This book gives a thorough and self contained presentation of H¹, its known isomorphic invariants and a complete classification of H¹ on spaces of homogeneous type. The necessary background is developed from scratch. This includes a detailed discussion of the Haar system, together with the operators that can be built from it. Complete proofs are given for the classical martingale inequalities, and for large deviation inequalities. Complex interpolation is treated. Througout, special attention is given to the combinatorial methods developed in the field. An entire chapter is devoted to study the combinatorics of coloured dyadic Intervals.
Author(s): Paul F.X. Müller
Edition: 1
Year: 2005
Language: English
Pages: 453
Contents......Page 7
Preface......Page 10
The Haar basis......Page 14
Khintchine’s inequality......Page 19
Burkholder’s inequality......Page 26
The Walsh system......Page 32
1.2 Dyadic H[sup(1)]......Page 47
Fefferman’s inequality......Page 48
Sharp maximal functions......Page 58
Square functions in L[sup(p)]......Page 65
Multipliers into SL[sup(∞)]......Page 81
1.4 Martingales and biorthogonal systems......Page 86
Martingale inequalities......Page 87
Biorthogonal systems......Page 94
Multipliers and paraproducts......Page 101
Rearrangement operators and Calderón–Zygmund kernels......Page 105
Orthogonal projections......Page 117
1.6 Notes......Page 125
2.1 Complemented subspaces......Page 129
Three-valued martingale differences......Page 136
Rosenthal’s space......Page 142
Weighted intersections......Page 149
2.2 Pełczyński’s decomposition method......Page 151
Infinite direct sums......Page 152
H[sup(1)] with values in l[sup(2)][sub(n)]......Page 154
2.3 Interpolation of operators......Page 156
Calderón's product......Page 160
Pisier’s norm on H[sup(1)]......Page 165
Dual estimates......Page 168
Analytic families of operators......Page 171
2.4 Notes......Page 178
3.1 The Carleson packing condition......Page 181
Generations of dyadic intervals......Page 182
The Gamlen–Gaudet construction......Page 188
Jones’s compatibility condition and colored intervals......Page 193
The first step towards the uniform approximation property......Page 205
3.3 Rearrangement operators......Page 208
Rearrangement operators on BMO......Page 210
Rearrangement operators on L[sup(1)]......Page 236
3.4 Notes......Page 240
4.1 Maurey’s isomorphism......Page 241
Operations on martingale differences......Page 242
Martingale transform techniques......Page 245
Classification of martingale H[sup(1)] spaces......Page 254
Classification of weighted intersections......Page 264
The theorem of Gamlen and Gaudet......Page 269
Related open problems......Page 272
4.4 Notes......Page 277
Existence, abundance......Page 278
Johnson’s factorization......Page 282
Intrinsic description......Page 288
5.2 Complemented copies of H[sup(1)][sub(n)]......Page 293
Dichotomies......Page 294
Intrinsic description......Page 306
H[sup(1)] with values in l[sup(2)]......Page 312
5.3 The uniform approximation property of BMO......Page 319
Splitting the Haar support......Page 323
UAP data with large Haar coefficients......Page 325
UAP data with small Haar coefficients......Page 327
General UAP data......Page 353
5.4 Notes......Page 355
6.1 Basic similarities between H[sup(1)] and H[sup(1)][sub(at)]......Page 358
Maximal functions and atoms......Page 362
Square functions......Page 366
6.2 Carleson’s biorthogonal system......Page 371
Definition......Page 373
Carleson coefficients versus Haar coefficients......Page 377
The compensation argument......Page 385
Unconditional basis in H[sup(1)][sub(at)]......Page 402
6.3 Spaces of homogeneous type......Page 408
Lipschitz partitions of unity......Page 415
Estimates for molecules......Page 416
6.4 Orthogonal projections in atomic H[sup(1)] spaces......Page 418
The square root of the Gram matrix......Page 429
6.5 Martingale approximation in atomic H[sup(1)] spaces......Page 432
6.6 Notes......Page 441
Bibliography......Page 443
List of Symbols......Page 458
H......Page 460
Q......Page 461
W......Page 462
