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Singular integral operators. Extended and partly modified translation from the German by Albrecht Böttcher and Reinhard Lehmann. (English) Zbl 0612.47024
Berlin etc.: Springer-Verlag. 528 p. DM 98.00 (Orig. Akademie-Verlag, Berlin) (1986).

The book is translated with some modifications from the German edition (1980; Zbl 0442.47027). It remains to indicate only the changes, incorporated by the translation.

In Chapter XI the new § 12 deals with the recent results on weighted norm inequalities for one-dimensional singular integral operators, due to R. Hunt. B. Muckenhoupt, R. Wheeden.

In Chapter XII the new § 7 is added, where polysingular integral operators with continuous coefficients are considered in the space ${L}_{p}\left({ℝ}^{2}\right)$ (results of I. Simonenko and V. Pilidi). In the same chapter § 8 deals with some basic notions for pseudo-differential operators in Sobolev-spaces ${W}_{2}^{\left(1\right)}\left({ℝ}^{m}\right)$ (the boundedness, composition formula, parametrix,...).

In Chapter XVII, § 3 two new subsections (3.6 and 3.7) deal with the polynomial approximation method for singular integral equations over a finite interval with discontinuous coefficients. In the same chapter the new § 4 describes the spline collocation method for a one-dimensional singular integral equation.

In many other places new results can be found that have appeared within the 5 years after the first edition. References are extended on 150 new items.

Reviewer: R.Duduchava

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 47-02 Research monographs (operator theory) 45E05 Integral equations with kernels of Cauchy type 47Gxx Integral, integro-differential, and pseudodifferential operators 45P05 Integral operators