Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators

Award-winning monograph of the Ferran Sunyer i Balaguer Prize 1997.

This book is a self-contained exposition of the spectral theory of Toeplitz operators with piecewise continuous symbols and singular integral operators with piecewise continuous coefficients. It includes an introduction to Carleson curves, Muckenhoupt weights, weighted norm inequalities, local principles, Wiener-Hopf factorization, and Banach algebras generated by idempotents. Some basic phenomena in the field and the techniques for treating them came to be understood only in recent years and are comprehensively presented here for the first time.
The material has been polished in an effort to make advanced topics accessible to a broad readership. The book is addressed to a wide audience of students and mathematicians interested in real and complex analysis, functional analysis and operator theory.

1 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 Fredholmoperators.- 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.