Fractional Integrals and Derivatives: Theory and Applications

Foreword xv
Preface to the English edition xvii
Preface xix
Introduction xxiii
Notation of the main forms of fractional integrals and derivatives xxv
Brief historical outline xxvii

Chapter 1 - Fractional Integrals and Derivatives on an Interval 1
§ 1. Preliminaries 1
1.1. The spaces $H^{\lambda}$ and $H^{\lambda}(\rho)$ 1
1.2. The spaces $L_{p}$ and $L_{p}(\rho)$ 7
1.3. Some special functions 14
1.4. Integral transforms 23

§ 2. Riemann-Liouville Fractional Integrals and Derivatives 28
2.1. The Abel integral equation 29
2.2. On the solvability of the Abel equation in the space of integrable functions 30
2.3. Definition of fractional integrals and derivatives and their simple properties 33
2.4. Fractional integrals and derivatives of complex order 38
2.5. Fractional integrals of some elementary functions 40
2.6. Fractional integration and differentiation as reciprocal operations 43
2.7. Composition formulae. Connection with semigroups of operators 46

§ 3. The Fractional Integrals of Hölder and Summable Functions 53
3.1. Mapping properties in the space $H^{\lambda}$ 53
3.2. Mapping properties in the space $H_{0}^{\lambda}(\rho)$ 57
3.3. Mapping properties in the space $L_{p}$ 66
3.4. Mapping properties in the space $L_{p}(\rho)$ 70

§ 4. Bibliographical Remarks and Additional Information to Chapter 1 82
4.1. Historical notes 82
4.2. Survey of other results (relating to §§ 1--3) 84

Chapter 2 - Fractional Integrals and Derivatives on the Real Axis and Half- Axis 93
§ 5. The Main Properties of Fractional Integrals and Derivatives 93
5.1. Definitions and elementary properties 93
5.2. Fractional integrals of Hölderian functions 98
5.3. Fractional integrals of summable functions 102
5.4. The Marchaud fractional derivative 109
5.5. The finite part of integrals due to Hadamard 112
5.6. Properties of finite differences and Marchaud fractional derivatives of order $\alpha>1$ 116
5.7. Connection with fractional power of operators 120

§ 6. Representation of Function by Fractional Integrals of $L_{p}$-Functions 122
6.1. The space $I^{\alpha}(L_{p})$ 122
6.2. Inversion of fractional integrals of $L_{p}$-functions 123
6.3. Characterization of the space $I^{\alpha}(L_{p})$ 127
6.4. Sufficiency conditions for the representability of functions by fractional integrals. 131
6.5. On the integral modulus of continuity of $I^{\alpha}(L_{p})$-functions 136

§ 7. Integral Transforms of Fractional Integrals and Derivatives 137
7.1. The Fourier transform 137
7.2. The Laplace transform 140
7.3. The Mellin transform 142

§ 8. Fractional Integrals and Derivatives of Generalized Functions 145
8.1. Preliminary ideas 145
8.2. The case of the axis $R^{1}$ Lizorkin's space of test functions 146
8.3. Schwartz's approach 154
8.4. The case of the half-axis. The approach via the adjoint operator 155
8.5. McBride's spaces 157
8.6. The case of an interval 159

§ 9. Bibliographical Remarks and Additional Information to Chapter 2 160
9.1. Historical notes 160
9.2. Survey of other results (relating to §§ 5--8) 163
9.3. Tables of fractional integrals and derivatives 172

Chapter 3 - Further Properties of Fractional Integrals and Derivatives 175
§ 10. Compositions of Fractional Integrals and Derivatives with Weights 175
10.1. Compositions of two one-sided integrals with power weights 176
10.2. Compositions of two-sided integrals with power weights 189
10.3. Compositions of several integrals with power weights 191
10.4. Compositions with exponential and power-exponential weights 195

§ 11. Connection between Fractional Integrals and the Singular Operator 199
11.1. The singular operator $S$ 199
11.2. The case of the whole line 202
11.3. The case of an interval and a half-axis 204
11.4. Some other composition relations 210

§ 12. Fractional Integrals of the Potential Type 213
12.1. The real axis. The Riesz and Feller potentials 214
12.2. On the "truncation" of the Riesz potential to the half-axis 218
12.3. The case of the half-axis 221
12.4. The case of a finite interval 222

§ 13. Functions Representable by Fractional Integrals on an Interval 224
13.1. The Marchaud fractional derivative on an interval 224
13.2. Characterization of fractional integrals of functions in $L_{p}$ 229
13.3. Continuation, restriction and "sewing" of fractional integrals 234
13.4. Characterization of fractional integrals of Hölderian functions 238
13.5. Fractional integration in the union of weighted Holder spaces 246
13.6. Fractional integrals and derivatives of functions with a prescribed continuity modulus 249

§ 14. Miscellaneous Results for Fractional Integro-differentiation of Functions of a Real Variable 254
14.1. Lipschitz spaces $H_{p}^{\lambda}$ and $\tilde{H}_{p}^{\lambda}$ 254
14.2. Mapping properties of fractional integration in $H_{p}^{\lambda}$ 256
14.3. Fractional integrals and derivatives of functions which are given on the whole line and belong to $H_{p}^{\lambda}$ on every finite interval 261
14.4. Fractional derivatives of absolutely continuous functions 267
14.5. The Riesz mean value theorem and inequalities for fractional integrals and derivatives 270
14.6. Fractional integration and the summation of series and integrals 275

§ 15. The Generalized Leibniz Rule 277
15.1. Fractional integro-differentiation of analytic functions on the real axis 277
15.2. The generalized Leibniz rule 280

§ 16. Asymptotic Expansions of Fractional Integrals 285
16.1. Definitions and properties of asymptotic expansions 285
16.2. The case of a power asymptotic expansion 287
16.3. The case of a power-logarithmic asymptotic expansion 294
16.4. The case of a power-exponential asymptotic expansion 297
16.5. The asymptotic solution of Abel's equation 299

§ 17. Bibliographical Remarks and Additional Information to Chapter 3 301
17.1. Historical notes 301
17.2. Survey of other results (relating to §§ 10--16) 305

Chapter 4 - Other Forms of Fractional Integrals and Derivatives 321
§ 18. Direct Modifications and Generalizations of Riemann-Liouville Fractional Integrals 321
18.1. Erdelyi-Kober-type operators 322
18.2. Fractional integrals of a function by another function 325
18.3. Hadamard fractional integro-differentiation 329
18.4. One-dimensional modification of Bessel fractional integro-differentiation and the spaces $H^{s,p}=L_{p}^{s}$ 333
18.5. The Chen fractional integral 338
18.6. Dzherbashyan's generalized fractional integral 344

§ 19. Weyl Fractional Integrals and Derivatives of Periodic Functions 347
19.1. Definitions. Connections with Fourier series 347
19.2. Elementary properties of Weyl fractional integrals 352
19.3. Other forms of fractional integration of periodic functions 354
19.4. The coincidence of Weyl and Marchaud fractional derivatives 356
19.5. The representability of periodic functions by the Weyl fractional integral 358
19.6. Weyl fractional integration and differentiation in the space of Hölderian functions 361
19.7. Weyl fractional integrals and derivatives of periodic functions in $H_{p}^{\lambda}$ 367
19.8. The Bernstein inequality for fractional integrals of trigono-metric polynomials 368

§ 20. An Approach to Fractional Integro-differentiation via Fractional Differences (The Grünwald-Letnikov Approach) 371
20.1. Differences of a fractional order and their properties 371
20.2. Coincidence of the Grünwald-Letnikov derivative with the Marchaud derivative. The periodic case 376
20.3. Coincidence of the Grünwald-Letnikov derivative with the Marchaud derivative. The non-periodic case 382
20.4. Grünwald-Letnikov fractional differentiation on a finite interval 385

§ 21. Operators with Power-Logarithmic Kernels 388
21.1. Mapping properties in the space $H^{\lambda}$ 389
21.2. Mapping properties in the space $H_{0}^{\lambda}(\rho)$ 396
21.3. Mapping properties in the space $L_{p}$ 401
21.4. Mapping properties in the space $L_{p}(\rho)$ 404
21.5. Asymptotic expansions 411

§ 22. Fractional Integrals and Derivatives in the Complex Plane 414
22.1. Definitions and the main properties of fractional integro-differentiation in the complex plane Fractional integro-differentiation of analytic functions 420
22.3. Generalization of fractional integro-differentiation of analytic functions 426

§ 23. Bibliographical Remarks and Additional Information to Chapter 4 431
23.1. Historical notes 431
23.2. Survey of other results (relating to §§ 18--22) 436
23.3. Answers to some questions put at the Conference on Fractional Calculus (New Haven, 1974) 455

Chapter 5 - Fractional Integro-differentiation of Functions of Many Variables 457
§ 24. Partial and Mixed Integrals and Derivatives of Fractional Order 458
24.1. The multidimensional Abel integral equation 458
24.2. Partial and mixed fractional integrals and derivatives 459
24.3. The case of two variables. Tensor product of operators 463
24.4. Mapping properties of fractional integration operators in the spaces $L_{\tilde{p}}(R^{n})$ (with mixed norm) 464
24.5. Connection with a singular integral 466
24.6. Partial and mixed fractional derivatives in the Marchaud form 468
24.7. Characterization of fractional integrals of functions in $L_{\tilde{p}}(R^{2})$ 471
24.8. Integral transform of fractional integrals and derivatives 473
24.9. Lizorkin function space invariant relative to fractional integro-differentiation 475
24.10. Fractional derivatives and integrals of periodic functions of many variables 476
24.11. Grünwald-Letnikov fractional differentiation 479
24.12. Operators of the polypotential type 480

§ 25. Riesz Fractional Integro-differentiation 483
25.1 Preliminaries 484
25.2. The Riesz potential and its Fourier transform. Invariant Lizorkin space 489
25.3. Mapping properties of the operator $I^{\alpha}$ in the spaces $L_{p}(R^{n})$ and $L_{p}(R^{n};\rho)$ 494
25.4. Riesz differentiation (hypersingular integrals) 498
25.5. Unilateral Riesz potentials 502

§ 26. Hypersingular Integrals and the Space of Riesz Potentials 505
26.1. Investigation of the normalizing constants $d_{n,l}(\alpha)$ as functions of the parameter $\alpha$ 505
26.2. Convergence of the hypersingular integral for smooth functions and diminution of order $l$ to $l>2[\alpha/2]$ in the case of a non-centered difference 510
26.3. The hypersingular integral as an inverse of a Riesz potential 512
26.4. Hypersingular integrals with homogeneous characteristics 518
26.5. Hypersingular integral with a homogeneous characteristic as a convolution with the distribution 525
26.6. Representation of differential operators in partial derivatives by hypersingular integrals 527
26.7. The space $I^{\alpha}(L_{p})$ of Riesz potentials and its characterization in terms of hypersingular integrals. The space $L_{p,r}^\alpha(R^{n})$ 532

§ 27. Bessel Fractional Integro-differentiation 538
27.1. The Bessel kernel and its properties 538
27.2. Connections with Poisson, Gauss-Weierstrass and metaharmonic continuation semigroups 541
27.3. The space of Bessel potentials 543
27.4. The realization of $(E-\Delta)^{\alpha/2}$, $\alpha>0$, in terms of hypersingular integrals 547

§ 28. Other Forms of Multidimensional Fractional Integro-differentiation 554
28.1. Riesz potential with Lorentz distance (hyperbolic Riesz potentials) 555
28.2. Parabolic potentials 562
28.3. The realization of the fractional powers $\left(-\Delta_{x}+\frac{\partial}{\partial t}\right)^{\alpha/2}$ and ($\left(\operatorname{E}-\Delta_{x}+\frac{\partial}{\partial t}\right)^{\alpha/2}$, $\alpha>0$, in terms of a hypersingular integral 565
28.4. Pyramidal analogues of mixed fractional integrals and derivatives 538

§ 29. Bibliographical Remarks and Additional Information to Chapter 5 580
29.1. Historical notes 580
29.2. Survey of other results (relating to §§ 24--28) 584

Chapter 6 - Applications to Integral Equations of the First Kind with Power and Power-Logarithmic Kernels 605
§ 30. The Generalized Abel Integral Equation 606
30.1. The dominant singular integral equation 606
30.2. The generalized Abel equation on the whole axis 610
30.3. The generalized Abel equation on an interval 616
30.4. The case of constant coefficients 622

§ 31. The Noether Nature of the Equation of the First Kind with Power-Type Kernels 629
31.1. Preliminaries on Noether operators 630
31.2. The equation on the axis 634
31.3. Equations on a finite interval 646
31.4. On the stability of solutions 657

§ 32. Equations with Power-Logarithmic Kernels 659
32.1. Special Volterra functions and some of their properties 661
32.2. The solution of equations with integer non-negative powers of logarithms 664
32.3. The solution of equations with real powers of logarithms 667

§ 33. The Noether Nature of Equations of the First Kind with Power-Logarithmic Kernels 672
33.1. Imbedding theorems for the ranges of the operators $I_{\alpha+}^{\alpha,\beta}$ and $I_{b-}^{\alpha,\beta}$ 673
33.2. Connection between the operators with power-logarithmic kernels and singular operator 674
33.3. The Noether nature of equation (33.1) 681

§ 34. Bibliographical Remarks and Additional Information to Chapter 6 684
34.1. Historical notes 684
34.2. Survey of other results (relating to §§ 30--33) 687

Chapter 7 - Integral Equations of the First Kind with Special Functions as Kernels 695
§ 35. Some Equations with Homogeneous Kernels Involving Gauss and Legendre Functions 696
35.1. Equations with the Gauss function 696
35.2. Equations with the Legendre function 699

§ 36. Fractional Integrals and Derivatives as Integral Transforms 703
36.1. Definition of the $G$-transform. The spaces $\mathcal{M}_{c,\gamma}^{-1}(L)$ and $L_{2}^{(c,\gamma)}$ and their characterization 704
36.2. Existence, mapping properties and representations of the $G$-transform 709
36.3. Factorization of the $G$-transform 713
36.4. Inversion of the $G$-transform 716
36.5. The mapping properties, factorization and inversion of fractional integrals in the spaces $\mathcal{M}_{c,\gamma}^{-1}(L)$ and $I_{2}^{(c,\gamma)}$ 720
36.6. Other examples of factorization 722
36.7. Mapping properties of the $G$-transform on fractional integrals and derivatives 726
36.8. Index laws for fractional integrals and derivatives 727

§ 37. Equations with Non-Homogeneous Kernels 730
37.1. Equations with difference kernels 731
37.2. Generalized operators of Hankel and Erdelyi-Kober transforms 737
37.3. Non-convolution operators with Bessel functions in kernels 741
37.4. Equation of compositional type 746
37.5. The $W$-transform and its inversion 752
37.6. Application of fractional integrals to the inversion of the $W$-transform 758

§ 38. Applications of Fractional Integro-differentiation to the Investigation of Dual Integral Equations 761
38.1. Dual Equations 762
38.2. Triple equations 768

§ 39. Bibliographical Remarks and Additional Information to Chapter 7 772
39.1. Historical notes 772
39.2. Survey of other results (relating to §§ 35--38) 775

Chapter 8 - Applications to Differential Equations 795
§ 40. Integral Representations for Solution of Partial Differential Equations of the Second Order via Analytic Functions and Their Applications to Boundary Value Problems 795
40.1. Preliminaries 796
40.2. The representation of solutions of generalized Helmholtz two-axially symmetric equation 800
40.3. Boundary value problems for the generalized Helmholtz two-axially symmetric equation 809

§ 41. Euler-Poisson-Darboux Equation 812
41.1. Representations for solutions of the Euler-Poisson-Darboux equation 813
41.2. Classical and generalized solutions of the Cauchy problem 819
41.3. The half-homogeneous Cauchy problem in multidimensional half-space 823
41.4. The weighted Dirichlet and Neumann problems in a half-plane 826

§ 42. Ordinary Differential Equations of Fractional Order 829
42.1. The Cauchy-type problem for differential equations and systems of differential equations of fractional order of general form 830
42.2. The Cauchy-type problem for linear differential equation of fractional order 837
42.3. The Dirichlet-type problem for linear differential equation of fractional order 843
42.4. Solution of the linear differential equation of fractional order with constant coefficients in the space of generalized functions 846
42.5. The application of fractional differentiation to differential equations of integer order 849

§ 43. Bibliographical Remarks and Additional Information to Chapter 8 856
43.1. Historical notes 856
43.2. Survey of other results (relating to §§ 40--42) 858

Bibliography 873
Author Index 953
Subject Index 965
Index of Symbols 973