calderon pair      427
Relatively monotone      428
Resonant measure space      45 51 116
Restricted weak type      §4.5 231 255 385
Retracts      54
Riesz convexity theorem for bilinear forms      189
Riesz convexity theorem for operators      §4.1 192
Riesz — Fischer inequality for the Fourier transform      200
Riesz — Fischer norm      304 320 329
Riesz — Fischer property      6 32
Riesz — Fischer space      304
Riesz — Thorin convexity theorem      §4.2 196
Riesz — Thorin convexity theorem, real version      199
Ryff's theorem      82
Ryff's theorem, counterexample      85
Separability      §1.5
Separability and reflexivity      30
Separable measure      27

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