Singular Integral Operators

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.