Subharmonic functions, vol.1

Cover......Page 1
Series......Page 2
Title page......Page 3
Copyright page......Page 4
0. Preface......Page 5
Acknowledgements......Page 13
Contents......Page 15
1.1 Basic Results from Set Theory......Page 19
1.2.1 Semicontinuous functions......Page 22
1.2.2 The classes $C^n$ and $A$......Page 27
1.3 Convex Functions......Page 29
1.4.1 The Lebesgue integral......Page 33
1.4.2 Surface integrals......Page 36
1.4.3 Domains and their frontier surfaces......Page 39
1.4.4 Green\'s theorem......Page 40
1.5.1 Green\'s function and Poisson\'s integral......Page 43
1.5.2 The maximum principle for harmonic functions......Page 47
1.5.4 The problem of Dirichlet for a hyperball......Page 49
1.5.5 The mean-value property......Page 51
1.5.6 Harnack\'s inequality and Harnack\'s theorem......Page 53
1.5.7 Conclusion......Page 56
2.1 Definition and Simple Examples......Page 58
2.2 Jensen\'s Inequality......Page 60
2.3 Some Further Classes of Subharmonic Functions......Page 64
2.4 The Maximum Principle......Page 65
2.5 S.H. Functions and the Poisson Integral......Page 67
2.6 Perron\'s Method and the Problem of Dirichlet......Page 73
2.6.1 Harmonicity......Page 74
2.6.2 Boundary behaviour......Page 76
2.6.3 Conditions for regularity and construction of the barrier function......Page 79
2.7 Convexity Theorems......Page 81
2.7.1 Some applications......Page 85
2.7.2 Harmonic extensions......Page 88
2.8 Subordination......Page 92
3.0 Introduction......Page 99
3.1 Measure and Integration......Page 100
3.2 Linear Functional......Page 102
3.3 Construction of Lebesgue Measure and Integrals, (F.Riesz\'s Theorem)......Page 106
3.3.1......Page 108
3.3.2......Page 109
3.3.3......Page 112
3.4 Repeated Integrals and Fubinfs Theorem......Page 114
3.4.1 Convolution transforms......Page 117
3.4.2......Page 120
3.5.1......Page 122
3.5.2......Page 125
3.5.3......Page 128
3.5.4 Proof of Riesz\' s Theorem......Page 130
3.6 Harmonic Measure......Page 132
3.6.1......Page 134
3.7 The Green\'s Function and the Poisson-Jensen Formula......Page 137
3.7.1......Page 138
3.7.2......Page 139
3.7.3......Page 140
3.8 Harmonic Extensions and Least Harmonic Majorants......Page 141
3.9 Nevanlinna Theory......Page 143
3.10 Bounded Subharmonic Functions in $R^m$......Page 146
3.10.1......Page 149
3.10.2......Page 150
4.1 The Weierstrass Representation Theorem......Page 154
4.1.1......Page 156
4.1.2......Page 158
4.2 Hadamard\'s Representation Theorem......Page 160
4.2.1......Page 162
4.2.2......Page 164
4.3 Relations Between $T(r)$ and $B(r)$......Page 165
4.3.1 Two examples......Page 167
4.4 Relations Between $N(r)$ and $T(r)$......Page 169
4.5 Functions of Order Less Than One......Page 173
4.5.1 A sharp inequality connecting $N(r)$ and $B(r)$......Page 175
4.5.2......Page 176
4.5.3 The sharp bound for $\\delta(u)$, statement of results......Page 179
4.5.4 Proof of theorem 4.9......Page 184
4.5.5 Proof of theorem 4.10......Page 187
4.6 Tracts and Asymptotic Values......Page 188
4.6.1 Preliminary results......Page 189
4.6.2......Page 192
4.6.3 Components $C(K)$ in domains......Page 194
4.6.4 Tracts and growth......Page 201
4.6.5 Iversen\'s Theorem......Page 203
4.6.6 Construction of an asymptotic path......Page 205
4.6.7 Growth on asymptotic paths......Page 210
4.6.8 Three examples......Page 214
5.1 Potentials and $\\alpha$-capacity......Page 219
5.1.1 Weak convergence......Page 223
5.2 Conductor Potentials and Capacity......Page 226
5.2.1 The nature of the conductor potential......Page 229
5.3 Polar Sets......Page 234
5.4 Capacity and Hausdorff Measure......Page 238
5.4.1 The main comparison theorems......Page 243
5.4.2 An application to bounded regular functions......Page 247
5.5 The Extended Maximum or Phragmén-Lindeloef Principle......Page 250
5.5.1 Uniqueness of the conductor potential......Page 253
5.5.2 Polar sets as null sets......Page 255
5.6 Polar Sets and the Problem of Dirichlet......Page 257
5.7 Generalized Harmonic Extensions and Green\'s Function......Page 264
5.7.1 Harmonic extensions......Page 265
5.7.2 The generalized Green\'s function......Page 267
5.7.3 The symmetry property of the Green\'s function......Page 273
5.7.4 The extended Green\'s function and the Poisson-Jensen formula......Page 274
5.8 Capacitability and Strong Subadditivity......Page 276
5.8.1 Strong subadditivity......Page 277
5.8.2 Outer capacities......Page 281
5.8.3 Capacitability......Page 287
5.9 Sets where s.h. Functions Become Infinite......Page 291
References......Page 295
Index......Page 300