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.


32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32T99 Pseudoconvex domains
Full Text: DOI


[1] Kerzman, N., Hölder and \(L_p\) estimates on the solutions of \(\overline{\partial} = f\) in a strongly pseudoconvex domains, Communications on Pure and Applied Mathematic, 24, 301-379 (1971) · Zbl 0217.13202
[2] Krantz, S. G., Characterizations of various domains of holomorphy via \(\overset{-}{\partial}\) estimates and applications to a problem of Kohn, Illinois Journal of Mathematics, 23, 2, 267-285 (1979) · Zbl 0394.32009
[3] McNeal, J., On sharp Hölder estimates for the solutions of the \(\overset{-}{\partial}\) equations, Several complex variables and complex geometry, Proceedings of Symposia in Pure Mathematics, 52, 277-285 (1989), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0747.32010
[4] Range, R. M., On Hölder estimates for \(\overset{-}{\partial} u = f\) on weakly pseudoconvex domains, Proceedings of International Conference on Onseveral Complex Variables
[5] Straube, E. J., A Remark on hölder smoothing and subellipticity of the \(∂\)-neumann operator, Communications in Partial Differential Equations, 20, 1-2, 267-275 (1995) · Zbl 0826.35084 · doi:10.1080/03605309508821093
[6] You, Y. H., Necessary conditions for Hölder regularity gain of \(\overline{\partial}\) equation in \(C^n\)
[7] Cho, S., Boundary behavior of the Bergman kernel function on some pseudo convex domains in \(C^n\), Transactions of the American Mathematical Society, 345, 2, 803-817 (1994) · Zbl 0813.32023
[8] Catlin, D. W., Boundary invariants of pseudoconvex domains, Annals of Mathematics, 120, 3, 529-586 (1984) · Zbl 0583.32048 · doi:10.2307/1971087
[9] Catlin, D. W., Estimates of invariant metrics on pseudoconvex domains of dimension two, Mathematische Zeitschrift, 200, 3, 429-466 (1989) · Zbl 0661.32030 · doi:10.1007/bf01215657
[10] Cho, S., Estimates of invariant metrics on some pseudoconvex domains in \(C^n\), Journal of the Korean Mathematical Society, 32, 661-678 (1995) · Zbl 0857.32012
[11] Cho, S., Extension of complex structures on weakly pseudoconvex compact complex manifolds with boundary, Mathematische Zeitschrift, 211, 1, 105-120 (1992) · Zbl 0759.32012 · doi:10.1007/bf02571421
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.