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# Integral Operators In Spaces Of Summable Functions

## byM.a. Krasnosel'skii, P.p. Zabreyko, E.i. Pustylnik

### Paperback | November 8, 2011

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The investigation of many mathematical problems is significantly simplified if it is possible to reduce them to equations involving continuous or com pletely continuous operators in function spaces. In particular, this is true for non-linear boundary value problems and for integro-differential and integral equations. To effect a transformation to equations with continuous or completely continuous operators, it is usually necessary to reduce the original problem to one involving integral equations. Here, negative and fractional powers of those unbounded differential operators which constitute 'principal parts' of the original problem, are used in an essential way. Next there is chosen or constructed a function space in which the corresponding integral oper ator possesses sufficiently good properties. Once such a space is found, the original problem can often be analyzed by applying general theorems (Fredholm theorems in the study of linear equations, fixed point principles in the study of non-linear equations, methods of the theory of cones in the study of positive solutions, etc.). In other words, the investigation of many problems is effectively divided into three independent parts: transformation to an integral equation, investi gation of the corresponding integral expression as an operator acting in function spaces, and, finally, application of general methods of functional analysis to the investigation of the linear and non-linear equations.

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Title:Integral Operators In Spaces Of Summable FunctionsFormat:PaperbackPublished:November 8, 2011Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9401015449

ISBN - 13:9789401015448

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Table of Contents

1. Linear operators in L? spaces.- 1. The space L?.- 1.1. Description of the spaces.- 1.2. Criteria for compactness.- 1.3. Continuous linear functionals and weak convergence.- 1.4. Semi-ordering in the spaces S and L?.- 1.5. Projections and bases of Haar type.- 1.6. Operators in the spaces L?.- 2. Continuous linear operators.- 2.1. Linear operators.- 2.2. Regular operators.- 2.3. The M. Riesz interpolation theorem.- 2.4. Interpolation theorems for regular operators.- 2.5. Classes of L-characteristics of linear operators.- 2.6. On a property of regular operators.- 2.7. The Marcinkiewics interpolation theorem.- 3. Compact linear operators.- 3.1. Compact linear operators.- 3.2. Compactness and adjoint operators.- 3.3. Properties of operators compact in measure.- 3.4. Interpolation properties of compactness.- 3.5. Strongly continuous linear operators.- 2. Continuity and compactness of linear integral operators.- 4. General theorems on continuity on integral operators.- 4.1. Linear integral operators.- 4.2. Regular operators.- 4.3. Example of a non-regular operator.- 4.4. The adjoint operator.- 4.5. Operators with symmetric kernels.- 4.6. Products of integral operators.- 4.7. Truncations of kernels of integral operators.- 5. General theorems on compactness of integral operators.- 5.1. Problem setting.- 5.2. Regular operators acting from Lo to L?0 and from L?0 to L1.- 5.3. Regular operators acting from L?0 to L?0 where 0 0).- 6.7. Integral operators acting from L? To c.- 6.8. ?0-Cobounded linear operators.- 6.9. Compactness of ?0-cobounded operators.- 6.10. Interpolation properties of u0-boundedness.- 6.11. On weakly compact operators in l1.- 7. Integral operators with kernels satisfying conditions of kantorovic type.- 7.1. Simplest criteria.- 7.2. Theorems with intermediate conditions.- 7.3. Lemmas.- 7.4. Applications of theorems on adjoint operators.- 7.5. Fundamental theorems.- 7.6. Conditions of 'Kantorovic' type.- 7.7. Summability of kernels of integral operators.- 8. Operators of potential type.- 8.1. Definitions.- 8.2. Simplest theorems on continuity and compactness of potentials.- 8.3. Interpolation theorem of Stein-Weiss.- 8.4. Limit theorems on continuity of potentials.- 8.5. Operators of potential type.- 8.6. The logarithmic potential.- 8.7. Iterates of operators of potential type.- 8.8. Generalizations to the case of distinct dimensions.- 8.9. Potentials with respect to non-Lebesgue measure.- 3. Fractional powers of selfadjoint operators.- 9. Splitting of linear operators.- 9.1. Square root of selfadjoint operators.- 9.2. Splitting of an operator.- 9.3. L-Characteristic of a square root.- 9.4. Representation of compact operators.- 9.5. Square root of integral operator.- 9.6. Example.- 9.7. Investigation of integral operators by means of properties of iterated kernels.- 9.8. Remark on Mercers' theorem.- 10. Fractional powers of bounded operators.- 10.1. The spectral function.- 10.2. Fractional powers of bounded selfadjoint operators.- 10.3. The fundamental theorem.- 10.4. Operators in real spaces.- 10.5. Fractional powers of compact operators.- 10.6. L-Characteristics of fractional powers of operators.- 10.7. Fractional powers of integral operators.- 11. Unbounded selfadjoint operators.- 11.1. Closed operators.- 11.2. Adjoint operators.- 11.3. Integration with respect to spectral functions.- 11.4. The fundamental theorem on spectral representation of unbounded selfadjoint operators.- 11.5. Functions of selfadjoint operators.- 11.6. Commuting selfadjoint operators.- 11.7. Integrals of operator-functions.- 11.8. Integral representation of fractional powers of an operator.- 12. Properties of fractional powers of unbounded operators.- 12.1. Problem setting.- 12.2. The moment inequality for fractional powers.- 12.3. Subordinate operators.- 12.4. Subordination of fractional powers.- 12.5. Heinz' first inequality.- 12.6. Heinz' second inequality.- 12.7. Fractional powers of projected operators.- 12.8. On a special class of selfadjoint operators.- 12.9. Theorems on splitting.- 12.10. Theorems on fractional powers.- 12.11. L-Characteristic of fractional powers.- 4. Fractional powers of operators of positive type.- 13. Semi-groups of operators.- 13.1. Vector-functions and operator-functions.- 13.2. Unbounded operators.- 13.3. Resolvents.- 13.4. Definition of a semi-group.- 13.5. Generator of a semi-group.- 13.6. Theorem of Hille-Phillips-Miyadera.- 13.7. Analytic-semi-groups.- 13.8. Estimates for the operators AnT(t).- 14. Fractional powers of positive-type operators.- 14.1. Positive-type operators.- 14.2. Negative fractional powers.- 14.3. Positive fractional powers.- 14.4. A moment inequality.- 14.5. Operators subordinate to fractional powers of a positive- type operator.- 14.6. General theorems on subordination.- 14.7. Estimates for elements of the form BA??x.- 14.8. Comparison of fractional powers of two operators.- 14.9. Fractional powers of positive-type generators.- 14.10. Compactness of fractional powers.- 14.11. Supplementary remarks.- 15. Moment inequalities and L-characteristics of fractional powers.- 15.1. Lorentz spaces.- 15.2. Linear operators.- 15.3. Interpolation theorems.- 15.4. Fundamental theorems.- 15.5. L-Characteristics of fractional powers.- 15.6. One more theorem on compactness.- 16. Fractional powers of elliptic operators.- 16.1. Elliptic differential expressions.- 16.2. Elliptic operators.- 16.3. Positive-type elliptic operators.- 16.4. Multiplicative inequalities and fractional powers of elliptic operators.- 16.5. L-Characteristics of negative fractional powers of elliptic operators.- 16.6. Further theorems.- 16.7. On integral representations of fractional powers of elliptic operators.- 5. Non-linear integral operators.- 17. The superposition operators.- 17.1. On functions which are continuous in one variable.- 17.2. Simplest properties of the superposition operator.- 17.3. Fundamental theorems.- 17.4. Examples.- 17.5. General form of L-characteristics of superposition operators.- 17.6. Uniform continuity of the superposition operator.- 17.7. Improvement of superposition operators.- 17.8. Supplementary remarks.- 18. Conditions for continuity of integral operators.- 18.1. Definitions and simple properties.- 18.2. Conditions for continuity of Uryson operators.- 18.3. General theorem on continuity of Uryson operators.- 18.4. On a property of Uryson operators.- 18.5. Regular Uryson operators.- 18.6. Special examples.- 18.7. Uryson operators with values in the space of bounded functions.- 18.8. On uniform continuity of Uryson operators.- 19. Conditions for complete continuity of an Uryson operator.- 19.1. Problem setting.- 19.2. Hammerstein operators.- 19.3. Complete continuity of regular Uryson operators acting from L0 to L?, ? ? (0, 1].- 19.4. Complete continuity of regular Uryson operators acting from L? to L?, ? > 0, 0 ? ? ? 1.- 19.5. Special criteria for complete continuity.- 19.6. On L-characteristics of Uryson operators.- 19.7. Weakening of singularities.- 19.8. On two criteria for compactness (in measure) of operators.- 19.9. Complete continuity of Uryson operators with values in L0.- 20. Differentiation of non-linear operators.- 20.1. Derivative of a non-linear operator.- 20.2. General form of the derivative of a superposition operator.- 20.3. Conditions for the differentiability of a superposition operator on the whole space.- 20.4. Sufficient criteria for the differentiability of a superposition operator.- 20.5. Differentiability of superposition operators on dense sets.- 20.6. Derivatives of Hammerstein operators.- 20.7. Derivatives of Uryson operators.- 20.8. A general theorem.- 20.9. Partial criteria for differentiability of Uryson operators.- 20.10. Differentiability of Uryson operators at distinguished points.- 20.11. Asymptotic derivatives of non-linear operators.- 20.12. On higher order derivatives.- 6. Some applications.- 21. Equations with completely continuous operators.- 21.1. Linear equations.- 21.2. On approximate solutions of equations.- 21.3. Existence of solutions of non-linear integral equations.- 21.4. Eigenfunctions of non-linear integral operators.- 22. Convergence of Fouriers' method.- 22.1. General theorems on convergence of Fouriers' method.- 22.2. Convergence of Fourier series with respect to eigenfunctions of elliptic operators.- 22.3. Fouriers' method for hyperbolic equations.- 22.4. Fouriers' method for parabolic equations.- 23. Translation operators along trajectories of differential equations.- 23.1. Linear equations.- 23.2. The Cauchy operator.- 23.3. Non-linear equations.- 23.4. Equations with unbounded non-linearities.- 23.5. The translation operator.- 23.6. Differentiability of the translation operator.- 23.7. The quasi-translation operator.- 23.8. Equations with variable operators.- 23.9. The translation operator and periodic solutions of parabolic equations.- Index of terminologies.- Index of notations.- Author index.