Advances in Harmonic Analysis and Operator Theory: The Stefan Samko Anniversary Volume

Cover......Page 1
The Stefan Samko Anniversary Volume......Page 4
Contents......Page 6
Preface......Page 8
2. Scientific origin from BVP and SIE, 1965–1974......Page 10
3.1.1. Relations between left- and right-hand sided fractional integration......Page 12
3.1.3. In collaboration with Bertram Ross......Page 13
3.2. Multidimensional FC......Page 14
4. Equations with involutive operators, 1970–1977......Page 15
5. Function spaces of fractional smoothness, influence of Steklov Mathematical Institute......Page 17
5.1. Hypersingular integrals and spaces of the type of Riesz potentials......Page 18
5.2. Potential type operators with homogeneous kernels......Page 19
5.3. Spherical HSI and potentials......Page 21
6.1.1. Approximative inverses for the fractional type operators......Page 22
6.1.3. Miscellaneous......Page 23
6.2. Equations with involutive operators, continued......Page 25
6.3. Variable Exponent Analysis: 1993–2003......Page 27
6.4. Variable Exponent Analysis in collaboration with V. Kokilashvili, 2001–present......Page 28
6.5.1. More on weighted estimates of potential operators......Page 31
6.5.2. Studies related to HSI and the range Iα(•) (Lp(•)) in case of variable exponents......Page 32
6.5.3. Morrey and Campanato spaces......Page 35
6.5.4. PDO in variable exponent setting......Page 36
6.5.5. Miscellaneous in variable exponent analysis......Page 37
7. Miscellaneous......Page 40
References......Page 44
1.1. Elements of the classification......Page 58
Ordinary fractional differential equations......Page 59
Treating problems:......Page 60
2.1. Development of fractional calculus......Page 61
2.2. Development of the theory of first-order integral equations......Page 62
2.4. Development of the theory of special functions......Page 63
3. The role of Professor S.G. Samko in the creation and development of the theory of fractional differential equations......Page 64
3.2. Abel integral equations and their generalizations......Page 65
3.5. Fractional integro-differentiation......Page 66
3.7. The theory of (one- and multidimensional) potential type operators......Page 67
References......Page 68
1. Introduction......Page 73
2. A solution formula......Page 75
3. L∞-time decay......Page 78
4. Energy flow......Page 79
References......Page 83
1. Introduction......Page 85
2. Preliminaries......Page 87
3. The main results......Page 90
4. Proofs......Page 91
5. The non-decreasing case......Page 94
References......Page 96
1. Introduction......Page 98
2. Differential operators for pseudoanalytic functions......Page 99
3. Boundary value problems and Bauer operators......Page 101
4. Poly-pseudoanalytic functions......Page 104
References......Page 105
1. Introduction......Page 107
2. Wiener-Hopf factorization......Page 108
3. AP factorization......Page 116
4. Factorization on compact abelian groups......Page 119
References......Page 123
1. Introduction......Page 129
2. Existence of least energy solutions of (1.7)......Page 133
3. Sharp estimate of CH......Page 136
4. Further remarks on CH......Page 139
5. Extensions to quasilinear equations......Page 140
References......Page 142
1. Introduction......Page 144
2. Basic equations......Page 146
3.1. Standard map: Fixed and periodic points......Page 148
3.2. Phase space at low K (stable (0,0) fixed point)......Page 149
3.3. Phase space at Kc1 < K < Kc21 (stable T = 2 antisymmetric trajectory)......Page 151
Remarks......Page 152
3.4. Phase space at K > Kc2......Page 153
3.5. Cascade of bifurcations type trajectories......Page 154
3.7. The fractional dissipative standard map (FDSM)......Page 155
4. Conclusion......Page 156
References......Page 157
1. Introduction......Page 161
2. Existence and uniqueness of the solution......Page 162
3. Dependence on the problem parameters......Page 163
4. Numerical examples......Page 168
Acknowledgment......Page 176
References......Page 177
On the Boundedness of the Fractional Maximal Operator, Riesz Potential and Their Commutators in Generalized Morrey Spaces......Page 178
1. Introduction......Page 179
2. Morrey spaces......Page 180
3. Generalized Morrey spaces......Page 181
4.1. Spanne type result......Page 182
4.2. Adams type result......Page 184
5.2. Adams type result......Page 186
6.1. Spanne type result......Page 188
6.2. Adams type result......Page 191
7.1. Spanne type result......Page 193
7.2. Adams type result......Page 196
8.1. Marcinkiewicz operator......Page 197
8.2. Fractional powers of the some analytic semigroups......Page 199
References......Page 200
1. Introduction......Page 203
2. Preliminaries results......Page 207
3.1. Existence......Page 209
3.2. Estimate for the solution......Page 212
References......Page 214
1. Introduction and statement of results......Page 217
2. Proofs......Page 226
References......Page 232
1. The Riemann boundary value problem for analytic functions......Page 234
1.1. The class Lp),θ (Γ; ω)......Page 236
1.2. The classes Wp),θ (Γ) and Kp),θ (Γ)......Page 237
1.3. One property of the operator SΓ......Page 238
1.4. Problem (1) in the class Kp),),θ (Γ) for continuous G......Page 239
1.5. On a weight function from Wp),θ (Γ)......Page 240
1.6. Problem (1) with a piecewise-continuous coefficient G(t) and gεLp),θ (Γ)......Page 244
2. The Dirichlet problem......Page 248
Acknowledgement......Page 250
References......Page 251
1. Introduction......Page 253
2. Preliminaries......Page 254
3. Power rule formulas......Page 258
4. Properties of the operators......Page 261
5. Initial value problems and exponentials......Page 263
6. Summary......Page 266
References......Page 267
1. Introduction......Page 269
2. Notation and preliminaries......Page 272
3. Formulation and analysis of an auxiliary problem......Page 276
4. A functional analytic representation theorem for the solution of problem (1.1)......Page 283
References......Page 287
1. Introduction......Page 290
2. Preliminaries......Page 291
3.1. Boundedness......Page 292
3.2. Integration by parts formulas......Page 294
3.3. A fundamental variational problem of variable fractional order......Page 295
4. Illustrative examples......Page 297
References......Page 298
Improving Bounds for Singular Operators via Sharp Reverse Holder Inequality for A∞......Page 301
1. Introduction......Page 302
2.1. Coifman–Fefferman inequalities......Page 306
2.2. Mixed A1–A strong and weak norm inequalities for commutators......Page 307
3.1. Rearrangement type estimates......Page 308
3.2. Pointwise inequalities......Page 310
4.1. Coifman–Fefferman inequalities......Page 311
4.1.1. Proof of the key rearrangement estimate......Page 312
4.2. Mixed A1-A∞ strong and weak norm inequalities for commutators......Page 315
References......Page 318
1. Introduction......Page 320
2.1. Pseudodifferential operators on R......Page 322
2.2. Mellin pseudodifferential operators......Page 324
3.1. Curves, weights, coefficients......Page 327
3.2. Simonenko local principle......Page 329
4.1. Operators of the double-layer potentials type......Page 330
4.2.1. Fredholm property......Page 334
4.3. Integral operators of the Dirichlet problem......Page 336
4.4. Integral operators of the Neumann problem......Page 340
References......Page 342
1. Introduction......Page 345
2.1. Grand Lebesgue spaces......Page 346
3. Grand grand Morrey spaces and the reduction lemma......Page 347
4.1. Maximal operator in grand grand Morrey spaces......Page 349
4.2. Singular integral operators in grand grand Morrey spaces......Page 350
References......Page 351
1. Introduction......Page 353
2.1. The Mellin transform of fractional derivatives......Page 354
2.2. Fractional calculus......Page 356
3.1. Recurrence relation for the coefficients of the series solution......Page 357
3.2. The Mellin transform method for solving the fractional Bessel equation......Page 360
4. Existence and uniqueness of solutions......Page 364
References......Page 365
1. Introduction......Page 367
2. Definitions and preliminary results......Page 369
3. Nonsingular integrals in generalized Morrey spaces......Page 372
4. The Dirichlet problem......Page 376
References......Page 381
2. Preliminaries......Page 383
3. Riesz-Thorin-Stein-Weiss theorem for Morrey spaces......Page 384
Acknowledgment......Page 387
References......Page 388