Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability

Naslovnica
Cambridge University Press, 25. velj 1999. - Broj stranica: 343
The main theme of this book is the study of geometric properties of general sets and measures in euc lidean space. Examples to which this theory applies include fractal-type objects such as strange attractors for dynamical systems, and those fractals used as models in the sciences.The author provides a firm and unified foundation for the subject and develops all the main tools used in its study, such as covering theorems, Hausdorff measures and their relations to Riesz capacities and Fourier transforms. Thus the book is essentially self-contained for graduate students in mathematics; it is primarily targetted at them and researchers.The last third of the book is devoted to the Besicovitch-Federer theory of rectifiable sets, which form in a sense the largest class of subsets of euclidean space possessing many of the properties of smooth surfaces. These sets have wide application; for example they are central in the higher-dimensional calculus of variations. Their relations to complex analysis and singular integrals are also studied.
 

Što ljudi govore - Napišite recenziju

Na uobičajenim mjestima nismo pronašli nikakve recenzije.

Sadržaj

1 General measure theory
7
Measures
8
Integrals
13
Image measures
15
Weak convergence
18
Approximate identities
19
Exercises
22
2 Covering and differentiation
23
Porosity and Hausdorff dimension
156
Exercises
158
12 The Fourier transform and its applications
159
The Fourier transform and energies
162
Distance sets
165
Borel subrings of R
166
Fourier dimension and Salem sets
168
Exercises
169

Vitalis covering theorem for the Lebesgue measure
26
Besicovitchs covering theorem
28
Vitalis covering theorem for Radon measures
34
Differentiation of measures
35
HardyLittlewood maximal function
40
Measures in infinite dimensional spaces
42
Exercises
43
3 Invariant measures Haar measure
44
Uniformly distributed measures
45
The orthogonal group
46
The Grassmannian of mplanes
48
The isometry group
52
The affine subspaces
53
4 Hausdorff measures and dimension
54
Hausdorff measures
55
Hausdorff dimension
58
Generalized Hausdorff measures
59
Cantor sets
60
Selfsimilar and related sets
65
Limit sets of Möbius groups
69
Dynamical systems and Julia sets
71
Harmonic measure
72
Exercises
73
5 Other measures and dimensions
75
Net measures
76
Packing dimensions and measures
81
Integralgeometric measures
86
Exercises
88
6 Density theorems for Hausdorff and packing measures
89
A density theorem for spherical measures
92
Densities of Radon measures
94
Density theorems for packing measures
95
Remarks related to densities
98
Exercises
99
7 Lipschitz maps
100
A Sardtype theorem
103
Hausdorff measures of level sets
104
The lower density of Lipschitz images
105
Remarks on Lipschitz maps
106
Exercises
107
8 Energies capacities and subsets of finite measure
109
Capacities and Hausdorff measures
110
Frostmans lemma in Rn
112
Dimensions of product sets
115
Weighted Hausdorff measures
117
Frostmans lemma in compact metric spaces
120
Existence of subsets with finite Hausdorff measure
121
Exercises
124
9 Orthogonal projections
126
Orthogonal projections capacities and Hausdorff dimension
127
Selfsimilar sets with overlap
134
Brownian motion
136
Exercises
138
10 Intersections with planes
139
Plane sections capacities and Hausdorff measures
142
Exercises
145
11 Local structure of sdimensional sets and measures
146
Conical densities
152
13 Intersections of general sets
171
Hausdorff dimension and capacities of intersections
177
Examples and remarks
180
Exercises
182
14 Tangent measures and densities
184
Preliminary results on tangent measures
186
Densities and tangent measures
189
suniform measures
191
Marstrands theorem
192
A metric on measures
194
Tangent measures to tangent measures are tangent measures
196
Proof of Theorem 1111
198
Remarks
200
15 Rectifiable sets and approximate tangent planes
202
mrectifiable sets
203
Linear approximation properties
205
Rectifiability and measures in cones
208
Approximate tangent planes
212
Remarks on rectifiability
214
Uniform rectifiability
215
Exercises
218
16 Rectifiability weak linear approximation and tangent measures
220
Weak linear approximation densities and projections
222
Rectifiability and tangent measures
228
Exercises
230
17 Rectifiability and densities
231
Rectifiability and density one
240
Preisss theorem
241
Rectifiability and packing measures
247
Exercises
249
18 Rectifiability and orthogonal projections
250
Remarks on projections
258
Besicovitch sets
260
Exercises
264
19 Rectifiability and analytic capacity in the complex plane
265
Analytic capacity Riesz capacity and Hausdorff measures
267
Cauchy transforms of complex measures
269
Cauchy transforms and tangent measures
273
Analytic capacity and rectifiability
275
Various remarks
276
Exercises
279
20 Rectifiability and singular integrals Basic singular integrals
281
Symmetric measures
283
Existence of principal values and tangent measures
284
Symmetric measures with density bounds
285
Existence of principal values implies rectifiability
288
Lpboundedness and weak 11 inequalities
289
A duality method for weak 11
292
A smoothing of singular integral operators
295
Kolmogorovs inequality
298
Collars inequality
299
Rectifiability implies existence of principal values
301
Exercises
304
References
305
List of notation
334
Index of terminology
337
Autorska prava

Ostala izdanja - Prikaži sve

Uobičajeni izrazi i fraze

Bibliografski podaci