The Koppelman-Leray-Norguet Formula For Strictly Pseudoconvex Domain With Non-smooth Boundary On Complex Manifold And Its Applications

It is well known that the integral representations and their applications for(O,q) differential form in C~n have been deeply studied.But the research for integral representations on complex manifolds began in 1980s.Most of the results,so far,are concerned with Stein manifolds.In the early 1990s,B.Berndtsson studied the theory of integral representations on general complex manifolds,under a suitable condition gained in a quite general integral kernel.Using this kernel,obtained the Koppelman formula on complex manifolds.Based on it,Tongde Zhong got the Koppelman-Leray-Norguet formula of type(p,q) on a bounded domain D with piecewise C~1 smooth boundaries in a complex manifold,and gave the continuous solutions of(?)-equations on D under a suitable condition.ln this paper,by using the Hermitian metric and Chern connection,we study the case of a strictly pseudoconvex domain D with non-smooth boundaries in a complex manifold.By constructing a new integral kernel,we obtain a new Koppelman-Leray-Norguet formula of type(p,q) on D,and get the continuous solutions of(?)-equations on D under a suitable condition. The new formula doesn't involve integrals on the boundary,thus one can avoid complex estimations of the boundary integrals,and the density of integral may be not defined on the boundary but only in the domain.As some applications,we discuss the Koppelman-Leray-Norguet formula of type(p,q) for general strictly pseudoeonvex polyhedrons(unnecessarily non-degenerate) on Stein manifolds,also get the continuous solutions of(?)-equations under a suitable condition.The whole dissertation contains three chapters:In the first chapter,the author introducs some definitions and notations on complex manifolds,including the Berndtsson kernel,piecewise C~1 smooth boundaries.Furthermore there are some important basic lemmas and so on. In the second chapter,By means of constructing new kernel,the author gives a new Koppelman-Leray-Norguet formula on a strictly pseudoconvex domain with non-smooth boundaries.We also gives two examples.In the third chapter,As some applications,we discuss the Koppelman-Leray -Norguet formula of type(p,q) for general strictly pseudoconvex polyhe-drons (unnecessary non-degenerate) on Stein manifolds,also get the continuous solutions of(?)-equations under a suitable condition.