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C~r(U). (ii) For each f~c00 (D) such that af is also continuous on D, over U, 'r we have the representation if 1~r~n-1 ( 1. 16) if r=n. Proof. Choose a C00 function X on G:n with supp X c:c: D and X= 1 in U. 4 (i) and the Koppelman formula ( 1. 8). 17. Theorem. Let E be a holomorphic vector bundle over a compact ndimensional complex manifold X. Then there exist linear operators n C~ O

Accordingly we use the notations p~D(x) and PaD if x is a c2 point of aD. 11. Lemma. Suppose that the following two conditions are fulfilled: (a) G ~ ~ is a domain and u is a real-valued which is convex; 62 c2 function on G (b) X is an n-dimensional complex manifold and is a c2 map such that the functions g 2 , ... ,gk are stictly plurisubharmonic on X and, for some 1~q~n, ~ 1 is q-convex. Then the following two assertions hold true: (i) If, for all XEG, min 1

By dt 1 ... d~. By~ 2: £ RN+1 with 47 For every strictly increasing collection K=(k1 , ... ,k1 ) of integers we set O~k 1 < ... : L_ tk s=1 We orient the manifolds ~K s = 1}. 3) Then, for any strictly increasing collection K=(k 1 , ... ,k 1 ) of integers 1~k 1 < ...

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