This volume consists of twenty peer-reviewed papers from the special sessions on pseudodifferential operators and on generalized functions and asymptotics at the Eighth Congress of ISAAC held at the Peoples' Friendship University of Russia in Moscow on August 22-27, 2011. The category of papers on pseudo-differential operators contains such topics as elliptic operators assigned to diffeomorphisms of smooth manifolds, analysis on singular manifolds with edges, heat kernels and Green functions of sub-Laplacians on the Heisenberg group and Lie groups with more complexities than but closely related to the Heisenberg group, L p-boundedness of pseudo-differential operators on the torus, and pseudo-differential operators related to time-frequency analysis. The second group of papers contains various classes of distributions and algebras of generalized functions with applications in linear and nonlinear differential equations, initial value problems and boundary value problems, stochastic and Malliavin-type differential equations. This second group of papers is related to the third collection of papers via the setting of Colombeau-type spaces and algebras in which microlocal analysis is developed by means of techniques in asymptotics. The volume contains the synergies of the three areas treated and is a useful complement to its predecessors published in the same series. Read more... Elliptic theory for operators associated with diffeomorphisms of smooth manifolds / A. Savin amd B. Sternin -- The singular functions of branching edge asymptotics / B.-W. Schulze and L. Tepoyan -- The heat kernel and green function of the sub-Laplacian on the Heisenberg group / X. Duan -- Metaplectic equivalence of the hierarchical twisted Laplacian / Shahla Molahajloo, L. Rodino and M.W. Wong -- The heat kernel and green function of a sub-Laplacian on the hierarchical Heisenberg group / S. Molahajloo and M.W. Wong -- L [superscript] p-bounds for pseudo-differential operators on the torus / J. Delgado -- Multiplication properties in Gelfand-Shilov pseudo-differential calculus / J. Toft -- Operator invariance / L. Cohen -- Initial value problems in the time-frequency domain / L. Galleani -- Polycaloric distributions and the generalized heat operator / V. Catană -- Smoothing effect and Fredholm property for rirst-order hyperbolic PDEs / I. Kmit -- A note on wave-front sets of Roumieu type ultradistributions / K. Johansson, S. Pilipović, N. Teofanov and J. Toft -- Ordinary differential equations in algebras of generalized functions / E. Erlacher and M. -- Asymptotically almost periodic generalized functions / C. Bouzar and M. Taha Khalladi -- Wave equations and symmetric first-order systems in case of low regularity / C. Hanel, G. Hörmann, C. Spreitzer and R. Steinbauer -- Concept of delta-shock type solutions to systems of conservation laws and the Rankine-Hugoniot conditions / V.M. Shelkovich -- Classes of generalized functions with finite type regularities / S. Pilipović, D. Scarpalézos and J. Vindas -- The wave equation with a discontinuous coefficient depending on time only: generalized solutions and propagation of singularities / H.Geguchi, G. Hörmann and M. Oberguggenberger -- Generalized solutions of abstract stochastic problems / I.V. Melnikova and M.A. Alshanskiy -- Nonhomogeneous first-order linear Malliavin type differential equation / T. Levajković and D. Seles̆i
Author(s): Shahla Molahajloo; Stevan Pilipović; Joachim Toft; Man Wah Wong (eds.)
Series: Operator theory, advances and applications, v.231
Publisher: Birkhauser
Year: 2013
Language: English
Pages: 371
City: Basel ; New York
Tags: Математика;Дифференциальные уравнения;
Cover......Page 1
Pseudo-Differential Operators, Generalized Functions and Asymptotics......Page 4
Contents......Page 6
Preface......Page 8
Introduction......Page 10
1.1. Main definitions......Page 15
1.2. Symbols of operators......Page 16
1.4. Examples......Page 18
2.1. Isometric actions......Page 20
2.2. General actions......Page 25
3.1. Main definitions......Page 28
3.2. Pseudodifferential uniformization......Page 29
References......Page 32
Introduction......Page 36
1.1. Edge spaces and specific operator-valued symbols......Page 38
1.2. Characterization of singular functions......Page 43
2.1. Wedge spaces with branching edge asymptotics......Page 47
2.2. The Sobolev regularity of coefficients in branching edge asymptotics......Page 55
References......Page 61
1. Introduction......Page 63
2. The sub-Laplacian on H and twisted Laplacians......Page 66
3. Convolutions on the Heisenberg group and twisted convolutions......Page 70
4. Fourier–Wigner transforms of Hermite functions......Page 71
5. Wigner transforms and Weyl transforms......Page 73
6. Heat kernels of twisted Laplacians and the sub-Laplacian on H......Page 77
7. The Green functions for the twisted Laplacians and the sub-Laplacian on H......Page 81
References......Page 82
1. Introduction......Page 84
2. Hierarchical Wigner transforms......Page 86
3. A unitary equivalence......Page 88
References......Page 90
1. Introduction......Page 92
2. The hierarchical Heisenberg group......Page 94
3. Hierarchical Wigner transforms and hierarchical Weyl transforms......Page 97
4. Spectral analysis of λ-hierarchical twisted Laplacians......Page 100
5. Twisted convolutions......Page 101
6. Heat kernels for λ-hierarchical twisted Laplacians......Page 103
7. Green functions for λ-hierarchical twisted Laplacians......Page 104
8. The heat kernel for the sub-Laplacian......Page 105
9. The Green function for the sub-Laplacian......Page 106
References......Page 108
1. Introduction......Page 110
2. Basics on pseudo-differential calculus on the torus......Page 111
3.Lp (Tn) estimates......Page 115
References......Page 122
0. Introduction......Page 124
1. Preliminaries......Page 129
2. Twisted convolution on modulation spaces and Lebesgue spaces......Page 139
3. Gelfand–Shilov tempered vector spaces......Page 143
4. Schatten–von Neumann classes and pseudo-differential operators......Page 152
5. Young inequalities for weighted Schatten–von Neumann classes......Page 161
6. Some consequences......Page 169
Appendix......Page 171
References......Page 176
1. Introduction......Page 180
2. How does time invariance arise?......Page 181
3. How does the system function arise?......Page 182
4. The transform domain......Page 184
5. Relation between the system function and the invariance operator......Page 185
6.1. Series connection......Page 186
6.2. Parallel connection......Page 187
9. Summary......Page 188
9.1. Discrete case......Page 189
10.1. Example 1: LTI......Page 190
10.2.1. Transform domain......Page 192
References......Page 194
1. Introduction......Page 195
2. Transformation to the time-frequency domain......Page 196
3. Example......Page 199
References......Page 202
Introduction......Page 204
1. Some preliminaries concerning the generalized heat equation and its parametrix......Page 206
2. Polycaloric distributions and their caloric component parts......Page 208
3. The first extension for poisson formula for 2-caloric distributions......Page 212
4. The first extension for the Poisson formula for p-caloric distributions......Page 214
5. The second extension for the Poisson formula for 2-caloric distributions......Page 215
6. The second extension for the Poisson formula for p-caloric distributions......Page 216
7. The generalized iterated heat operator......Page 218
References......Page 222
1. Introduction......Page 223
2. Smoothing effect......Page 224
2.1. Classical boundary conditions......Page 227
2.2. Integral boundary conditions in age structured population models......Page 231
2.3. Dissipative boundary conditions and periodic problems......Page 233
3. Fredholm solvability of periodic problems......Page 236
References......Page 240
0. Introduction......Page 243
0.2. Test function spaces and their duals......Page 244
0.3. Fourier–Lebesgue spaces......Page 246
1. Wave-front sets of Fourier–Lebesgue type in some spaces of ultradistributions......Page 247
2. Wave-front sets for convolutions in Fourier–Lebesgue spaces......Page 250
3.1. Modulation spaces......Page 251
3.2. Wave-front sets with respect to modulation spaces......Page 254
References......Page 255
1. Introduction......Page 257
2. Notation and preliminaries......Page 258
3. Local existence and uniqueness results for ODEs......Page 261
4. A Frobenius theorem in generalized functions......Page 270
References......Page 273
1. Introduction......Page 275
2. Asymptotically almost periodic functions and distributions......Page 276
3. Almost periodic generalized functions......Page 277
4. Asymptotically almost periodic generalized functions......Page 279
References......Page 285
1. Introduction......Page 287
2. Basic notions and spaces......Page 288
3. Transformation between equations and systems......Page 290
4. Existence and uniqueness for the Cauchy problems......Page 294
References......Page 298
1.1. L∞-generalized solutions......Page 301
1.2. Generalized solution with the δ-function singularities......Page 302
2. Zero-pressure gas dynamics (1.3)......Page 303
3. New class of systems of conservation laws admitting δ-shocks......Page 306
References......Page 308
1. Introduction......Page 310
2.1. Hölder–Zygmund spaces......Page 312
3. Classes of generalized functions with finite type regularities......Page 313
4. Characterization of local regularity through association......Page 314
4.1. Characterization of Cr*,loc - the role of gk,-s......Page 315
4.2. Regularity via association......Page 316
5. Global Zygmund-type spaces and algebras......Page 318
5.1. The algebra gL∞ (Rd)......Page 319
5.2. Global Zygmund classes......Page 320
5.3. Hölder-type spaces and algebras of generalized functions......Page 323
References......Page 324
1. Introduction......Page 0
2. The Colombeau theory of generalized functions......Page 328
3. Existence and uniqueness of generalized solutions......Page 330
4. Propagation of singularities......Page 332
5. Slow scale coefficients and regularity along the refracted ray......Page 336
6. Associated distributions......Page 338
References......Page 341
1. Introduction......Page 343
2.1. Integrated semigroups......Page 344
2.3. Spaces of abstract distributions. White noise in spaces of abstract distributions......Page 345
2.4. Spaces of abstract stochastic distributions. Singular white noise......Page 346
3. Spaces of generalized functions in t and w......Page 349
References......Page 354
1. Introduction......Page 355
2. Notions and notations......Page 356
2.1. Generalized stochastic processes......Page 357
2.3. The Malliavin derivative within chaos expansion......Page 359
3. Nonhomogeneous first-order linear equation......Page 360
References......Page 370
