This book presents recent developments in the fractional calculus of functions of one and several real variables, and shows the relation of this field to a variety of areas in pure and applied mathematics.

1. Preliminaries. 1.1. Integral Inequalities and Maximal Functions. 1.2. Integral Operators with Homogeneous Kernels. 1.3. Gamma and Beta Functions. 1.4. Analyticity of Functions Represented by Integrals. 1.5. Analytic Continuation of Integrals with Power Singularity. 1.6. Spherical Harmonics and Related Topics. 1.7. The Fourier Transform and Lp-Multipliers. 1.8. Approximate Identities and Related Results. 1.9. Distributions. 1.10. The Semyanistyi-Lizorkin Spaces. 1.11. Some Useful Integrals. 2. Basics of One-Dimensional Fractional Integration. 2.1. Definitions and Simplest Properties. 2.2. Fractional Derivatives and Abel’s Integral Equation. 2.3. Mapping Properties on Lp- and Hölder Spaces. Preliminaries. 2.4. Integrals of the Potential Type. 2.5. Factorization Formulas. 2.6. Fractional Integrals on the Half-Line. 2.7. Fractional Integrals and Potentials on the Real Line. 2.8. Fractional Integrals of Distributions. 3. Comparison of Ranges and Mapping Properties. 3.1. Singular Integrals in the Spaces with a Power Weight. 3.2. The Case of a Finite Interval. 3.3. The Case of a Half-Line. 3.4. The Case of the Entire Real Line. 3.5. On the Ranges of Riesz Potentials. 3.6. Restriction and Extension. 3.7. Mapping Properties in Weighted Lp and Hölder Spaces. 4. Local Properties and the Critical Exponent α = 1/p. 4.1. Some Local Estimates. 4.2. The Relationship Between the Left- and Right-Sided Integrals. 4.3. The BMO Approach. 4.4. The Spaces Defined by Asymptotics of the Norm. 4.5. The Spaces of the Local Type. 5. Marchaud’s Method. 5.1. The Generalized Finite Differences. 5.2. Analytic Continuation via Finite Differences. 5.3. Marchaud’s Derivatives in the Semyanistyi-Lizorkin Space. 5.4. More General Function Spaces. 5.5. Fractional Integrals of the Pure Imaginary Order. 5.6. A Generalization of Marchaud’s Method. 6. Fractional Integrals and Wavelet Transforms. 6.1. On the Calder´on Reproducing Formula. 6.2. Wavelet Type Integrals with a Complex Parameter. 6.3. Wavelet Type Representation of Fractional Derivatives. 6.4. Lp-Theorems. 7. Potentials on Rn. 7.1. Riesz Potentials. 7.2. Helmholtz Potentials. 7.3. Bessel Potentials. 8. One-Sided Riesz Potentials. 8.1. Definitions and Basic Properties. 8.2. Inversion Formulas. 8.3. Restriction and Extension. 8.4. Factorization Formula and Relations Between Potentials. 8.5. Inversion of Riesz Potentials on a Half-Space. 9. One-Sided Helmholtz Potentials. 9.1. Kernels of the Poisson Type. 9.2. Some Properties of the One-Sided Helmholtz Potentials. 9.3. Inversion Formulas. 9.4. Restriction and Extension. 9.5. Factorization and Further Properties. 9.6. Inversion of the Helmholtz Potentials on a Half-Space. 10. Ball Fractional Integrals. 10.1. Definitions, Mapping Properties, and Factorization. 10.2. Harmonic Analysis. 10.3. Inversion Formulas. 10.4. Traces on the Spheres. 10.5. The Restriction Problem. 10.6. Inversion of Riesz Potentials over the Ball and its Exterior. 11. Fractional Integrals on the Unit Sphere. 11.1. Approximate Identities. 11.2. Inversion of the Spherical Riesz Potentials. 11.3. Spherical Potentials and Poisson Integrals. 11.4. Spherical Wavelet Transforms. 12. Fractional Integrals in Integral Geometry. 12.1. The k-Plane Transforms on Rn . 12.2. Funk Transforms on the Unit Sphere. 12.3. Integral Geometry in the Real Hyperbolic Space. 13. Garding-Gindikin Integrals and Radon Transforms. 13.1. Some Prerequisites from Matrix Analysis. 13.2. Garding-Gindikin Fractional Integrals. 13.3. Matrix Planes and Radon Transforms. 13.4. Radon Transforms of Radial Functions. 13.5. The General Case. 13.6. Inversion of Radon Transforms. Appendix: On Operators Commuting with Rotations and Dilations.