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Inhaltsangabe1 Carleson curves.- 1.1 Definitions and examples.- 1.2 Growth of the argument.- 1.3 Seifullayev bounds.- 1.4 Submultiplicative functions.- 1.5 The W transform.- 1.6 Spirality indices.- 1.7 Notes and comments.- 2 Muckenhoupt weights.- 2.1 Definitions.- 2.2 Power weights.- 2.3 The logarithm of a Muckenhoupt weight.- 2.4 Symmetric and periodic reproduction.- 2.5 Portions versus arcs.- 2.6 The maximal operator.- 2.7 The reverse Hölder inequality.- 2.8 Stability of Muckenhoupt weights.- 2.9 Muckenhoupt condition and W transform.- 2.10 Oscillating weights.- 2.11 Notes and comments.- 3 Interaction between curve and weight.- 3.1 Moduli of complex powers.- 3.2 U and V transforms.- 3.3 Muckenhoupt condition and U transform.- 3.4 Indicator set and U transform.- 3.5 Indicator functions.- 3.6 Indices of powerlikeness.- 3.7 Shape of the indicator functions.- 3.8 Indicator functions of prescribed shape.- 3.9 Notes and comments.- 4 Boundedness of the Cauchy singular integral.- 4.1 The Cauchy singular integral.- 4.2 Necessary conditions for boundedness.- 4.3 Special curves and weights.- 4.4 Brief survey of results on general curves and weights.- 4.5 Composing curves and weights.- 4.6 Notes and comments.- 5 Weighted norm inequalities.- 5.1 Again the maximal operator.- 5.2 The Calderón-Zygmund decomposition.- 5.3 Cotlar's inequality.- 5.4 Good ? inequalities.- 5.5 Modified maximal operators.- 5.6 The maximal singular integral operator.- 5.7 Lipschitz curves.- 5.8 Measures in the plane.- 5.9 Cotlar's inequality in the plane.- 5.10 Maximal singular integrals in the plane.- 5.11 Approximation by Lipschitz curves.- 5.12 Completing the puzzle.- 5.13 Notes and comments.- 6 General properties of Toeplitz operators.- 6.1 Smirnov classes.- 6.2 Weighted Hardy spaces.- 6.3 Fredholm operators.- 6.4 Toeplitz operators.- 6.5 Adjoints.- 6.6 Two basic theorems.- 6.7 Hankel operators.- 6.8 Continuous symbols.- 6.9 Classical Toeplitz matrices.- 6.10 Separation of discontinuities.- 6.11 Localization.- 6.12 Wiener-Hopf factorization.- 6.13 Notes and comments.- 7 Piecewise continuous symbols.- 7.1 Local representatives.- 7.2 Fredholm criterion.- 7.3 Leaves and essential spectrum.- 7.4 Metamorphosis of leaves.- 7.5 Logarithmic leaves.- 7.6 General leaves.- 7.7 Index and spectrum.- 7.8 Semi-Fredholmness.- 7.9 Notes and comments.- 8 Banach algebras.- 8.1 General theorems.- 8.2 Operators of local type.- 8.3 Algebras generated by idempotents.- 8.4 An N projections theorem.- 8.5 Algebras associated with Jordan curves.- 8.6 Notes and comments.- 9 Composed curves.- 9.1 Extending Carleson stars.- 9.2 Extending Muckenhoupt weights.- 9.3 Operators on flowers.- 9.4 Local algebras.- 9.5 Symbol calculus.- 9.6 Essential spectrum of the Cauchy singular integral.- 9.7 Notes and comments.- 10 Further results.- 10.1 Matrix case.- 10.2 Index formulas.- 10.3 Kernel and cokernel dimensions.- 10.4 Spectrum of the Cauchy singular integral.- 10.5 Orlicz spaces.- 10.6 Mellin techniques.- 10.7 Wiener-Hopf integral operators.- 10.8 Zero-order pseudodifferential operators.- 10.9 Conformal welding and Haseman's problem.- 10.10 Notes and comments.

Winner of the Sunyer Prize in 1997, this volume presents current developments in an active area of mathematics - real and complex analysis, functional analysis and operator theory. It is self-contained in order to make the material accessible to graduate students as well as researchers.