In [DHOh, OhT, Oh2, Oh3], extension theorems for weighted square-integrable holomopshic functions that are defined on intersections of lower-dimensional affine subspaces with a pseudoconvex domain D were proved on the basis of L2-estimates for the 9-operator. (See also [Oh2, De; Bou, Mv] for generaliza- tions to holomorphic differential forms with values in certain vector bundles.) They have proved to be useful in many applications, among them the behavior of the Bergman kernel [DH2, McN, JP] and the construction of integral kemels for the 9-equation, [BonD] . It is therefore of interest to have proofs for such extension results that are as elementary as possible. For the theorem of Ohsawa and Takegoshi (see [OhT]), such new proofs have been given, for instance, in [Bs; McN, Siu] and also by T. Ohsawa himself (oral communication). Our goal here is to give also an elemen- tary proof for the refined extension theorem of Ohsawa [Oh3] that allows so -called negligible weights in the extension. Our proof will be free of tools from Kahler geometry.
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