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Heinonen J. — Lectures on Analysis on Metric Spaces
Heinonen J. — Lectures on Analysis on Metric Spaces

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Название: Lectures on Analysis on Metric Spaces

Автор: Heinonen J.

Аннотация:

Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order calculus. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev space theories have been instrumental in this development. The purpose of this book is to communicate some of the recent work in the area while preparing the reader to study more substantial, related articles. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is relatively recent and appears for the first time in book format. There are plenty of exercises. The book is well suited for self-study, or as a text in a graduate course or seminar. The material is relevant to anyone who is interested in analysis and geometry in nonsmooth settings.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1 edition

Год издания: 2001

Количество страниц: 141

Добавлена в каталог: 28.03.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
"Thick" families of curves      29 68
$L^{1, n}$-extension property      65
$W^{1, p}$-extension domain      40
$\lambda$-lemnia      96
(1, 1)-Poincare inequality      73 76
(1, n)-Poincare inequality, connection with n-Loewner condition      70
(1, p)-Poincare inequality      69
3-manifolds      120
4-manifold      98 121
Absolutely continuous on almost every line      38 40
Admissible functions      51
Admissible metrics      51
Ahlfors regular space      62 112
Antoine's necklace      116
Arc length parametrization      50
Ascoli's theorem      85
Assouad dimension      81 104 113 125
Assouad embedding theorem      98 112 121
Axiom of Choice      3
Banach space      8 99 120
Basic covering theorem      2 11
Besicovitch — Federer covering theorem      7 12
Bessel potentials      23
Bi-Lipschitz      8 71 78
Bi-Lipschitz embedding      99
Bi-Lipschitz embedding in Euclidean space      98 102 124
Bi-Lipschitz equivalence      110 113
bi-Lipschitz map      78 116
Borel regular      3
Bounded turning      120
Bourdon — Pajot space      67 75 99 123
Broad domain      66
Brownian motion      88
BV function      25 see
Calderon differentiability theorem      47
Cantor set      67 108 118 121
Cantor ternary set      60 62 88 92 114 121 122
Capacity      56 66
Carnot group      67 76 123
Carnot metric      67 76
Cauchy sequences      81
Chain condition      30
Chain condition in geodesic spaces      72
Characteristic function      11
Cheeger — Sobolev space      56
Coarea formula      24
Codifferential      16
Coloring      101 102
Compact-open topology      86
Complex dynamics      95 107
conformal      51 95
Conformal dimension      121
Conformal gauge      119
Conformal gauge of Cantor set, characterization of      121
Conformal gauge, flat      124
Conformal gauge, fractal      122
Conformal gauge, of one-dimensional spaces, characterization of      120
Conformal modulus      52
Conformal modulus of ring domains      52
Conformal modulus, conformal invariance of      51
Conformal modulus, historical remarks on      54
Continuum      59 64
Continuum, nondegenerate      59
Convolution      17 37
Covering      1
Covering dimension      81 see
Covering function      81
Curve      49
Directionally limited set      7
Dirichlet problem      108
Dirichlet spaces      56
Disjointed      2
distribution      14
Distribution function      12
Divergence      15
Doubling      3 98 118 119
Doubling measure      3 82 107
Doubling measure and quasisymmetrically thick sets      117
Doubling measure on quasimetric space      110
Doubling measure, deformation by      109 112
Doubling metric space      81—83
Doubling metric space, quasisymmetric invariance of      82
Egg yolk principle      93
Elliptic partial differential equations      108
Flat conformal gauge      124
Fractal conformal gauge      122
Frostman's lemma      62
Gauge, conformal      119
Gauge, metric      121
Gauge, topological      119
Geodesic      70 120
Geodesic space      31
Gromov boundary      120
Gromov hyperbolic space      120
H = W theorem      18 26 40
Hajlasz — Sobolev space      35 49 56
Hajlasz — Sobolev space, local versions of      41
Hajlasz — Sobolev space, Poincare inequality for      39
Hardy — Littlewood maximal function theorem      10 21 37
Harmonic measure      88 108
Hausdorff content      61 74
Hausdorff dimension      60 62 81 92 103 107 116
Hausdorff dimension of Loewner spaces      63
Hausdorff dimension, relation to topological dimension      62
Hausdorff dimension, ways to estimate      61
Hausdorff measure      60
Heisenberg group      8 67 76 99 123
Hilbert space      8 99
Hodge star operator      15
Hoelder continuous      89
Hoelder continuous function      18 44
Holomorphic motion      95 97
Homogeneous measure      103 107
Homogeneously dense      97
Hyperbolic geometry      88
Hyperbolic metric      96
Inner regular      3
Integrable      3
integral      3
Internal distance      66 95
Interpolation argument      12
Isometric embedding, in Banach spaces      99
Isoperimetric dimension      25
Isoperimetric inequality      24 49
Isoperimetric profile      25
John domain      66 95
Jones's Sobolev extension theorem      66 94
Julia set      88 96
Kirzsbraun's theorem      44
Klee trick      89
Kleinian group      89 95 107
Koebe distortion theorem      93
Kuratowski embedding      99
Laakso space      67 75 99 123
Lebesgue point      47
Lebesgue's Differentiation Theorem      4 10 12
Length      50
length function      50
Length space      71
Lie group      67
Limit set      88 96
Line integral      50
Linear local connectivity      64—67 94 95
Linear local connectivity of Loewner spaces      64
Lipschitz function      40 43
Lipschitz function, differentiability almost everywhere of      47
Lipschitz function, extensions of      44 48
Local Sobolev space      14
Locally integrable      5
Locally quasiconvex      57
Locally rectifiable      50
Loewner function      59 68 94
Loewner function, asymptotics of      64
Loewner space      59 63 65 67 73 75 122
Loewner space, example of      60 65
Loewner space, Hausdorff dimension of      63
Loewner space, linear local commectivity of      64
Loewner space, quasiconvexity of      64
Lower semicontinuous      45
Lusin theorem for Sobolev functions      40
Mandelbrot set      96
Mapping problem      95 97
Mass distribution principle      104 106
Maximal function      10 74
Mazur's theorem      36
McShane extension theorem      43
Mean value      5 10
Measure      3
Metric covering dimension      81 see
Metric doubling measure      113 118
Metric doubling measure and uniform disconnectivity      115
Metric gauge      121
Minimal upper gradient      58
Modulus      50
Modulus of continuity      45
Net      101 104
Newtonian spaces      56
Newtonian spaces, reflexivity of      56
Normal family      85
Normed space      7
Orthonormal basis      100
Outer regular      3
p-capacity      56
p-capacity, equality with p-modulus      57
p-modulus      50
p-modulus of ring domains      53
Pansu differentiability theorem      99
Pencil of curves      30
Poincare inequality      17 28 39 56
Poincare inequality for Lipschitz functions      75
Poincare inequality for measurable functions      75
Poincare inequality, self-improvement and      30
Porous      116
Potential estimate      20
Power quasisymmetric      89
Proper      58 70 86 120
Pullback measure      107 113
QED domain      66
Quasiconformal map      52
Quasiconformal map, local quasisymmetry of      92
Quasiconvex      57 66 70
Quasiconvex of Loewner spaces      64
Quasidisks      95
Quasiextremal distance domain      66
Quasimetric      109 118
Quasimetric, determined by doubling measures      112
Quasimoebius map      94 96
Quasiregular maps      58
Quasisymmetric      78 112 113
Quasisymmetric map      107
Quasisymmetric map and Cauchy sequences      81
Quasisymmetric map and holomorphic motions      95
Quasisymmetric map in Euclidean domains      92
Quasisymmetric map on quasimetric space      110
Quasisymmetric map, absolute continuity of      117
Quasisymmetric map, basic properties of      79
Quasisymmetric map, compactness properties of      85
Quasisymmetric map, equicontinuity of      85
Quasisymmetric map, extension of      89
Quasisymmetric map, singular      107
Quasisymmetrically embeddable, in Euclidean space      98
Quasisymmetrically thick      108 117
Quasiultrametric      110
Rademacher theorem      47 48
Rademacher — Stepanoff theorem      47
Radon measure      7 12 17
Rational numbers      103
Real hyperbolic space      65
Rectifiable      50
Rectifiable curves      60
Reflexive Banach space      56
Regular map      102
Regular space      62
Removable set for Sobolev functions      40
Removable set for Sobolev spaces      58
Restricted maximal function      11 69
Ricci curvature, nonnegative      75
Riemann sphere      107
Riemann surface      107
Riemannian manifold      8 15 51 75 92
Riesz potential      20 34 69 74
Schwartz lemma      96
Self-improvement      30
Sierpinski carpet      125
Singular integral operators      102
Slit disk      40
Slodkowski's theorem      96
Snowflake      110 121
Snowflaked metric space      78 98
Sobolev conjugate      18
Sobolev embedding theorem      18 47 65
Sobolev function, differentiability almost everywhere of      47
Sobolev function, Hajlasz inequality for      34
Sobolev inequalities      18
Sobolev inequalities, connection with isoperimetric inequalities      24
Sobolev space      15
Sobolev space of differential forms      15
Sobolev space, Hajlasz formulation      36
Sobolev — Poincare inequality      49
Starlike domain      40
Stokes theorem      16
Sub-Riemannian geometry      26
Subharmonic functions      88
Subinvariance principle      97
Teichmueller spaces      87
Tensor product      100
Topological dimension      62
Topological gauge      119
Topological gauge, dimension of      125
Transfinite induction      101
Trudinger's inequality      18
Twice-punctured plane      96
Ultrametric      110 114
Uniform convexity      36
Uniform domain      66 94 95 125
Uniform metric dimension      81
Uniformly continuous function      45
Uniformly disconnected      118
Uniformly disconnected spaces      114
Uniformly disconnected spaces, quasisymmetric invariance of      114
Uniformly perfect      88 103 112 121
Uniformly perfect, quasisymmetric invariance of      89
Uniformly perfect, quasisymmetry and      89
Upper gradient      55
Urysohn's theorem      119
Vitali covering theorem      3
Vitali space      6
Vitali — Caratheodory theorem      51
Vol'berg — Konyagin theorem      104
Volume growth      25 61 64 65
Weak (1, p)-Poincare inequality      68
Weak differential      16
Weak gradient      15 47 65
Weak partial derivative      14
Weak quasisymmetry      83
Weakly quasisymmetric      79
Well-ordering principle      101
Zorn's lemma      2 101
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