闭光滑流形上的奇异积分方程

<正> §1.前言 自Giraud G.以来,已有不少关于闭光滑流形上的奇异积分和奇异积分方程的研究(见10]),但利用多复变函数的Cauchy型积分作为工具者,至今不多(如3—8]),而在一维奇异积分方程论中,复变函数的Cauchy型积分起着基本的作用.本文试以定理在多复变函数论中的拓广为基础,讨论闭光滑流形上奇异积分的合成和奇异积分方程的求解,其方法和结论,都是与Giraud G.等人的工作全然不同的.

SINGULAR INTEGRAL EQUATIONS ON A CLOSED SMOOTH MANIFOLD

Let be a bounded domain in the 2n-dimensional space of n complex variables z = (z_1,…,z_n), n≥2, and its boundary Ω be a (2n-1)-dimensional smooth orientable manifold of class C~2. K(ζ, ξ)denotes the Bochner-Martinelli kernel, where ζξ∈Ω; dS denotes the volume element of Ω; |ζ- ξ|——the Euclidean distance between ζ and ξ.In this paper, we have proved:Theorem (Transformation formula). If φ(ξ, η)is a continuous complex-valued function defined on Ω, which satisfies Holder condition with index α(0<α<1) with respect to ξ and η, and if ζ∈Ω, then we have where the integral on the left and the inner integral of the first term on the right take on principle values, and the inner integral on the right is uniformly O(|ζ-η|~(-(2n-1-α/2)))dS_η with respect to ζ and η on Ω, when |ζ-η|→0; therefore the outer integral is an ordinary integral.With this formula, we have given the regularization theorem of the singular integral equations with Bochner-Martinelli kernel on Ω.