Uniform Estimates Of Solutions Of (?)-Equations On Local Q-Convex Wedges In Stein Manifolds

It is well known that a Stein manifold is a very important manifold on which there are many nonconstant holomorphic functions. Cⁿ is just a Stein manifold. So it is very natural to research into complex analysis in several variables on Stein manifolds. The integral representation method is one of main methods of complex analysis in several variables. In the past, many integral formulas for solving the (?)-equations on Cⁿ and Stein manifolds were obtained. Based on these integral formulas, the Holder estimates and uniform estimates of solutions of the (?)-equations for (0, s) differential forms were also given. C. Laurent-Thiebaut k J. Lettered^[10] introduced local q-convex wedges in Cⁿ, which are extensions of piecewise smooth pseudoconvex domains, obtained the homotopy formula and the uniform estimates for the Cauchy-Riemann equation on q-convex wedges in Cⁿ. By using the Hermitian metric and Chern connection^[13], Tongde Zhong^[14] obtained the homotopy formlas for (r,s) differential forms and solutions of (?)-equation on local q-convex wedges in Stein manifolds. On the basis of [10,14], by means of the ideas of C. Laurent-Thiebaut & J. Leiterer^[10] and trick of Range& Siu^[8] , the author obtains the uniform estimates of the solutions of (?)-equations for (r, s) differential forms on local q-convex wedges in Stein manifolds.The whole dissertation includes three chapters. The aim of this paper is to generalize the uniform estimates of solutions of (?)-equations on local q-convex wedges in Cⁿ to Stein manifolds. Suppose that X is a Stein manifold of complex dimension n and D (?)(?) X is an open set.In the first chapter, the author introduces some definitions, the basic lemma and notations on Stein manifolds, including a Stein manifold, the complex tangent bundle, the complex cotangent bundle, fibre, and the most important basic lemma and so on. In the second chapter, the results of Tongde Zhong^[14] are introduced. The local q-convex wedges in stein manifolds are defined. Then, by means of the Hermitian metric and Chern connection^[13] ,Tongde Zhong^[14] constructed a Leray map, and obtained the homotopy formulas and solutions of (?)-equation on local q -convex wedges in stein manifolds.The integral formula does not involve boundary integrals, so one can avoid complex estimates of the boundary integrals. Moreover, a local q-convex wedges in Stein manifolds is an extension of piecewise smooth pseudoconvex domain, so the homotopy formula has its generalization meaning, it has important applications in uniform estimates of (?)-equation and holomorphic extension of CR-manifolds.In the third chapter,by means of the ideas of C. Laurent-Thiebaut and J. Leiterer^[10], and by introducing the Hermitian metric and Chern connection^[13] and using of the localization technique, the author admits some estimate of the integral operator H for (r,s)(r > 0,s > 0) differential forms on local q-convex wedges in Stein manifolds. Then by means of the trick of Range-Siu trick^[8] the authors complicatedly calculate the uniform estimates of solutions of (?)-equation on local q-convex wedges in Stein manifolds.