关于Cⁿ中奇异积分的两个问题

关于C n 中奇异积分的两个问题 TWO PROBLEMS ON SINGULAR INTEGRALS IN C n 马道玮 Ma Daowei first-author <正> 自陆启铿和钟同德研究了Bochner—Mattinclli的积分表示[1]的边界性质[2]以后,关于闭光滑流形上的奇异积分的研究已有不少。研究了边值的连续特性[3]。孙继广则得到了闭光滑流形上的奇异积分的Poincaré-Bertrand置换公式[5]。钟同德研究了特征流形上的奇异积分[4],并给出了相应的Poincaré-Bertrand置换公式[6]。（但是,作者认为,这个公式的证明中庄交换积分次序时有疏忽之处。）本文的目的之一是讨论闭光滑流形上的一个奇异积分的连续特性（§1的定理）,这个积分在讨论奇异积分方程的正则化时是常遇到的。本文的第二个目的是对特征流形上的奇异积分的Poincaré-Bertrand置换公式给出个一不同于[6]中方法的证明。 Since Lu Qikeng and Zhong Tongde studied the boundary property of Bochner-Martinelli’s integral representation[2],there has been a lot of work on singular integals on closed smooth manifolds. Zhong further considered singular integrals on characteristic manifolds and gave the corresponding Poincaré-Bertrand formula[6] . In the first part of this paper, we prove the continuity property of a singular integral on a closed smooth manifold, one often meets this integral when considering the regularization of singular integral equations. Let Ω denote a （2n-1）-dimensional closed smooth orientable manifold of class C 2 in 2n-dimensional Euclidean space R 2 n, K（ξ,η） denote Bochner-Martinelli kernel and dSe be the volumn element of Ω. The function φ（ξ,η） is defined by K（ξ,η） = φ（ξ,η）dS ε . The following theorm is proved. Theorem. Suppose f（η） ∈H （α,Ω）, 0 < α≤1, ζ∈Ω, define g（ξ）= |ξ-ξ| 2n-1 f（η）φ（ξ,η）K（η,ζ）, ξ∈Ω, then g（ξ）∈H（α/2, Ω）. In the second part, we prove the Pomcaré-Bertrand formula （34）, where Ω= Ω 1 ×Ω 2 , Ω 1 and Ω 2 are closed smooth orientable manifolds of dimension （2n-1） and （2m-1） in Euclidean spaces of dimension 2n and 2m repectively, ×Ω 2 ×Ω 1 ×Ω 2 ). This formula was first given by Znong Tongde in 1980[6]. （But, the author thinks, there are some flaws in its proof.） We reprove it in a way which is entirely different from that given in [6] . 1983-03-01 2021-04-01 3