Analysis of and on uniformly rectifiable sets

Cover......Page 1
Title......Page 2
Table of Contents......Page 4
Preface......Page 8
Notation and Conventions......Page 10
PART I: Background Information and the Statements of the Main Results......Page 12
1.1 Review from geometric measure theory......Page 13
1.2 Review of some topics concerning singular integral operators and rectifiability......Page 17
1.3 Review of some aspects of Littlewood-Paley theory in connection with rectifiability......Page 26
1.4 Various characterizations of uniform rectifiability......Page 31
1.5 The weak geometric lemma and its relatives......Page 36
2.1 The results of Part II......Page 41
2.2 Bilateral approximation from a functorial point of view......Page 47
2.3 The results of Part III......Page 52
2.4 A rapid description of Part IV......Page 60
3.1 Cubes......Page 63
3.2 Corona decompositions......Page 65
3.3 Generalized corona decompositions......Page 73
PART II: New Geometrical Conditions Related to Uniform Rectifiability......Page 77
1.1 The weak connectedness condition......Page 79
1.2 The weaker local symmetry condition (d=1)......Page 87
1.3 Weak constant density for one-dimensional sets......Page 96
1.4 The weak 'two points on spheres' condition......Page 103
2.1 Introduction; the corona method......Page 107
2.2 Big projections in codimension 1......Page 114
2.3 Big projections in the higher codimension case......Page 120
2.4 The local convexity condition LCV......Page 130
2.5 The weaker local convexity condition WLCV......Page 136
2.6 Weak starlikeness......Page 139
2.7 Some questions about variants of the LCV and the LS......Page 141
3.1 The WHIP, the WTP, and uniform rectifiability......Page 145
3.2 The WHIP and weaker versions of the BWGL......Page 148
3.3 The weak exterior convexity condition and the GWEC......Page 151
3.4 The weak-no-mugs, weak-no-boxes, and weak-no-reels conditions......Page 157
3.5 The proof of Theorem 3.9 (part 1)......Page 164
3.6 Part 2 of the proof: The stopping-time argument......Page 175
4.1 Introduction......Page 193
4.2 Labellings......Page 197
4.3 The derivation of Theorem 4.9 from Theorem 4.31......Page 206
PART III: Applications......Page 215
1.1 Preliminaries......Page 217
1.2 Step one......Page 218
1.3 Step two......Page 222
1.4 An abstraction of  3......Page 224
2.1 Some general comments about square function estimates......Page 227
2.2 Uniform rectifiability implies the USFE when d=1......Page 229
2.3 From square function estimates to uniform rectifiability: Preliminary reductions and the plan of the proof......Page 236
2.4 The proof of Lemma 2.36......Page 239
2.5 A topological lemma......Page 242
2.6 The main step in the proof of Proposition 2.38......Page 244
2.7 The end of the proof of Proposition 2.38......Page 254
3.1 A brief review of Clifford analysis......Page 259
3.2 Clifford analysis and square function estimates......Page 261
3.3 From square functions to uniform rectifiability: Preliminary reductions......Page 262
3.4 Cauchy flatness implies rectifiability......Page 263
3.5 The analogue of Proposition 2.59......Page 266
3.6 Cauchy flatness implies weak flatness......Page 271
3.7 Weak flatness implies exterior convexity......Page 275
3.8 Some remarks about the higher-codimension case......Page 277
4.1 The direct estimates......Page 279
4.2 The converse when d = 1......Page 289
4.3 A more abstract version of the WALA......Page 303
5.1 Compactness will only get you so far......Page 307
5.2 The codimension 1 case, part 1......Page 311
5.3 A general lemma about Carleson packing conditions......Page 315
5.4 The codimension 1 case, part 2......Page 316
5.5 The weak dyadic density condition......Page 317
PART IV: Direct Arguments for Some Stability Results......Page 322
1.1 The statements......Page 323
1.2 A John-Nirenberg-Stromberg lemma for Carleson packing conditions......Page 325
1.3 Two lemmas on approximations of regular sets by (/-planes......Page 328
1.4 The proof of the theorems......Page 334
2.1 Corona decompositions revisited......Page 337
2.2 Corona constructions and Lipschitz functions......Page 338
2.4 Preliminaries......Page 346
2.5 The proof of Lemma 2.38......Page 350
References......Page 355
Table of Selected Notation......Page 359
Table of Acronyms......Page 361
Table of Theorems......Page 363
Index......Page 365