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On sharp Hölder estimates of the Cauchy-Riemann equation on pseudoconvex domains in \(\mathbb{C}^n\) with one degenerate eigenvalue. (English) Zbl 1351.32070

Let \(L^\infty_{(0,1)}(\Omega)\) and \(\Lambda_\delta(\Omega)\) denote the space of \((0,1)\)-forms with \(L^\infty(\Omega)\) coefficients and the space of Hölder functions on \(\Omega\) of class \(\delta\), respectively.
In this paper, the authors study sharp Hölder estimates for the \(\overline\partial\)-problem on smooth bounded pseudoconvex domains with a Levi form that has at most one degenerate eigenvalue. To be more precise, the authors prove the following result.
Theorem. Let \(\Omega\) be a smooth bounded pseudoconvex domain in \(\mathbb C^n\) and the Levi form of \(b\Omega\) have \(n-1\) positive eigenvalues at \(z_0\in b\Omega\). Furthermore, assume that there is a smooth complex curve through \(z_0\) with order of contact greater than or equal to \(\eta\). Assume that there exists a neighborhood \(U\) of \(z_0\) and \(C>0\) such that whenever \(\alpha\in L^\infty_{(0,1)}(\Omega)\) with \(\overline\partial \alpha=0\) there exists \(u\in \Lambda_\delta(\Omega\cap U)\) such that \(\partial U=\alpha\) and \(\| u\|_{\Lambda_\delta(\Omega\cap U)}\leq C \| \alpha\|_{L^\infty_{(0,1)}(\Omega)}\). Then \(\delta\leq 1/\eta\).
Hölder estimates for \(\overline\partial\) have been studied on strongly pseudoconvex domains in \(\mathbb C^n\), pseudoconvex finite-type domains in \(\mathbb C^2\), finite-type convex domains in \(\mathbb C^n\), etc.

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32T99 Pseudoconvex domains
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