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Duality between $$A^ \infty$$ and $$A^{-\infty}$$ on domains with nondegenerate corners. (English) Zbl 0834.32002
Curto, Raúl E. (ed.) et al., Multivariable operator theory. A joint summer research conference on multivariable operator theory, July 10-18, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 185, 77-87 (1995).
S. Bell [Ann. Math., II. Ser. 114, 103-113 (1981; Zbl 0458.32011)] showed that on any smoothly bounded domain in $$\mathbb{C}^n$$, the usual $$L^2$$ pairing on holomorphic functions extends naturally to a pairing between the space $$A^\infty$$ of holomorphic functions smooth up to the boundary and the space $$A^{- \infty}$$ of holomorphic functions that blow up at the boundary no faster than an inverse power of the distance to the boundary. The author now shows that such a pairing can be defined on piecewise smooth domains with complex-nondegenerate corners. However, the pairing need not be a duality pairing, and the author gives a procedure (based on a certain eigenvalue problem on bordered Riemann surfaces) for constructing examples in which there exist $$A^{- \infty}$$ functions orthogonal to all of $$A^\infty$$.
For the entire collection see [Zbl 0819.00022].

##### MSC:
 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators 32D15 Continuation of analytic objects in several complex variables 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)