Singular Integral Operators, 1986

The present edition differs from the original German one mainly in the following addi­ tional material: weighted norm inequalities for maximal functions and singular opera­ tors (§ 12, Chap. XI), polysingular integral operators and pseudo-differential operators (§§ 7, 8, Chap. XII), and spline approximation methods for solving singular integral equations (§ 4, Chap. XVII). Furthermore, we added two subsections on polynomial approximation methods for singular integral equations over an interval or with dis­ continuous coefficients (Nos. 3.6 and 3.7, Chap. XVII). In many places we incorporated new results which, in the vast majority, are from the last five years after publishing the German edition (note that the references are enlarged by about 150 new titles). S. G. Mikhlin wrote §§ 7, 8, Chap. XII, and the other additions were drawn up by S. Prossdorf. We wish to express our deepest gratitude to Dr. A. Bottcher and Dr. R. Lehmann who together translated the text into English carefully and with remarkable expertise.

I. Basic facts from functional analysis.- II. The one-dimensional singular integral.- III. One-dimensional singular integral equations with continuous coefficients on closed curves.- IV. One-dimensional singular integral equations with discontinuous coefficients.- V. Systems of one-dimensional singular equations.- VI. One-dimensional singular equations with degenerate symbol.- VII. Some problems leading to singular integral equations.- VIII. Some further subsidiaries.- IX. Singular integrals of higher dimensions in spaces with a uniform metric.- X. The symbol of higher dimensional singular integral operators.- XI. Singular integral operators in spaces with integral metric.- XII. Multidimensional singular integral equations.- XIII. Singular equations on smooth manifolds without boundary.- XIV. Systems of multidimensional singular equations.- XV. The localization principle. Singular operators on manifolds with boundary.- XVI. Multidimensional singular equations with degenerate symbol.- XVII. Methods for the approximate solution of one-dimensional singular integral equations.- XVIII. Approximate solution ot multidimensional singular integral equations.- References.- Symbols and notations.- Name index.