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Lectures on singular integral operators. Expository lectures from the CBMS regional conference held at the University of Montana, Missoula, MT (USA) from August 28-September 1, 1989. (English) Zbl 0745.42008
Regional Conference Series in Mathematics. 77. Providence, RI: American Mathematical Society (AMS). ix, 132 p. (1990).
In the 1970’s, singular integral operators concerned convolutions with honest kernels containing a singularity. The techniques used to study these operators involved Fourier transforms, maximal functions, and geometry. This changed during the 1980’s, as a much wider collection of operators was shown to fall under the theory of singular integral operators. Involved in this change was the realization that Littlewood- Paley theory could be interpreted in terms of vector-valued singular integrals and an attempt to understand the Cauchy integral on Lipschitz curves.
Today, the study of singular integral operators involves standard kernels, paraproducts, (para-) accretive functions, and the celebrated \(T(1)\) and \(T(b)\) theorems. This beautiful little book introduces all of these ideas and more. Numerous examples spread throughout the text show the breadth of applications which are now encompassed in this theory, ranging from the Hilbert transform and the Cauchy integral to square roots of accretive second order partial differential operators. Anyone interested in learning this emerging theory should read this book. This is not an easy task as the material is presented at a sophisticated level. The proofs of the major results are generally complete though not at all trivial and there are many results which are left to the reader to prove. What one does get is a context which shows were the major pieces of the theory come from and how they fit together. The bibliography is extensive and citations appear throughout the text.
One of the motivating forces for this work was the Cauchy integral. It is studied several times, as it is attacked from various points of view.
It is amazing the number of topics considered in a monograph of less than 130 pages. In addition to the ones mentioned above, the author considers Carleson measures, Calderón commutators, and spaces of homogeneous type.

MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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