Fractional Integrals, Potentials, and Radon Transforms (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

✍ Scribed by Boris Rubin

• Publisher: Chapman and Hall/CRC
• Year: 2024
• Tongue: English
• Leaves: 565
• Edition: 2
• Category: Library

✦ Synopsis

Fractional Integrals, Potentials, and Radon Transforms, Second Edition 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. In this thoroughly revised new edition, the book aims to explore how fractional integrals occur in the study of diverse Radon type transforms in integral geometry.

Beyond some basic properties of fractional integrals in one and many dimensions, this book also contains a mathematical theory of certain important weakly singular integral equations of the first kind arising in mechanics, diffraction theory and other areas of mathematical physics. The author focuses on explicit inversion formulae that can be obtained by making use of the classical Marchaud’s approach and its generalization, leading to wavelet type representations.

New to this Edition

• Two new chapters and a new appendix, related to Radon transforms and harmonic analysis of linear operators commuting with rotations and dilations have been added.
• Contains new exercises and bibliographical notes along with a thoroughly expanded list of references.

This book is suitable for mathematical physicists and pure mathematicians researching in the area of integral equations, integral transforms, and related harmonic analysis.

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface to the Second Edition
Preface to the First Edition
Notation and Conventions
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
1.6.1. Definitions
1.6.2. Harmonic Analysis
1.6.3. Spherical Polynomials and the Funk-Hecke Formula
1.6.4. Spherical Convolutions and the Poisson Integral
1.6.5. Spherical and Bi-Spherical Coordinates
1.6.6. Jackson Kernels
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
2.6.1. Some Weighted Norm Inequalities
2.6.2. The Mellin Transform and Multipliers
2.6.3. Decreasing Fractional Integrals and Wavelet Measures
2.6.4. Fractional Derivatives on the Half-Line
2.6.5. Fractional Integrals of the Erdélyi-Kober Type
2.7. Fractional Integrals and Potentials on the Real Line
2.7.1. The Fourier Transform of Fractional Integrals
2.7.2. The Fourier Transform of Feller Potentials
2.8. Fractional Integrals of Distributions
2.8.1. Distributions on the Real Line
2.8.2. Distributions on the Positive Half-Line
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
3.6.1. The Restriction Problems
3.6.2. The Extension Problems
3.6.3. Multiplication by the Characteristic Function and Additivity
3.7. Mapping Properties in Weighted Lp and Hölder Spaces
3.7.1. Preliminaries
3.7.2. Estimates of the “Lp → Lq” Type
3.7.3. Estimates of the “Lp → Hɑ–1/p” Type
3.7.4. Estimates of the “Hμ → Hμ+ɑ” Type
4. Local Properties and the Critical Exponent ɑ = 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 Calderón Reproducing Formula
6.2. Wavelet Type Integrals with a Complex Parameter
6.3. Wavelet Type Representation of Fractional Derivatives
6.3.1. The case Re ɑ = 0
6.4. Lp-Theorems
7. Potentials on Rn
7.1. Riesz Potentials
7.1.1. Basic Properties
7.1.2. Factorization
7.1.3. Explicit Inversion Formulas
7.1.4. Approximate Inversion
7.2. Helmholtz Potentials
7.3. Bessel Potentials
7.3.1. On Metaharmonic Continuation of Functions
7.3.2. Inversion of 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
10.3.1. Formal Calculations
10.3.2. Inversion of Bɑ±φ in Weighted Lp-Spaces
10.3.3. The Case 0 < ɑ < 1
10.3.4. Inversion of Bɑc±φ
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
12.1.1. Preliminaries
12.1.2. The k-Plane Transforms of Radial Functions
12.1.3. Weighted Equalities and Existence
12.1.4. Fuglede's Formula
12.1.5. Shifted k-Plane Transforms and Intertwining Operators
12.1.6. Semyanistyi's Fractional Integrals
12.1.7. Inversion Methods
12.2. Funk Transforms on the Unit Sphere
12.2.1. Basic Properties. Averaging Operators
12.2.2. Inversion Formulas
12.2.3. The Convolution-Backprojection Method and Wavelet Transforms
12.3. Integral Geometry in the Real Hyperbolic Space
12.3.1. Preliminaries
12.3.2. Hyperbolic Convolutions and Spherical Means
12.3.3. Horospherical Radon Transforms
12.3.4. Inversion Formulas
12.3.5. The Totally Geodesic Transforms
13. Gårding-Gindikin Integrals and Radon Transforms
13.1. Some Prerequisites from Matrix Analysis
13.1.1. Gamma and Beta Functions of the Cone Pm
13.1.2. Stiefel Manifolds
13.1.3. The Laplace Transform
13.1.4. Differential Operators
13.2. Gårding-Gindikin Fractional Integrals
13.2.1. Preliminaries
13.2.2. Integrals of Integer and Half-Integer Order
13.2.3. Integrals of Locally Integrable Functions
13.2.4. Inversion Formulas
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
A. On Operators Commuting with Rotations and Dilations
A.1. Preliminaries and Main Results
A.1.1. Convolutions with Zonal Distributions
A.1.2. The Fourier Analysis
A.2. Proof of Theorem A.12
A.3. Proof of Lemma A.9
A.4. Proof of Theorem A.22
A.5. Proof of Theorems A.27 and A.28
A.5.1. The Coifman-Weiss Transference Method
A.5.2. Dimensionality Reduction
A.5.3. Multipliers
A.5.4. Transition to R2 and Completion of the Proof
Notes and Comments
Bibliography
Subject index