Abstract:

In this paper, firstly using different method and technique we derive the corresponding integral representation formulas of(0, q)(q > 0) differential forms for the two types of the bounded domains in complex submanifolds with codimension-m. Secondly we obtain the unified integral representation formulas of(0, q)(q > 0) differential forms for the general bounded domain in complex submanifold with codimension-m, which include Hatziafratis formula, i.e. Koppelman type integral formula for the bounded domain with smooth boundary in analytic varieties. In particular, when m = 0, we obtain the unified integral representation formulas of(0, q)(q > 0) differential forms for general bounded domain in Cn,which are the generalization and the embodiment of Koppelman-Leray formula. 

Key words: Complex submanifold, Analytic varieties, Unified formula, Extension, Differential form, Integral representation