By means of the Hermitian metric and Chern connection,Qiu [4] obtained the Koppelman-Leray-Norguet formula for(p,q)differential forms on an open set with C^1 piecewise smooth boundary on a Stein manifold,and under suitable conditions gave the solutions of■-equation on a Stein manifold.In this article,using the method of Range and Siu [5],under suitable conditions,the authors complicatedly calculate to give the uniform estimates of solutions of■-equation for(p,q)differential forms on a Stein manifold.
In this paper, the Laplacian on the holomorphic tangent bundle T 1,0 M of a complex manifold M endowed with a strongly pseudoconvex complex Finsler metric is defined and its explicit expression is obtained by using the Chern Finsler connection associated with (M, F ). Utilizing the initiated "Bochner technique", a vanishing theorem for vector fields on the holomorphic tangent bundle T 1,0 M is obtained.
First of all,using the relations(2.3),(2.4),and(2.5),we define a complex Clifford algebra W n and the Witt basis.Secondly,we utilize the Witt basis to define the operators ■ and ■∧ on Kaehler manifolds which act on W n-valued functions.In addition,the relation between above operators and Hodge-Laplace operator is argued.Then,the Borel-Pompeiu formulas for W n-valued functions are derived through designing a matrix Dirac operator and a 2 × 2 matrix-valued invariant integral kernel with the Witt basis.
In this paper,by the method of global analysis,the authors give a new global integral transformation formula and obtain the Plemelj formula with Hadamard principal value of higher-order partial derivatives for the integral of Bochner-Martinelli type on a closed piecewise smooth orientable manifold Cn.Moreover,the authors obtain the composition formula,Poincar'e-Bertrand extended formula of the corresponding singular integral.As the application of some results,the authors also study a higher-order Cauchy boundary problem and a regularization problem of higher-order linear complex differential singular integral equation with variable coefficients.
HUANG YuSheng1,& LIN LiangYu2 1Department of Mathematics,Putian College,Putian 351100,China
By using the Chern-Finsler connection and complex Finsler metric,the Bochner technique on strong Khler-Finsler manifolds is studied.For a strong Khler-Finsler manifold M,the authors first prove that there exists a system of local coordinate which is normalized at a point v ∈ M-=T 1,0M\o(M),and then the horizontal Laplace operator H for diffierential forms on PTM is defined by the horizontal part of the Chern-Finsler connection and its curvature tensor,and the horizontal Laplace operator H on holomorphic vector bundle over PTM is also defined.Finally,we get a Bochner vanishing theorem for diffierential forms on PTM.Moreover,the Bochner vanishing theorem on a holomorphic line bundle over PTM is also
