By L. A. Aizenberg and Sh. A. Dautov

The authors examine the matter of characterizing the outside differential kinds that are orthogonal to holomorphic capabilities (or varieties) in a site $D\subset {\mathbf C}^n$ with recognize to integration over the boundary, and a few similar questions. they provide a close account of the derivation of the Bochner-Martinelli-Koppelman necessary illustration of external differential kinds, which was once got in 1967 and has already discovered many very important purposes. They research the homes of $\overline \partial$-closed kinds of variety $(p, n - 1), 0\leq p\leq n - 1$, which turn into the duals (with admire to the orthogonality pointed out above) to holomorphic services (or varieties) in different advanced variables, and resemble holomorphic features of 1 advanced variable of their houses.

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4. 4. ~)(i)). m ;;;a. 1. q-I)(D) such that aa = y. PRooF. Consider the domain of holomorphy V introduced in 2°, and a y E C(~:~>(C") such that y =yonD and supp y c V. By (Ll), y = -I},q(V, fJy)- fJI},q-l(V, y ). 4. 36 I. INTEGRAL REPRESENTATION OF FORMS Now let q oe;; n- 1. 1(n I_ p )I. 1( n 1_ p )I. Dp,n-p(az, an. Since a-y = 0 on a(V\D), the second integral above vanishes by Stokes' theorem. 4. 5. ~ 1·"( CD), then g can be extended to C" as a function in cm(C"). _O a form a E Zt0;1~(C"), there exists an extension/ E cm(C") of g such that aj= a.

Z)- U0 ,0 (r, z')) = 0, and that this limit is approached uniformly since the radius of the ball B does not depend on z0 E aD. 7. If aD E C 1•"- and y E C(~~)(ilD), then y = y+ lao - y-lao on aD. 4. We note that, if y E C(ilD), then y ± are in general not extendable to i5 as continuous functions, and it is not possible to obtain an assertion analogous to Corollary 2. 7 for continuous forms. This is shown by the following example. ( 4 ) Let D be a domain such that aD contains a (2n- I)dimensional ball B lying in the plane {y,.

If we extend a 2 to a form a2 E c<;:~·-l)(en), then y f p. 3: 1\ y = f p. 1\ D aa2 = 1 p. 1\ az = 0, aD since a 2 lao =a- a 1 lao• and a, a 1 lao E A;_P(D). 3, a 2 = a2 = ap on CD, and fJ E c1;,n-z>(en). n-z>(D). 1) yields that y = y1 - p has the desired properties. 1 cannot be carried over to arbitrary domains because it is not true that all a-closed forms (which, as shown in I o, are orthogonal to holomorphic forms) are a-exact for every domain. 2. Let D = 0\ U 1 (0;), where 0 and the 0 1 are strictly pseudoconvex domains with cm+Z boundary, m;;;..