Differential Forms Orthogonal to Holomorphic Functions or by L. A. Aizenberg and Sh. A. Dautov

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;;;..

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