C ~ N-singular Integrals In Research

The study of the boundary behavior of Cauchy type integral in several complexvariables is one of the classical contents in several complex analysis. As well known, the theory of singular integral and singular integral equation in one complex variable has been extensively applied in the study of elastic mechanics and fluid mechanics. It is naturally to consider the problem of using Cauchy type integral in several complex variables to deal with high dimensional singular integral.In 1957, Lu Qikeng and Zhong Tongde first studied the boundary behavior of Bochner-Martinelli type integral. Afterwards many international and domestic mathematicians joined in this fields, such as, A.M. Kytmanov, E.M. Stein, N. Kerzman and Sheng Gong. In the past more than a half century, with the further development on the theory of the singular integral in several complex variables, it was combined with the other subjects of the several complex variables and the other branches of mathematics.Lu Qikeng and Zhong Tongde [52] studied the Cauchy principal value and the boundary behavior of the Cauchy type integral with Bochner-Martinelli kernelon the bounded domain with smooth boundary in C~n. Afterwards, Zhong Tongde [66] studied the boundary behavior of Bochner-Martinelli type integral in the topological product of domains, and obtained Plemelj formula. Sun Jiguang [64] obtained the Poincare-Bertrand permutation formula and composition formulaof the singular integral with B-M kernel. Sun Jiguang [64],Zhong Tongde [66] [71] studied the theory of singular integral and singular integral equation with B-M kernel.As for the boundary behavior of Bochner-Martinelli type integral on the closed piecewise smooth manifold, Lin Liangyu [44] [45] [46] made an exhaustive study on it, where the boundary behavior of Bochner-Martinelli type integral is more complicated.Chen Luping [22] studied the Cauchy principal value and the boundary behavior of the Cauchy integral on Stein manifold.A.M. Kytmanov [34] summarized the achievements and their applications about the Bochner-Martinelli type integral up to the end of 80' in last century. In [34], one can see that the contents of the Bochner-Martinelli type integral and its boundary behavior are rich and they have wide applications. This sufficientlyexhibits the important position of B-M kernel and Bochner-Martinelli type integral in several complex variables.From the brief review above, one sees that the study of the singular integral with B-M kernel is fairly complete in content. Comparatively speaking, the study of the singular integral with holomorphic kernel is incomplete. It is due to the fact that the holomorphic kernel in different kind of domains may be different, and the Cauchy principal value is different when the deleted neighborhood is different.W. Alt [5], N. Kerzman [32], E.M. Stein [60] respectively studied the principaland the boundary behavior of singular integral with Henkin-Rairez and Szego kernel in the strongly pseudoconvex domain with smooth boundary.Gong Sheng [19] systematically studied singular integral and singular integralequation in complex ball, four classical domains, and the strongly pseudoconvexdomain with smooth boundary. Lin Liangyu [45],Gong Dingdong [16][17] extended the theory of the singular integral and singular integral equation on the complex ball in [19] to the more general building domain of complex biballs.As for the the closed piecewise smooth strongly pseudoconvex manifold, the singular integral is remained to be studied.In this paper, the author mainly considers the singular integral in some unbounded domains in C~n. As well known, Lu Jianke [51] studied the singular integral with Cauchy kernel on the real line in C. The author considers the singular integral in two unbounded domains in C~n, and obtained some results different from those in bounded domains.As well known, the upper half plane in C is holomorphic equivalent to the unit disk. From the analysis in [51], the H~αfunctions on the real line in C are equivalent to those on the unit circle, therefore the boundary behavior of Cauchy type integral on the real axis is equivalent to the boundary behavior of on the unit circle. While, when n > 1, the upper half space D = {z∈C~n : Imz_n > 0} is not holomorphic equivalent to the unit ball B = {z∈C~n:|z|<1}in C~n, and the H~αfunctions on the real hyperplane in C~n are not equivalent to those on the complex sphere. Hence the singular integral on the real hyperplane in C~n is not a simple extension of the singular integral on the real axis in C.The key problem in Chapter 1 is the existence of the principal value of Cauchy type integral. The difficult is the proofs of Theorem 1.2.2 and Theorem 1.5.2, because here the domain is unbounded. The boundary behavior of Cauchy type integral at∞is more complicated than that in bounded domain.In Chapter 2, we consider the singular integral with holomorphic kernel in unbounded domain. The Siegel upper half spaceIt is worth speaking that Perkins Diaz, Katharine [12] studied the singular integral of a kind pseudoconvex domains in C~2, and obtained Plemelj formula and its L~p boundedness. The second chapter of this paper is different from [12]. Firstly, Szego kernel is used in [12], while Cauchy-Fantappie kernel is used here, and this two kernels are different, secondly, the only case of C~2 was studied in [12], and this method could not be simply extended to the case in C~n(n > 2), while, Chapter 2 in this paper deals with general case in C~n(n > 1).The contents of this paper.In Chapter 1, the author studies the Bochner-Martinelli type integral and its boundary behavior on the real hyperplane in C~n(n > 1). We define the principal value of B-M kernel type integral on the boundary, and prove its existence. We obtain Plemelj formula, and study the Holder continuity of B-M type integral.In Chapter 2, we study the singular integral in the generalized upper half space in DC~n. Here we use a holomorphic, kernel K(ζ,z) different from the B-M kernel U(ζ, z) in Chapter 1. We prove the integrablity of K(ζ, z) when z∈D in the sense of principal value, and the existence of the principal value of singular integral on the boundary. Further, we obtain Plemelj formula, and study the Holder continuity of B-M type integral. In Chapter 3, we study the regularization of the singular integral equation and the system of singular integral equations on the strongly pseudoconvex domainwith smooth boundary. We use the Cauchy principal value and the Plemelj formula which were obtained by Kytmanov A.M. and. Myslivets S.G. in [36] to deduce a composition formula and discuss the singular integral equation and the system of singular integral equations as well.