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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].

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.)