A smooth pseudoconvex domain in \(\mathbb C^ 2\) for which \(L^ \infty\)- estimates for \(\overline\partial\) do not hold
It has been known for some time that on a general pseudoconvex domain (bounded and with smooth boundary) one cannot solve \(\overline \partial\) with supnorm estimates [\textit{N. Sibony}, Invent. Math. 62, 235--242 (1980; Zbl 0429.32026)]. Likewise, counterexamples to \({\mathcal L}^p\)-estimates are known [\textit{J. E. Fornaess} and \textit{N. Sibony}, Proc. Symp. Pure Math. 52, Part 3, 129--163 (1991; Zbl 0827.32019)]. These counterexamples are for \(n\geq 3\) and \(p > 2\). In this paper, the author gives an example which is a smooth, bounded, pseudoconvex Hartogs domain in \(\mathbb C^2\), on which supnorm estimates fail for \(\overline \partial\) and for \(\overline \partial_b\). Since in \(\mathbb C^2\), \(\overline \partial_b\) is essentially symmetric (with the usual identifications), this implies failure of \({\mathcal L}^1\)- estimates for \(\overline \partial_b\) as well. The failure of supnorm estimates for \(\overline \partial\) on the Hartogs domain is reduced to failure of certain weighted supnorm estimates for \(\overline \partial\) on the (one-dimensional) base of the Hartogs domain. There is a more recent survey article by the author [Aspects Math. E 26, 27--42 (1994; Zbl 0823.32007)] that contains further results (also concerning Hölder estimates). For example, there is a Hartogs domain in \(\mathbb C^2\), bounded, pseudoconvex, and with smooth boundary, so that \(\overline \partial_b\) does not have closed range in \({\mathcal L}^p\) for any \(p\neq 2\).
scientific article; zbMATH DE number 556410