Trends in harmonic analysis

Cover......Page 1
Trends in Harmonic Analysis......Page 3
Preface......Page 6
Acknowledgements......Page 7
Contents......Page 8
The Shifted Wave Equation on Damek-Ricci Spaces and on Homogeneous Trees......Page 10
1 Introduction......Page 11
2 Spherical Analysis on Damek-Ricci Spaces......Page 12
3 Ásgeirsson's Mean Value Theorem and the Shifted Wave Equation on Damek-Ricci Spaces......Page 17
4 The Shifted Wave Equation on Homogeneous Trees......Page 22
References......Page 33
1 Introduction and Statement of the Results......Page 35
2 Proof of the Theorems......Page 37
References......Page 40
A Koksma-Hlawka Inequality for Simplices......Page 41
1 Introduction......Page 42
2 Parallelepipeds......Page 45
3 Simplices......Page 51
References......Page 54
1 Introduction......Page 55
2 The Continuous Bounded Cohomology of G=Isom(Hn)......Page 59
3.1 Notation and Definitions......Page 63
3.2 Transfer Maps......Page 64
3.2.2 The Transfer Map taudR:H•(N,partial N)->H•c(G,Repsilon)......Page 65
3.2.3 Commutativity of the Transfer Maps......Page 66
3.2.4 Definition of the Transfer Map for Relative Cohomology......Page 67
3.2.5 Commutativity of the Lower Square......Page 68
3.3 Properties of Vol(rho)......Page 69
4 On the Proof of Theorem 1.1......Page 70
4.1 Step 1: The Equivariant Boundary Map......Page 71
4.2 Step 2: Mapping Regular Simplices to Regular Simplices......Page 72
4.3 Step 3: The Boundary Map is an Isometry......Page 79
References......Page 82
The Algebras Generated by the Laplace Operators in a Semi-homogeneous Tree......Page 85
1 Introduction......Page 86
2 Preliminaries and Notation......Page 87
3 Commutative Algebras of Functions on the Vertices of a Semi-homogeneous Tree......Page 88
4 Commutative Algebras of Functions on the Edges of a Semi-homogeneous Tree......Page 91
References......Page 98
1 Introduction......Page 99
2.1 The Space of Configurations and the Shift Action......Page 101
2.2 Cellular Automata......Page 102
2.3 Composition of Cellular Automata......Page 104
2.5 Uniform Spaces and the Generalized Curtis-Hedlund Theorem......Page 106
2.7 Induction and Restriction of Cellular Automata......Page 108
3 Cellular Automata over Concrete Categories......Page 109
3.1 Concrete Categories......Page 110
3.2 Cellular Automata over Concrete Categories......Page 113
4 Projective Sequences of Sets......Page 117
5.1 Algebraic Subsets......Page 119
5.2 Subalgebraic Subsets......Page 120
5.3 The Subalgebraic Intersection Property......Page 121
5.4 Projective Sequences of Algebraic Sets......Page 123
6 The Closed Image Property......Page 124
7.1 Injectivity and Surjectivity in Concrete Categories......Page 126
7.2 Surjunctive Categories......Page 127
7.3 Surjunctive Groups......Page 128
7.5 Surjunctivity of Residually Finite Groups......Page 129
8.1 The Subdiagonal Intersection Property......Page 131
8.2 Reversibility of C-Cellular Automata......Page 136
References......Page 139
1 Introduction......Page 142
2 Proof of Theorem 1.1......Page 145
References......Page 153
1 Introduction......Page 154
1.1 Notation and Background......Page 156
2 Differential Operators and Symbols......Page 158
2.2 First-Order Differential Operators......Page 159
3 Distributions and Weak Differentiability......Page 160
3.1 Mollifiers and Smooth Approximation......Page 162
3.2 Integration and Differentiation......Page 171
4 Reversible Sub-Finsler Geometry......Page 173
4.1 Topologies on M......Page 177
4.2 Distance, Rectifiability and Length......Page 179
4.3 Completeness......Page 182
4.4 Subunit Vector Fields and Hörmander's Condition......Page 183
5 The Control Distance for a Differential Operator......Page 188
5.1 The Weak Differentiability of Lipschitz Functions......Page 189
5.2 Equivalent Definitions of the Control Distance......Page 193
6 The L2 Theory: Formal and Essential Self-adjointness......Page 196
7 Finite Propagation Speed......Page 197
7.1 Propagation and the Control Distance......Page 199
7.2 Second-Order Operators......Page 201
8.1 Preliminaries on Multilinear Algebra......Page 202
8.2 Riemannian Manifolds......Page 203
8.3 Hermitean Complex Manifolds......Page 204
8.4 CR Manifolds......Page 205
8.5 Sub-Riemannian Structures......Page 206
8.6 Non-Riemannian Propagation......Page 207
Construction of the Sub-Riemannian Structure......Page 208
Hörmander's Condition......Page 209
A Smooth Curve with Nonsmooth Arc-Length......Page 210
References......Page 211
1 Starting from the Unit Disc......Page 213
1.1.1 Qualitative Results for Holomorphic Functions......Page 214
1.1.2 Quantitative Results for Holomorphic Functions......Page 217
Littlewood's Question......Page 218
Question (Littlewood, 1927)......Page 219
Question (Rudin, 1979)......Page 223
Question (Rudin, 1979)......Page 225
Question h......Page 228
Question h in the unit disc......Page 229
Zygmund's Vision......Page 230
The (Classical) Dirichlet Problem......Page 231
1.2.3 The Dirichlet Problem for the Unit Disc......Page 232
An Inversion Problem for f->Pf......Page 233
Motivations......Page 234
The Classical Solution Operator......Page 235
The Generalized Solution Operator......Page 236
Regular Points for the Dirichlet Problem......Page 237
Harmonic Measure......Page 238
Analytic Boundary Subsets are omega-measurable......Page 239
A Weak Form of the Dirichlet Problem......Page 240
Korányi's Results......Page 243
D'Angelo's Domains......Page 244
The Adapted Approach Regions for CDA......Page 245
A Few Words of Explanation......Page 246
References......Page 248
1 Introduction......Page 251
2.1 Groups Acting on Rooted Trees......Page 254
2.2 Schreier Graphs......Page 255
2.3 Self-similar Groups and Automata......Page 256
2.4 Matrix Recursion......Page 257
3.1 Limit Spaces of Contracting Self-similar Groups......Page 258
3.2 Iterated Monodromy Groups......Page 261
3.3 Post-Critically Finite Self-similar Fractals......Page 264
4.1 Dirichlet Forms and Laplacians on Finite Sets......Page 265
4.2 Dirichlet Forms on Locally Compact Metric Spaces......Page 268
4.3 Harmonic Structures......Page 269
5 Examples......Page 272
5.2 The Circle R/Z via the Adding Machine......Page 273
5.3 The Basilica Fractal via the Group IMG(z2-1)......Page 275
5.4 The Dendrite Fractal via the Group IMG(z2+i)......Page 277
References......Page 280
1 Introduction......Page 282
2 Analysis at Infinity......Page 283
3 Cumulants and Edgeworth Expansion......Page 287
4 Parameters Estimates......Page 290
Appendix......Page 292
References......Page 294
1 Introduction......Page 296
2.1 Labeled Chamber Complexes......Page 297
2.3 Coxeter Complexes......Page 298
2.5 Regularity and Parameter System......Page 299
2.6 Affine Buildings......Page 300
2.7 Root Systems......Page 301
2.8 Affine Weyl Group of a Root System......Page 303
2.10 Geometric Realization of an Affine Coxeter Complex......Page 304
2.11 Extended Affine Weyl Group of R......Page 306
2.12 Automorphisms of A and D......Page 307
2.14 Subgroups of G......Page 309
2.15 Geometric Realization of an Affine Building......Page 310
2.16 Parameter System of R......Page 314
2.17 The Algebra H(C)......Page 316
2.18 Chamber and Vertex Regularity of the Building......Page 319
2.19 Partial Ordering on A......Page 321
2.21 Extended Chambers......Page 323
3.1 Sectors of A......Page 324
3.2 Maximal Boundary......Page 325
3.3 Retraction rhoxomega......Page 326
3.4 Topologies on the Maximal Boundary......Page 332
3.5 Probability Measures on the Maximal Boundary......Page 333
4.1 Walls......Page 335
4.2 The alpha-Boundary Omegaalpha......Page 337
4.3 Trees at Infinity......Page 338
4.4 Orthogonal Decomposition with Respect to a Root alpha......Page 342
4.5 Topologies on Omegaalpha......Page 343
4.7 Topologies and Probability Measures on the Trees at Infinity......Page 344
4.8 Decomposition of the Measure nux......Page 345
5.2 The Fundamental Character chi0 of A......Page 346
5.3 Probability Measures on the Boundaries......Page 351
5.4  Poisson Kernel and Poisson Transform......Page 352
6.1 The Algebra H(Delta)......Page 355
6.2 Eigenvalue of the Algebra H(Delta) Associated with a Character chi......Page 356
6.3 Preliminary Results......Page 358
6.4 W-Invariance of the Eigenvalues......Page 362
6.5 Technical Results About the Poisson Transform......Page 364
7.1 Convolution Operators on A......Page 367
7.2 The Hecke Algebra on A......Page 368
7.3 Operators Ãlambda......Page 369
7.4 Satake Isomorphism......Page 372
7.5 Characterization of the Eigenvalues of the Algebra H(Delta)......Page 373
References......Page 374
1 Introduction......Page 375
2.1 Notation......Page 376
2.2 Conformal Vector Fields......Page 378
2.3 Prolongation of the Differential Equations......Page 379
2.4 The Lie Algebra of Conformal Vector Fields and the Group of Conformal Maps......Page 382
References......Page 384
1 Introduction......Page 385
2 Stochastic Completeness vs. the Feller Property......Page 386
3 Model Manifolds and Comparison Results......Page 390
4 Ends and Further Geometric Conditions for the Feller Property......Page 394
5 Applications to Geometry and PDE's......Page 397
5.1 Isometric Immersions......Page 398
5.2 Conformal Deformations......Page 399
5.3 Compact Support Property of Bounded Solutions of PDEs......Page 400
References......Page 401
1 Introduction......Page 403
2 Proofs of the Results......Page 405
References......Page 411
1 Introduction......Page 412
2 The Inclusion H1atH1max,h......Page 416
3 The Converse Inclusion......Page 421
4 The Poisson Maximal Hardy Space......Page 423
References......Page 426
1 Introduction......Page 428
2 Proof of Theorem 1.1 for Second Order Riesz Transforms......Page 435
3 The Pointwise Control of T by M°T Fails for the Hilbert Transform......Page 436
4 Proof of Theorem 1.2 for First Order Riesz Transforms......Page 438
5 Necessary Conditions for the L2 Estimate of Tf by Tf......Page 442
6 Sufficient Conditions for the L2 Estimate of Tf by Tf......Page 446
References......Page 449