Singular Integral Operators

ISBN-13: 9783642648922 / Angielski / Miękka / 2013 / 528 str.

Solomon G. Mikhlin; Siegfried Prossdorf; A. Bottcher

Opis

The present edition differs from the original German one mainly in the following addi- tional material: weighted norm inequalities for maximal functions and singular opera- tors ( 12, Chap. XI), polysingular integral operators and pseudo-differential operators ( 7, 8, Chap. XII), and spline approximation methods for solving singular integral equations ( 4, Chap. XVII). Furthermore, we added two subsections on polynomial approximation methods for singular integral equations over an interval or with dis- continuous coefficients (Nos. 3.6 and 3.7, Chap. XVII). In many places we incorporated new results which, in the vast majority, are from the last five years after publishing the German edition (note that the references are enlarged by about 150 new titles). S. G. Mikhlin wrote 7, 8, Chap. XII, and the other additions were drawn up by S. Prossdorf. We wish to express our deepest gratitude to Dr. A. Bottcher and Dr. R. Lehmann who together translated the text into English carefully and with remarkable expertise.

Spis treści

I. Basic facts from functional analysis.- § 1. Basic concepts.- § 2. Regularizatlon of operators.- § 3. Fredholm and semi-Fredholm operators on Banach spaces.- § 4. Fredholm and semi-Fredholm operators on linear topological spaces.- § 5. The symbol.- § 6. The symbol of the convolution operator.- II. The one-dimensional singular integral.- § 1. The singular integral and its simplest properties.- § 2. The boundedness of the singular integral operator on the space Lp(?).- § 3. The boundedness of the singular integral operator on the space Lp with weight.- § 4. Further properties of the singular integral operator.- 4.1. Integral operators with weak singularity.- 4.2. Two theorems on commutators.- 4.3. The Poincaré-Bertrand commutation formula.- 4.4. The singular integral operator on the space H?(?).- § 5. Operators related to the Cauchy singular integral.- 5.1. The adjoint singular integral operator.- 5.2. The singular integral with Hilbert kernel.- 5.3. The projections generated by the singular integral.- § 6. The singular integral operator on spaces of differentiable functions.- III. One-dimensional singular integral equations with continuous coefficients on closed curves.- § 1. Abstract singular operators.- 1.1. Paired operators.- 1.2. Abstract singular operators.- § 2. Singular integral operators with rational coefficients.- § 3. Singular integral operators with continuous coefficients.- § 4. Singular integral operators on the space H?(?).- § 5. Factorization of continuous functions.- 5.1. Factorization in R-algebras.- 5.2. Factorization in algebras with two norms.- 6.3. Generalized factorization of continuous functions.- § 6. Effective solution of singular integral equations with continuous coefficients.- § 7. The case of a composite curve system.- IV. One-dimensional singular integral equations with discontinuous coefficients.- § 1. Preliminaries.- 1.1. Alteration of the integration curve.- 1.2. Separation of the singularities.- § 2. Singular equations with bounded measurable coefficients.- 2.1. Necessary conditions for the Fredholm property.- 2.2. Theorems on the kernel and cokernel.- 2.3. Reduction to the case of an invertible operator.- § 3. Generalized factorization of bounded measurable functions and the effective solution singular equations.- 3.1. Generalized factorization in the space Lp(?, ?).- 3.2. Effective solution of singular equations with bounded measurable coefficients.- § 4. Singular equations with piecewise continuous coefficients on closed curves.- § 5. Singular equations with piecewise continuous coefficients on non-closed curves.- § 6. Singular equations with piecewise continuous coefficients on the real line.- § 7. Norm estimates for the singular integral operator.- V. Systems of one-dimensional singular equations.- § 1. Two theorems on operator matrices.- § 2. Systems of singular integral equations with continuous coefficients on closed curves.- § 3. Factorization of matrix functions.- 3.1. General factorization theorems.- 3.2. Canonical factorization of matrix functions.- § 4. Generalized factorization of continuous matrix functions and its application.- § 5. Systems of singular integral equations with bounded measurable coefficients.- § 6. Systems of singular integral equations with piecewise continuous coefficients.- 6.1. The case of closed curves.- 6.2. The case of non-closed curves.- 6.3. The definition of the symbol.- § 7. Productsums of singular operators with piecewise continuous coefficients.- § 8. The algebra generated by singular operators with piecewise continuous coefficients.- VI. One-dimensional singular equations with degenerate symbol.- § 1. Reduction of an operator with finite index to a Fredholm operator.- § 2. Factorization of abstract singular operators.- § 3. Some classes of differentiable functions.- § 4. Function spaces.- § 5. Singular operators with degenerate continuous coefficients.- § 6. Singular operators with degenerate piecewise continuous coefficients.- § 7. Singular operators with degenerate coefficients on non-closed curves.- § 8. Singular operators with degenerate measurable coefficients.- § 9. Singular matrix operators with degenerate coefficients.- § 10. Singular operators on the spaces C?(?) and C??(?).- § 11. Singular matrix operators with degenerate coefficients of constant rank.- VII. Some problems leading to singular integral equations.- § 1. Singular integro-differential equations.- § 2. The generalized Riemann-Hilbert-Poincaré problem.- § 3. A boundary value problem for elliptic systems of first order in the plane.- § 4. On algebraic operators.- § 5. Singular integral equations with Carleman shift.- § 6. The Tricomi equation.- VIII. Some further subsidiaries.- § 1. Stereographic projection.- § 2. Some function spaces.- § 3. Weakly singular integral operators.- § 4. On the powers of the Beltrami operator.- IX. Singular integrals of higher dimensions in spaces with a uniform metric.- § 1. Basic notions.- § 2. Singular integrals over an arbitrary integration manifold.- § 3. The Zygmund inequality.- § 4. Consequences of the Zygmund inequality.- § 5. The order of a singular integral at infinity.- § 6. Singular integrals in some other spaces with a uniform metric.- X. The symbol of higher dimensional singular integral operators.- § 1. The Fourier transform of a singular kernel. The symbol.- § 2. Expansion of the symbol into a series with respect to spherical functions.- § 3. Transformation of the symbol under a substitution of variables.- § 4. Transformation of the symbol under inversion.- § 5. A theorem on the boundedness of the singular operator.- § 6. On series with respect to spherical functions.- § 7. Differentiability properties of the symbol and the characteristic.- § 8. The symbol ring.- XI. Singular integral operators in spaces with integral metric.- § 1. An extension of the notion of a singular integral.- § 2. Criteria for boundedness in L2(Rm).- § 3. The theorem of Calderón and Zygmund.- § 4. Some further results.- § 5. Singular integrals in weighted spaces. Stein’s theorem.- § 6. Singular integrals in weighted spaces. The theorems of Plamenevski and Haikin.- § 7. A multiplication rule for symbols.- § 8. The adjoint singular operator.- § 9. Singular operators in Sobolev spaces.- § 10. Factor ring of singular operators.- § 11. Higher derivatives of the volume potential.- § 12. Weighted norm inequalities for maximal functions and singular integral operators.- XII. Multidimensional singular integral equations.- § 1. The constant symbol case.- § 2. The general case and Noether theorems.- § 3. Equivalent regularization. The index theorem.- § 4. A necessary condition for the existence of a regularizer.- § 5. Singular equations in Sobolev spaces.- § 6. Singular equations in test and in distribution spaces.- § 7. Polysingular integral operators.- § 8. Pseudo-differential operators.- XIII. Singular equations on smooth manifolds without boundary.- § 1. Manifolds.- § 2. Singular operators on manifolds. The symbol.- § 3. Singular equations in Lp(?).- § 4. On the gradient of a harmonic function.- § 5. The oblique derivative problem.- § 6. On the boundedness of singular operators in Lipschitz spaces.- § 7. Singular integral equations in Lipschitz spaces.- XIV. Systems of multidimensional singular equations.- § 1. General remarks.- § 2. The index problem. Reduction to a more special case.- § 3. Computation of the index.- § 4. The case of a two-dimensional manifold.- § 5. Elementary cases with index zero.- § 6. Problems in static elasticity theory.- XV. The localization principle. Singular operators on manifolds with boundary.- § 1. Operators of local type.- § 2. Equivalence at a point and locally Fredholm operators.- § 3. The envelope of an operator family.- § 4. A theorem on the connection between Fredholm and locally Fredholm operators.- § 5. Homogeneous operators and translation invariant operators.- § 6. Canonical singular integrals with piecewise continuous symbols.- § 7. Generalized singular integrals.- § 8. Compound generalized singular operators.- § 9. Singular integral equations in domains with boundary.- § 10. A survey on the papers by Vishik and Eskin.- XVI. Multidimensional singular equations with degenerate symbol.- § 1. Convolution equations with degenerate symbol.- 1.1. Function spaces.- 1.2. Existence and general form of the solution of the singular integral equation.- 1.3. A correctly posed problem.- § 2. Further results on convolution operators with degenerate symbol.- § 3. A multidimensional analogue of the Cauchy singular integral operator and the corresponding paired operators.- 3.1. Generalized Cauchy-Riemann systems.- 3.2. A generalized Borel-Pompeiu formula.- 3.3. Further properties of the singular integral operator S.- 3.4. Fredholm properties of certain degenerate paired operators.- 3.5. Some generalizations.- XVII. Methods for the approximate solution of one-dimensional singular integral equations.- § 1. General theorems on the convergence of projection methods.- § 2. Projection methods for the solution of abstract singular equations.- § 3. Polynomial approximation methods for the solution of singular integral equations.- 3.1. The reduction method.- 3.2. The collocation method.- 3.3. The method of mechanical quadratures.- 3.4. The method of least squares.- 3.5. The case of a non-vanishing index.- 3.6. Discontinuous coefficients.- 3.7. Singular integral equations over an interval.- § 4. Spline approximation methods for the solution of singular integral equations.- 4.1. The polygonal method.- 4.2. Collocation methods with splines of arbitrary degree.- 4.3. The Galerkin method.- 4.4. Singular integral equations over an interval.- § 5. The approximate solution of singular integral equations with degenerate symbol.- 6.1. The reduction method.- 5.2. The collocation method.- 5.3. The method of mechanical quadratures.- 5.4. The method of least squares.- § 6. The approximate solution of systems of singular integral equations.- 6.1. Polynomial approximation methods.- 6.2. Discontinuous coefficients.- 6.3. Degenerate symbols.- 6.4. Spline approximation methods.- XVIII. Approximate solution ot multidimensional singular integral equations.- § 1. Approximate computation of singular integrals.- § 2. An iteration method.- § 3. The Bubnov-Galerkin and the least squares methods.- § 4. Coordinate functions connected with spherical functions.- § 5. The case of exactly known eigenfunctions.- § 6. Approximate construction of the eigenfunctions.- § 7. Constructing the approximations and estimating them.- § 8. Applications to one-dimensional singular equations.- § 9. Application of Hermite functions.- References.- Symbols and notations.- Name index.