Author | Gohberg, Israel. author |
---|---|
Title | One-Dimensional Linear Singular Integral Equations [electronic resource] : Volume II General Theory and Applications / by Israel Gohberg, Naum Krupnik |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1992 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8602-4 |
Descript | 232 p. online resource |
6 Preliminaries -- 6.1 The operator of singular integration -- 6.2 The space Lp(?,?) -- 6.3 Singular integral operators -- 6.4 The spaces $$L_{p}̂{ + }(\Gamma ,\rho ),L_{p}̂{ - }(\Gamma ,\rho ) and \mathop{{L_{p}̂{ - }}}\limitŝ{̂\circ } (\Gamma ,\rho )$$ -- 6.5 Factorization -- 6.6 One-sided invertibility of singular integral operators -- 6.7 Fredholm operators -- 6.8 The local principle for singular integral operators -- 6.9 The interpolation theorem -- 7 General theorems -- 7.1 Change of the curve -- 7.2 The quotient norm of singular integral operators -- 7.3 The principle of separation of singularities -- 7.4 A necessary condition -- 7.5 Theorems on kernel and cokernel of singular integral operators -- 7.6 Two theorems on connections between singular integral operators -- 7.7 Index cancellation and approximative inversion of singular integral operators -- 7.8 Exercises -- Comments and references -- 8 The generalized factorization of bounded measurable functions and its applications -- 8.1 Sketch of the problem -- 8.2 Functions admitting a generalized factorization with respect to a curve in Lp(?, ?) -- 8.3 Factorization in the spaces Lp(?, ?) -- 8.4 Application of the factorization to the inversion of singular integral operators -- 8.5 Exercises -- Comments and references -- 9 Singular integral operators with piecewise continuous coefficients and their applications -- 9.1 Non-singular functions and their index -- 9.2 Criteria for the generalized factorizability of power functions -- 9.3 The inversion of singular integral operators on a closed curve -- 9.4 Composed curves -- 9.5 Singular integral operators with continuous coefficients on a composed curve -- 9.6 The case of the real axis -- 9.7 Another method of inversion -- 9.8 Singular integral operators with regel functions coefficients -- 9.9 Estimates for the norms of the operators P?, Q? and S? -- 9.10 Singular operators on spaces H?o(?, ?) -- 9.11 Singular operators on symmetric spaces -- 9.12 Fredholm conditions in the case of arbitrary weights -- 9.13 Technical lemmas -- 9.14 Toeplitz and paired operators with piecewise continuous coefficients on the spaces lp and ?p -- 9.15 Some applications -- 9.16 Exercises -- Comments and references -- 10 Singular integral operators on non-simple curves -- 10.1 Technical lemmas -- 10.2 A preliminary theorem -- 10.3 The main theorem -- 10.4 Exercises -- Comments and references -- 11 Singular integral operators with coefficients having discontinuities of almost periodic type -- 11.1 Almost periodic functions and their factorization -- 11.2 Lemmas on functions with discontinuities of almost periodic type -- 11.3 The main theorem -- 11.4 Operators with continuous coefficients โ the degenerate case -- 11.5 Exercises -- Comments and references -- 12 Singular integral operators with bounded measurable coefficients -- 12.1 Singular operators with measurable coefficients in the space L2(?) -- 12.2 Necessary conditions in the space L2(?) -- 12.3 Lemmas -- 12.4 Singular operators with coefficients in ?p(?). Sufficient conditions -- 12.5 The Helson-Szegรถ theorem and its generalization -- 12.6 On the necessity of the condition a ? Sp -- 12.7 Extension of the class of coefficients -- 12.8 Exercises -- Comments and references -- 13 Exact constants in theorems on the boundedness of singular operators -- 13.1 Norm and quotient norm of the operator of singular integration -- 13.2 A second proof of Theorem 4.1 of Chapter 12 -- 13.3 Norm and quotient norm of the operator S? on weighted spaces -- 13.4 Conditions for Fredholmness in spaces Lp(?, ?) -- 13.5 Norms and quotient norm of the operator aI + bS? -- 13.6 Exercises -- Comments and references -- References