Boundedness of the Bergman projector and Bell's duality theorem (Q789573)

Let \(W^ sH(\Omega)\) denote the space of holomorphic functions in \(\Omega\) contained in the \(L^ 2\) Sobolev space \(W^ s(\Omega)\) of order \(s>0,\) and let \(W^{-s}\!\!_{cl}H(\Omega)\) denote the closure of \(L^ 2H(\Omega)=W^ 0H(\Omega)\) in \(W^{-s}H(\Omega)\), where \(\Omega\) is a bounded domain in \({\mathbb{C}}^ n\) with smooth boundary. This paper is concerned with the duality \(W^ sH(\Omega)\subset L^ 2H(\Omega)\subset W^ sH(\Omega)^*\) and the isomorphism \(\Lambda^ s:W^ sH(\Omega)\to W^ sH(\Omega)^*.\) Three theorems are contained. - Theorem 1: \(W^ sH(\Omega)^*=W^{-s}\!\!_{cl}H(\Omega)\) if and only if the condition \((R)^ s\!_ 0\) holds, i.e., the Bergman projector K maps \(W^ s\!_ 0(\Omega)\) to \(W^ s(\Omega)\) boundedly. - Theorem 2: If \((R)^ s\!_ 0\) holds, then \(\Lambda^ s\) is given by \(\Lambda^ sg(z)=(g,K(\cdot,z))_ s,\) where \(K(\cdot,\cdot)\) denotes the Bergman kernel and \((\cdot,\cdot)_ s\) stands for the scalar product in \(W^ s(\Omega)\). - Theorem 3. If \((R)^ s\!_ 0\) holds, then every element of \(W^{-s}\!\!_{cl}H(\Omega)\) admits a Fourier series expansion with respect to the eigenfunctions of \(\Lambda^ s\) regarded as a self-adjoint operator in \(L^ 2H(\Omega)\). Analogous results are also valid in a framework of the Szegö projector. In case \(\Omega\) is strictly pseudoconvex, results corresponding to theorems 1 and 2 have been known by \textit{S. R. Bell} [Ill. J. Math. 26, 19-26 (1982; Zbl 0475.32004)]. A similar generalization has been obtained independently by \textit{S. R. Bell} and \textit{H. P. Boas} [Math. Ann. 267, 473-478 (1984)]. It has since been pointed out by E. J. Straube that the boundedness of \(K:W^ s\!_ 0(\Omega)\to W^ s(\Omega)\) is equivalent to that of K in \(W^ s(\Omega)\).