Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators

Naslovnica
Springer Science & Business Media, 1997 - Broj stranica: 397

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.

 

Sadržaj

Carleson curves
1
12 Growth of the argument
8
13 Seifullayev bounds
11
14 Submultiplicative functions
13
15 The W transform
15
16 Spirality indices
18
17 Notes and comments
26
Muckenhoupt weights
27
62 Weighted Hardy spaces
207
63 Fredholm operators
211
64 Toeplitz operators
213
65 Adjoints
215
66 Two basic theorems
217
67 Hankel operators
219
68 Continuous symbols
221
69 Classical Toeplitz matrices
222

22 Power weights
30
23 The logarithm of a Muckenhoupt weight
32
24 Symmetric and periodic reproduction
35
25 Portions versus arcs
39
26 The maximal operator
44
27 The reverse Holder inequality
48
28 Stability of Muckenhoupt weights
56
29 Muckenhoupt condition and W transform
59
210 Oscillating weights
66
211 Notes and comments
68
Interaction between curve and weight
71
32 U and V transforms
73
33 Muckenhoupt condition and U transform
79
34 Indicator set and U transform
84
35 Indicator functions
90
36 Indices of powerlikeness
97
37 Shape of the indicator functions
101
38 Indicator functions of prescribed shape
105
39 Notes and comments
114
Boundedness of the Cauchy singular integral
117
42 Necessary conditions for boundedness
123
43 Special curves and weights
130
44 Brief survey of results on general curves and weights
137
45 Composing curves and weights
139
46 Notes and comments
144
Weighted norm inequalities
145
52 The CalderonZygmund decomposition
149
53 Cotlars inequality
151
54 Good A inequalities
157
55 Modified maximal operators
160
56 The maximal singular integral operator
165
57 Lipschitz curves
171
58 Measures in the plane
184
59 Cotlars inequality in the plane
187
510 Maximal singular integrals in the plane
190
511 Approximation by Lipschitz curves
194
512 Completing the puzzle
198
513 Notes and comments
199
General properties of Toeplitz operators
203
61 Smirnov classes
204
610 Separation of discontinuities
225
611 Localization
226
612 WienerHopf factorization
228
613 Notes and comments
231
Piecewise continuous symbols
235
72 Fredholm criterion
240
73 Leaves and essential spectrum
242
74 Metamorphosis of leaves
243
75 Logarithmic leaves
247
76 General leaves
252
77 Index and spectrum
258
78 SemiFredholmness
261
79 Notes and comments
265
Banach algebras
267
82 Operators of local type
270
83 Algebras generated by idempotents
273
84 An N projections theorem
275
85 Algebras associated with Jordan curves
288
86 Notes and comments
297
Composed curves
301
92 Extending Muckenhoupt weights
304
93 Operators on flowers
316
94 Local algebras
320
95 Symbol calculus
326
96 Essential spectrum of the Cauchy singular integral
329
97 Notes and comments
330
Further results
335
102 Index formulas
338
103 Kernel and cokernel dimensions
343
104 Spectrum of the Cauchy singular integral
344
105 Orlicz spaces
347
106 Mellin techniques
353
107 WienerHopf integral operators
365
108 Zeroorder pseudodifferential operators
370
109 Conformal welding and Hasemans problem
372
1010 Notes and comments
377
Bibliography
379
Index
391
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