Distributions and Operators

Cover\r......Page 1
Title Page\r......Page 4
Copyright Page\r......Page 5
Preface......Page 6
Contents......Page 9
Part I: Distributions and derivatives\r......Page 14
1.1 Introduction......Page 15
1.2 On the definition of distributions......Page 18
2.1 The space of test functions......Page 21
2.2 Some other function spaces......Page 28
2.3 Approximation theorems......Page 29
2.4 Partitions of unity......Page 35
Exercises for Chapter 2......Page 36
3.1 Distributions......Page 39
3.2 Rules of calculus for distributions......Page 43
3.3 Distributions with compact support......Page 48
3.4 Convolutions and coordinate changes......Page 52
3.5 The calculation rules and the weak* topology on D\'......Page 58
Exercises for Chapter 3......Page 62
Part II: Extensions and applications\r......Page 68
4.1 Realizations of di.erential operators......Page 69
4.2 Sobolev spaces......Page 73
4.3 The one-dimensional case......Page 80
4.4 Boundary value problems in higher dimensions......Page 89
Exercises for Chapter 4......Page 98
5.1 Rapidly decreasing functions......Page 107
5.2 Temperate distributions......Page 115
5.3 The Fourier transform on S\'......Page 117
5.4 Homogeneity......Page 122
5.5 Application to the Laplace operator......Page 126
5.6 Distributions associated with nonintegrable functions......Page 128
Exercises for Chapter 5......Page 132
6.1 Differential and pseudodifferential operators on R^n......Page 135
6.2 Sobolev spaces of arbitrary real order. The Sobolev theorem......Page 139
6.3 Dualities between Sobolev spaces. The Structure theorem......Page 144
6.4 Regularity theory for elliptic differential equations......Page 150
Exercises for Chapter 6......Page 156
Miscellaneous exercises (exam problems)......Page 159
Part III: Pseudodifferential operators\r......Page 174
7.1 Symbols and operators, mapping properties......Page 175
7.2 Negligible operators......Page 184
7.3 Composition rules......Page 190
7.4 Elliptic pseudodi.erential operators......Page 195
7.5 Strongly elliptic operators, the Garding inequality......Page 201
Exercises for Chapter 7......Page 205
8.1 Coordinate changes......Page 207
8.2 Operators on manifolds......Page 209
8.3 Fredholm theory, the index......Page 220
Exercises for Chapter 8......Page 227
Part IV: Boundary value problems\r......Page 230
9.1 Boundary maps for the half-space......Page 231
9.2 The Dirichlet problem for I-\\Delta on the half-space......Page 240
9.3 The Neumann problem for I-\\Delta on the half-space......Page 246
9.4 Other realizations of I-\\Delta......Page 248
9.5 Variable-coe.cient cases, higher orders, systems......Page 258
Exercises for Chapter 9......Page 260
10.1 The real formulation......Page 263
10.2 Fourier transform and Laguerre expansion of S_+......Page 278
10.3 The complex formulation......Page 287
10.4 Composition rules......Page 293
10.5 Continuity......Page 301
10.6 Elliptic {\\psi}dbo’s......Page 308
Exercises for Chapter 10......Page 312
11.1 The Calderón projector......Page 317
11.2 Application to boundary value problems......Page 329
11.3 The solution operator for the Dirichlet problem......Page 332
Exercises for Chapter 11......Page 338
Part V: Topics on Hilbert space operators\r......Page 348
12.1 Unbounded operators in Banach spaces......Page 349
12.2 Unbounded operators in Hilbert spaces......Page 352
12.3 Symmetric, selfadjoint and semibounded operators......Page 355
12.4 Operators associated with sesquilinear forms......Page 362
12.5 The Friedrichs extension......Page 371
12.6 More on variational operators......Page 373
Exercises for Chapter 12......Page 376
13.1 A universal parametrization of extensions......Page 385
13.2 The symmetric case......Page 395
13.3 Consequences for resolvents......Page 405
13.4 Boundary triplets and M-functions......Page 409
Exercises for Chapter 13......Page 414
14.1 Evolution equations......Page 417
14.2 Contraction semigroups in Banach spaces......Page 421
14.3 Contraction semigroups in Hilbert spaces......Page 428
14.4 Applications......Page 431
Exercises for Chapter 14......Page 433
Some notation and prerequisites......Page 435
Exercises for Appendix A......Page 441
B.1 Fréchet spaces......Page 443
B.2 Inductive limits of Fréchet spaces......Page 453
Exercises for Appendix B......Page 455
Function spaces on sets with smooth boundary......Page 459
Exercises for Appendix C......Page 462
Bibliography......Page 463
Index......Page 469