Singular Integral Operators

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