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Integrals of Cauchy type on the ball. (English) Zbl 0934.32004

Hong Kong: International Press Inc. iii, 304 p. (1993).
Summary: This book, the English version of the author’s Chinese monograph, from the early 1980’s, collects the works of the author and his colleagues in singular integrals of several complex variables since 1964. The kernels that give rise to such singular integrals on the boundaries of the domains are the Cauchy-Fantappiè kernels and Szegő kernels. A brief description of these kernels is given as a preliminary in the introduction of the book. In the first chapter, the author starts from the most important simply connected irreducible domain, i.e., the ball in \(\mathbb{C}^n\), and makes an extensive investigation of the integral of Cauchy type on the hypersphere. The author’s conclusion shows that, quite differently from the case of one complex variable, there are many ways to define the principal value of the Cauchy integral, and accordingly, various forms of the Plemelj formula are obtained. With these results as a tool, the author advances the theory of singular integral equations with constant coefficients and variable coefficients on the hypersphere in Chapter two. This was something completely new at that time. In Chapter three, the concept of the Hadamard principal value is extended to the space of several complex variables. The author defines the Hadamard principal value of singular integrals on the hypersphere, in terms of which, the limiting values of the derivative of the Cauchy integral can then be expressed. The next two chapters deal with the Cauchy integral on two kinds of classical domains defined by Prof. L. K. Hua. The author points out that the limiting value of the integral on the characteristic manifold is essentially different from that on the other parts of the boundary. Finally, in Chapter six, the Cauchy integrals defined by the Henkin-Ramirez kernel and the Stein-Kerzman kernel of the strictly pseudo-convex domains are studied, and a generalized Plemelj formula is obtained.

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
45E05 Integral equations with kernels of Cauchy type
32A55 Singular integrals of functions in several complex variables
32T15 Strongly pseudoconvex domains
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