
Advances in Harmonic Analysis and Operator Theory
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Content
- Intro
- Advances in Harmonic Analysis and Operator Theory
- The Stefan Samko Anniversary Volume
- Contents
- Preface
- Stefan G. Samko - Mathematician, Teacher and Man
- 1. Introduction
- 2. Scientific origin from BVP and SIE, 1965-1974
- 3. Research in Fractional Calculus (FC), 1967-1996
- 3.1. One-dimensional Fractional Calculus
- 3.1.1. Relations between left- and right-hand sided fractional integration
- 3.1.2. Estimates of moduli of continuity
- 3.1.3. In collaboration with Bertram Ross
- 3.1.4. Other
- 3.2. Multidimensional FC
- 4. Equations with involutive operators, 1970-1977
- 5. Function spaces of fractional smoothness, influence of Steklov Mathematical Institute
- 5.1. Hypersingular integrals and spaces of the type of Riesz potentials
- 5.2. Potential type operators with homogeneous kernels
- 5.3. Spherical HSI and potentials
- 6. Portugal period
- after 1995
- 6.1. FC continued
- constant exponents
- 6.1.1. Approximative inverses for the fractional type operators
- 6.1.2. Local nature of Riesz potential operators
- 6.1.3. Miscellaneous
- 6.2. Equations with involutive operators, continued
- 6.3. Variable Exponent Analysis: 1993-2003
- 6.4. Variable Exponent Analysis in collaboration with V. Kokilashvili, 2001-present
- 6.5. Variable Exponent Analysis, continued: 2004-present
- 6.5.1. More on weighted estimates of potential operators
- 6.5.2. Studies related to HSI and the range Ia() (Lp()) in case of variable exponents
- 6.5.3. Morrey and Campanato spaces
- 6.5.4. PDO in variable exponent setting
- 6.5.5. Miscellaneous in variable exponent analysis
- 7. Miscellaneous
- References
- The Role of S.G. Samko in the Establishing and Development of the Theory of Fractional Differential Equations and Related Integral Operators
- 1. Main aspects of the modern theory of fractional differential equations
- 1.1. Elements of the classification
- Ordinary fractional differential equations
- Fractional partial differential equations
- 1.2. Methods of investigation
- Treating problems:
- Types of solutions:
- Methods of solution:
- 2. Basic components of investigations related to fractional differential equations
- 2.1. Development of fractional calculus
- 2.2. Development of the theory of first-order integral equations
- 2.3. Development of methods of integral transforms
- 2.4. Development of the theory of special functions
- 2.5. Development of multidimensional fractional calculus
- 3. The role of Professor S.G. Samko in the creation and development of the theory of fractional differential equations
- 3.1. Singular integral equations and boundary value problems
- 3.2. Abel integral equations and their generalizations
- 3.3. Integral equations with weak singularities
- 3.4. Convolution type integral equations
- 3.5. Fractional integro-differentiation
- 3.6. Fractional powers of operators
- 3.7. The theory of (one- and multidimensional) potential type operators
- 4. Conclusion
- Acknowledgment
- References
- Energy Flow Above the Threshold of Tunnel Effect
- 1. Introduction
- 2. A solution formula
- 3. L8-time decay
- 4. Energy flow
- References
- Some New Hardy-type Integral Inequalities on Cones of Monotone Functions
- 1. Introduction
- 2. Preliminaries
- 3. The main results
- 4. Proofs
- 5. The non-decreasing case
- Acknowledgment
- References
- On a Boundary Value Problem for a Class of Generalized Analytic Functions
- 1. Introduction
- 2. Differential operators for pseudoanalytic functions
- 3. Boundary value problems and Bauer operators
- 4. Poly-pseudoanalytic functions
- Acknowledgement
- References
- The Factorization Problem: Some Known Results and Open Questions
- 1. Introduction
- 2. Wiener-Hopf factorization
- 3. AP factorization
- 4. Factorization on compact abelian groups
- References
- A Class of Sub-elliptic Equations on the Heisenberg Group and Related Interpolation Inequalities
- 1. Introduction
- 2. Existence of least energy solutions of (1.7)
- 3. Sharp estimate of CH
- 4. Further remarks on CH
- 5. Extensions to quasilinear equations
- Appendix
- References
- New Types of Solutions of Non-linear Fractional Differential Equations
- 1. Introduction
- 2. Basic equations
- 3. Fractional attractors
- 3.1. Standard map: Fixed and periodic points
- 3.2. Phase space at low K (stable (0,0) fixed point)
- 3.3. Phase space at Kc1 & K & Kc21 (stable T = 2 antisymmetric trajectory)
- Remarks
- 3.4. Phase space at K & Kc2
- 3.5. Cascade of bifurcations type trajectories
- 3.6. More FSM's attractors
- 3.7. The fractional dissipative standard map (FDSM)
- 4. Conclusion
- Acknowledgment
- References
- Stability, Structural Stability and Numerical Methods for Fractional Boundary Value Problems
- 1. Introduction
- 2. Existence and uniqueness of the solution
- 3. Dependence on the problem parameters
- 4. Numerical examples
- 5. Conclusions
- Acknowledgment
- References
- On the Boundedness of the Fractional Maximal Operator, Riesz Potential and Their Commutators in Generalized Morrey Spaces
- 1. Introduction
- 2. Morrey spaces
- 3. Generalized Morrey spaces
- 4. Boundedness of the fractional maximal operator in generalized Morrey spaces
- 4.1. Spanne type result
- 4.2. Adams type result
- 5. Riesz potential operator in the spaces Mp,f
- 5.1. Spanne type result
- 5.2. Adams type result
- 6. Commutators of fractional maximal operatorsin the spaces Mp,f
- 6.1. Spanne type result
- 6.2. Adams type result
- 7. Commutators of Riesz potential operators in the spaces Mp,f
- 7.1. Spanne type result
- 7.2. Adams type result
- 8. Some applications
- 8.1. Marcinkiewicz operator
- 8.2. Fractional powers of the some analytic semigroups
- Acknowledgment
- References
- Existence of Solutions of a Class of Nonlinear Singular Equations in Lorentz Spaces
- 1. Introduction
- 2. Preliminaries results
- 3. Proof of Theorem 1.1
- 3.1. Existence
- 3.2. Estimate for the solution
- References
- Growth of Schrödingerian SubharmonicFunctions Admitting Certain Lower Bounds
- 1. Introduction and statement of results
- 2. Proofs
- References
- The Riemann and Dirichlet Problems with Data from the Grand Lebesgue Spaces
- 1. The Riemann boundary value problem for analytic functions
- 1.1. The class Lp),? (G
- ?)
- 1.2. The classes Wp),? (G) and Kp),? (G)
- 1.3. One property of the operator SG
- 1.4. Problem (1) in the class Kp),),? (G) for continuous G
- 1.5. On a weight function from Wp),? (G)
- 1.6. Problem (1) with a piecewise-continuous coefficient G(t) and geLp),? (G)
- 2. The Dirichlet problem
- Acknowledgement
- References
- Overview of Fractional h-difference Operators
- 1. Introduction
- 2. Preliminaries
- 3. Power rule formulas
- 4. Properties of the operators
- 5. Initial value problems and exponentials
- 6. Summary
- Acknowledgment
- References
- A Singularly Perturbed Dirichlet Problem for the Poisson Equation in a Periodically Perforated Domain. A Functional Analytic Approach
- 1. Introduction
- 2. Notation and preliminaries
- 3. Formulation and analysis of an auxiliary problem
- 4. A functional analytic representation theorem for the solution of problem (1.1)
- Appendix A. A real analyticity result for a composition operator
- Acknowledgment
- References
- Fractional Variational Calculus of Variable Order
- 1. Introduction
- 2. Preliminaries
- 3. Main results
- 3.1. Boundedness
- 3.2. Integration by parts formulas
- 3.3. A fundamental variational problem of variable fractional order
- 4. Illustrative examples
- Acknowledgment
- References
- Improving Bounds for Singular Operators via Sharp Reverse Holder Inequality for A8
- 1. Introduction
- 2. Main results
- 2.1. Coifman-Fefferman inequalities
- 2.2. Mixed A1-A strong and weak norm inequalities for commutators
- 3. Background and preliminaries
- 3.1. Rearrangement type estimates
- 3.2. Pointwise inequalities
- 3.3. Building A1 weights from duality
- 4. About the proofs
- 4.1. Coifman-Fefferman inequalities
- 4.1.1. Proof of the key rearrangement estimate
- 4.2. Mixed A1-A8 strong and weak norm inequalities for commutators
- Acknowledgement
- References
- Potential Type Operators on Weighted Variable Exponent Lebesgue Spaces
- 1. Introduction
- 2. Pseudodifferential operators
- 2.1. Pseudodifferential operators on R
- 2.2. Mellin pseudodifferential operators
- 3. Operators on slowly oscillating curves
- 3.1. Curves, weights, coefficients
- 3.2. Simonenko local principle
- 4. Potential type operators in Lp() (G, w)
- 4.1. Operators of the double-layer potentials type
- 4.2. Fredholm property and essential spectrum of operators of double-layer potentials type
- 4.2.1. Fredholm property
- 4.2.2. Essential spectrum of the operator Dg,G
- 4.3. Integral operators of the Dirichlet problem
- 4.4. Integral operators of the Neumann problem
- References
- A Note on Boundedness of Operators in Grand Grand Morrey Spaces
- 1. Introduction
- Notation
- 2. Preliminaries
- 2.1. Grand Lebesgue spaces
- 2.2. Morrey spaces
- 3. Grand grand Morrey spaces and the reduction lemma
- 4. On boundedness of operators in the grand grand Morrey spaces
- 4.1. Maximal operator in grand grand Morrey spaces
- 4.2. Singular integral operators in grand grand Morrey spaces
- Acknowledgment
- References
- Operational Calculus for Bessel's Fractional Equation
- 1. Introduction
- 2. Preliminaries
- 2.1. The Mellin transform of fractional derivatives
- 2.2. Fractional calculus
- 3. Fractional Bessel equation
- 3.1. Recurrence relation for the coefficients of the series solution
- 3.2. The Mellin transform method for solving the fractional Bessel equation
- 4. Existence and uniqueness of solutions
- Acknowledgment
- References
- The Dirichlet Problem for Elliptic Equations with VMO Coefficients in Generalized Morrey Spaces
- 1. Introduction
- 2. Definitions and preliminary results
- 3. Nonsingular integrals in generalized Morrey spaces
- 4. The Dirichlet problem
- Acknowledgment
- References
- Riesz-Thorin-Stein-Weiss Interpolation Theorem in a Lebesgue-Morrey Setting
- 1. Introduction
- 2. Preliminaries
- 3. Riesz-Thorin-Stein-Weiss theorem for Morrey spaces
- Acknowledgment
- References
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