Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory: [Ferran Sunyer i Balaguer award winning monograph]
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Cham [u.a.]
Springer
2014
|
| Series: | Progress in Mathematics
307 |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 381 - 392 |
| Physical Description: | XIII, 396 S. Ill. |
| ISBN: | 9783319005959 9783319005966 |
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| 245 | 1 | 0 | |a Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory |b [Ferran Sunyer i Balaguer award winning monograph] |c Xavier Tolsa |
| 264 | 1 | |a Cham [u.a.] |b Springer |c 2014 | |
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| adam_text | Titel: Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory
Autor: Tolsa, Xavier
Jahr: 2014
Contents
Introduction 1
Basic notation 11
1 Analytic capacity 15
1.1 Introduction............................... 15
1.2 Definition and basic properties of analytic capacity......... 15
1.3 Removable sets and the Painlevé problem.............. 20
1.4 Two technical lemmas......................... 21
1.5 The Cauchy transform and Vitushkin s localization operator .... 22
1.6 Relationship with Hausdorff measures................ 28
1.7 Rectifiable sets and Vitushkin s conjecture.............. 34
1.8 Historical remarks, the semiadditivity of 7, etc............ 37
1.8.1 Some historical remarks.................... 37
1.8.2 Uniform approximation by rational functions and the semi-
additivity of analytic capacity................. 37
1.8.3 The approach by duality to analytic capacity........ 39
1.8.4 Other spaces and capacities.................. 40
1.8.5 Hausdorff measures....................... 41
2 Calderón- Zygmund theory with non-doubling measures 45
2.1 Introduction............................... 45
2.2 Preliminaries.............................. 47
2.3 Covering theorems and maximal operators.............. 49
2.4 Doubling cubes and balls ....................... 53
2.5 Some standard estimates........................ 55
2.6 Calderón-Zygmund decomposition .................. 57
2.7 Weak (1,1) boundedness of Calderón-Zygmund operators...... 60
2.8 Cotlar s inequality........................... 63
2.9 The good lambda method....................... 68
2.10 Historical remarks and further results ................ 73
x Contents
3 The Cauchy transform and Menger curvature 75
3.1 Introduction............................... 75
3.2 The curvature of a measure...................... 76
3.3 The Tl theorem for the Cauchy singular integral operator..... 81
3.4 The Cauchy transform on Lipschitz graphs ............. 84
3.5 The Cauchy transform on AD regular curves ............ 88
3.6 Curvature and Jones /J s ....................... 91
3.7 Historical remarks and further results ................ 98
3.7.1 The Cauchy transform and curvature............. 98
3.7.2 The Tl theorem........................ 98
3.7.3 The Cauchy transform on Lipschitz graphs and AD regular
curves.............................. 99
3.7.4 Application of the curvature method to other kernels .... 99
4 The capacity 7+ 103
4.1 Introduction............................... 103
4.2 Smoothing of the Cauchy kernel by mollification .......... 104
4.3 Dualization of the weak (1,1) inequality............... 107
4.4 The Denjoy conjecture......................... 110
4.5 Semiadditivity of 7+.......................... 112
4.6 Some potential theory for 7+ ..................... 115
4.6.1 A couple of technical lemmas about curvature........ 115
4.6.2 A maximum principle for curvature and for U^....... 117
4.6.3 A dual version for the capacity 7+.............. 120
4.7 The capacity 7+ of some Cantor sets................. 125
4.8 A quantitative version of Denjoy s conjecture............ 127
4.9 Relationship between 7+ and Wolff s potentials........... 129
4.10 Historical remarks and further results ................ 132
4.10.1 Denjoy s conjecture, the Davie-Øksendal dualization, and 7-)- . . 132
4.10.2 Analytic capacity and curvature ............... 132
4.10.3 The analytic capacity of the Cantor sets E(X) and Wolff s
potentials............................ 133
4.10.4 Capacities associated with the signed Riesz kernels..... 134
5 A Tò theorem of Nazarov, Treil and Volberg 137
5.1 Introduction............................... 137
5.2 The exceptional set S ......................... 138
5.3 The suppressed operators K ..................... 140
5.4 Dyadic lattices and the martingale decomposition.......... 143
5.4.1 The dyadic Carleson embedding theorem .......... 143
5.4.2 Random dyadic lattices.................... 145
5.4.3 Transit and terminal squares................. 146
5.4.4 The martingale decomposition ................ 146
5.5 Good and bad squares and functions................. 152
Contents xi
5.6 Estimates for good functions ..................... 153
5.6.1 The main lemma for good functions............. 153
5.6.2 Beginning of the proof of Lemma 5.13............ 154
5.7 Estimate of the sum S : distant squares............... 156
5.8 Estimate of the sum S4 ........................ 160
5.8.1 Splitting of S4 ......................... 160
5.8.2 Estimate of S^x via a paraproduct.............. 161
5.8.3 Estimate of S^fm ....................... 168
5.9 Estimate of 5*2 + 5 3: the diagonal term................ 172
5.10 Cotlar s inequality revisited...................... 176
5.11 The final probabilistic argument ................... 181
5.11.1 The low probability of bad squares and functions...... 181
5.11.2 The nice set G..............^ . . . ^...... 185
5.11.3 The functions Þ, tyw, and the operators K and C...... 187
5.11.4 The key estimates involving C and K............. 188
5.11.5 The final step.......................... 191
5.12 Historical remarks and further results ................ 193
5.12.1 The Tb theorem and analytic capacity............ 193
5.12.2 Other To-like theorems .................... 194
The comparability between 7 and 7+ 195
6.1 Introduction............................... 195
6.2 An argument for sets of finite length................. 196
6.3 Outline of the argument for proving that 7 » 7+.......... 200
6.4 An intermediate approximation in terms of 7+ ........... 202
6.5 Construction of the good measures p and u............. 206
6.5.1 The construction of p and v and the proof of (a) (c) .... 207
6.5.2 The exceptional set H and the proof of (e), (f)....... 208
6.5.3 Proof of (d)........................... 209
6.5.4 Proof of (g)........................... 213
6.6 Dyadic lattices and the exceptional sets H-p and T-p ........ 216
6.6.1 The construction of Hv.................... 216
6.6.2 The accretivity condition and the exceptional set Tp . . . . 217
6.6.3 The size of Hp U Tv...................... 217
6.7 Application of a Tb theorem...................... 218
6.8 The final induction argument..................... 220
6.9 Estimate of the Cauchy integral and AD regular curves ...... 221
6.9.1 Estimate of the Cauchy integral................ 221
6.9.2 Another proof of the L2 boundednnss of the Cauchy trans-
forms on AD regular curves.................. 225
6.10 Historical remarks and further results ................ 226
6.10.1 Analytic capacity........................ 226
6.10.2 Other capacities ........................ 228
xii Contents
7 Curvature and rectifiability 231
7.1 Introduction............................... 231
7.2 The quantitative version of the David-Léger theorem........ 233
7.3 The two squares condition, Jones /J s, and curvature........ 235
7.4 The sets Z, Ex and E2......................... 241
7.5 Construction of the Lipschitz graph.................. 243
7.6 E and F are close to each other.................... 253
7.7 The set E is small........................... 256
7.8 The set E2 is small........................... 258
7.8.1 The implications E2 big =» ||A ||2 big = c2{Hl IT) big . . . 258
7.8.2 The implication c2(7^1[r) big =» c2(/x) big.......... 259
7.9 Three applications........................... 276
7.9.1 A characterization of rectifiable sets in terms of pointwise
curvature............................ 276
7.9.2 A quantitative version of David s theorem.......... 277
7.9.3 Characterization of 7 in terms of measures with bounded
upper density.......................... 279
7.10 Curvature, /3 s, and rectifiability in the AD regular case...... 281
7.11 Historical remarks and further results ................ 286
7.11.1 About the results in this chapter............... 286
7.11.2 Rectifiability, uniform rectifiability and Riesz transforms in
arbitrary dimensions...................... 286
7.11.3 Other results in connection with curvature.......... 288
8 Principal values for the Cauchy transform 289
8.1 Introduction............................... 289
8.2 L2 boundedness implies existence of principal values........ 290
8.2.1 Reduction to a class of dense functions............ 290
8.2.2 Existence of principal values on Lipschitz graphs and recti-
fiable sets............................ 293
8.2.3 Plemelj formulas on Lipschitz graphs............. 296
8.2.4 Weak convergence and existence of principal values for mea-
sures with zero density..................... 300
8.2.5 The final argument for the proof of Theorem 8.1...... 307
8.3 Finiteness of the maximal Cauchy transform............. 309
8.4 Some consequences........................... 312
8.5 Historical remarks and further results ................ 314
8.5.1 Existence of principal values on rectifiable sets....... 314
8.5.2 From the analytic information to rectifiability........ 315
9 RBMO(n) and Hlatb{p) 319
9.1 Introduction............................... 319
9.2 The space RBMO(p).......................... 321
9.2.1 Introduction .......................... 321
Contents xiii
9.2.2 The coefficients KQ%R..................... 321
9.2.3 The definition of RBMO(p).................. 323
9.2.4 Equivalent definitions..................... 326
9.2.5 Boundedness of CZO s from L°°(p) to RBMO(ß)...... 330
9.2.6 Some examples......................... 334
9.3 The John-Nirenberg inequality.................... 336
9.4 The Hardy spaces HlJb{p) . ...................... 340
9.5 The duality H^l(p) - RBMO(p)................... 347
9.6 Another maximal operator and another covering theorem..... 353
9.7 The sharp maximal operator ..................... 356
9.8 Two interpolation theorems...................... 360
9.9 The Tl theorem for the Cauchy transform again.......... 365
9.10 The Tl theorem in terms of RBMO(p) and BMOp(u)....... 369
9.11 Historical remarks and further results ................ 376
9.11.1 About RBMO{p) and BMO(p)................ 376
9.11.2 Other related results...................... 376
9.11.3 To theorems........................... 378
Bibliography 381
Index 393
|
| any_adam_object | 1 |
| author | Tolsa, Xavier |
| author_facet | Tolsa, Xavier |
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| author_sort | Tolsa, Xavier |
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| dewey-sort | 3515.9 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| spelling | Tolsa, Xavier Verfasser aut Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory [Ferran Sunyer i Balaguer award winning monograph] Xavier Tolsa Cham [u.a.] Springer 2014 XIII, 396 S. Ill. txt rdacontent n rdamedia nc rdacarrier Progress in Mathematics 307 Literaturverz. S. 381 - 392 Progress in Mathematics 307 (DE-604)BV000004120 307 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027027685&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tolsa, Xavier Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory [Ferran Sunyer i Balaguer award winning monograph] Progress in Mathematics |
| title | Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory [Ferran Sunyer i Balaguer award winning monograph] |
| title_auth | Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory [Ferran Sunyer i Balaguer award winning monograph] |
| title_exact_search | Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory [Ferran Sunyer i Balaguer award winning monograph] |
| title_full | Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory [Ferran Sunyer i Balaguer award winning monograph] Xavier Tolsa |
| title_fullStr | Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory [Ferran Sunyer i Balaguer award winning monograph] Xavier Tolsa |
| title_full_unstemmed | Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory [Ferran Sunyer i Balaguer award winning monograph] Xavier Tolsa |
| title_short | Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory |
| title_sort | analytic capacity the cauchy transform and non homogeneous calderon zygmund theory ferran sunyer i balaguer award winning monograph |
| title_sub | [Ferran Sunyer i Balaguer award winning monograph] |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027027685&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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