By means of the invariant integral kernel (the Berndtsson kernel), the complex Finsler metric and the non-linear connection associated with the Chern-Finsler connection to research into the integral representation theory on complex Finsler manifolds, theKoppelman and Koppelman-Leray formulas are obtained, and the - -equations are solved.
QIU Chunhui & ZHONG Tongde School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
Suppose that D is a bounded domain with a piecewise C 1 smooth boundary in C n . Let ? ∈ C 1+α (?D). By using the Hadamard principal value of the higher order singular integral and solid angle coefficient method of points on the boundary, we give the Plemelj formula of the higher order singular integral with the Bochner–Martinelli kernel, which has integral density ?. Moreover, by means of the Plemelj formula and methods of complex partial differential equations, we discuss the corresponding Cauchy boundary value problem with the Bochner–Martinelli kernel on a closed piecewise smooth manifold and obtain its unique branch complex harmonic solution.
