Sobolev and Hölder estimates for \(\overline{\partial}\) on bounded convex domains in \(\mathbb{C}^2\)
The author proves the following two estimates for the Cauchy-Riemann operator \(\bar\partial\) on the bounded convex domain \(D\) in \(\mathbb C^2\) with smooth boundary. The first result is the Sobolev estimate. Theorem 1. There is an integral solution operator \(T\) for \(\bar\partial\) on \(D\) such that if \(f\in W^{p,\alpha}\), then \[ \|Tf\|_{W^{p,\alpha}} \leq C(\alpha, p)\|f\|_{W^{p,\alpha}} \] for all \(\alpha\geq 0\) and \(1<p<\infty\). The above theorem extends the \(L^p\) estimates of \textit{J. Polking} which was proved in [Proc. Symp. Pure Math. 52, Part 3, 309-322 (1991; Zbl 0742.32013)]. The second result is the Hölder estimate. Theorem 2. There is an integral solution operator \(T\) for \(\bar\partial\) on \(D\) such that if \(f\in \Lambda^{p,\alpha}\), then \[ \|Tf\|_{\Lambda^{p,\alpha}} \leq C(\alpha, p) \|f\|_{\Lambda^{p,\alpha}} \] for all \(\alpha> 0\) and \(1<p\leq \infty\). The above theorem also extends the result in [\textit{R. M. Range}, J. Geom. Anal. 2, No. 6, 575-584 (1992; Zbl 0768.32013)].
scientific article; zbMATH DE number 1193067