Cauchy Transform of Measures on the Unit Circle

J. A. Cima, A. Matheson, and W. T. Ross

Contents

1. Preliminaries

1. Lebesgue spaces
2. Borel measures
3. Some elementary functional analysis
4. Functional analysis on the space of measures
5. Non-tangential limits and angular derivatives
6. Poisson and conjugate Poisson integrals
7. The classical Hardy spaces
8. Interpolation and Carleson's theorem

2. The Cauchy transform as a function

1. General properties of Cauchy integrals
2. Cauchy integrals and H¹
3. Cauchy A-integrals
4. Fatou's jump theorem
5. Plemelj's formula
6. Tangential boundary behavior
7. Cauchy-Stieltjies integrals

3. The Cauchy transform as an operator

1. An early theorem of Privalov
2. Riesz's theorem
3. Bounded and vanishing mean oscillation
4. Kolmogorov's theorem
5. Weighted spaces
6. The Cauchy transform and duality
7. Best constants
8. The Hilbert transform

4. Topologies on the space of Cauchy transforms

1. The norm topology
2. The weak-* topology
3. The weak topology
4. Schauder basis

5. Which functiona are Cauchy integrals?

1. General remarks
2. A theorem of Havin
3. A theorem of Tumarkin
4. Alexandrov's characterization
5. Other representation theorems
6. Some geometric conditions

6. Multipliers and divisors

1. Multipliers and Toeplitz operators
2. Some necessary conditions
3. A theorem of Goluzina
4. Some sufficient conditions
5. The F-property
6. Multipliers and inner functions

7. Other operators on the space of Cauchy transforms

1. Some classical operators
2. The forward shift
3. The backward shift
4. Toeplitz operators
5. Composition operators
6. The Cesaro operator

8. The distribution function for Cauchy integrals

1. The Hilbert transform of a measure
2. Boole's theorem and its generalizations
3. A refinement of Boole's theorem
4. Measures on the circle
5. A theorem of Stein and Weiss

9. The backward shift on H²

1. H² as a Hilbert space
2. Beurling's theorem
3. A theorem of Douglas, Shapiro, and Shields
4. Spectral properties
5. Kernel functions
6. A density theorem
7. A theorem of Clark and Ahern
8. A basis for backward shift invariant subspaces
9. The compression of the shift

10. Clark measures

1. A quick review of notation
2. Some basic facts about Clark measures
3. Angular derivatives and point masses
4. Aleksandrov's disintegration theorem
5. Extensions of the disintegration theorem
6. Clark's theorem on perturbations
7. Some remarks on pure point spectra
8. Poltoratski's distribution theorem

11. The normalized Cauchy transform

1. Basic definition
2. Mapping properties of the normalized Cauchy transform
3. Function properties of the normalized Cauchy transform
4. A few remarks about the Borel transfomr
5. A closer look at the F-property