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Global regularity of the \(\overline{\partial}\)-Neumann problem on circular domains. (English) Zbl 0662.32016
We show that if \(D\subseteq {\mathbb{C}}^ n\), \(n\geq 2\), is a smooth bounded pseudoconvex circular domain with \(\sum^{n}_{i=1}z_ i\frac{\partial r}{\partial z_ i}\neq 0\) near the boundary, where r(z) is the defining function for D, then global regularity of the \({\bar \partial}\)-Neumann problem holds on D. More precisely, if given \(f\in W^ k_{p,q}(D)\), let \(u\in L^ 2_{q,p}(D)\) be the solution to the \({\bar \partial}\)- Neumann problem, \[ Q(u,v)=({\bar \partial}u,{\bar \partial}v)+({\vec \partial}u,{\vec \partial}v)=(f,v), \] for all \(v\in \tilde {\mathcal D}_{p,q}(D)\), then \(u\in W^ k_{p,q}(D)\) and \(\| u\|_ k\leq C_ k\| f\|_ k\), where \(W^ k_{p,q}(D)\) is the Sobolev space of order k on D and \(\tilde {\mathcal D}_{p,q}(D)\) is the completion of smooth (p,q)-forms with Neumann boundary conditions under Q. We also prove that same conclusions hold on any smooth bounded Reinhardt pseudoconvex domain. As an application we show that same results hold on Sibony’s domain that is a smooth bounded weakly pseudoconvex domain in \({\mathbb{C}}^ 3\), but fails to have sup-norm estimate for the \({\bar \partial}\)-equation.
Reviewer: S.Chen

MSC:
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32T99 Pseudoconvex domains
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
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References:
[1] Catlin, D.: Necessary conditions for subellipticity of the \(\bar \partial \) -Neumann problem. Ann. Math.117, 147-171 (1983) · Zbl 0552.32017
[2] Catlin, D.: Subelliptic estimates for the \(\bar \partial \) -Neumann problem on pseudoconvex domains. Ann. Math.126, 131-191 (1987) · Zbl 0627.32013
[3] Catlin, D.: Global regularity of the \(\bar \partial \) -Neumann problem. Proc. Sympos. Pure Math.41, Amer. Math. Soc., Providence, R.I. (1984) · Zbl 0578.32031
[4] Chen, S.C.: Regularity of the Bergman projection on domains with partial transverse symmetries. Math. Ann.277, 135-140 (1987) · Zbl 0603.35067
[5] Chen, S.C.: Global analytic hypoellipticity of the \(\bar \partial \) -Neumann problem on circular domains. Invent. Math.92, 173-185 (1988) · Zbl 0621.35067
[6] Chen, S.C.: Global real analyticity of solutions to the \(\bar \partial \) -Neumann problem on Reinhardt domains. Indiana Univ. Math. J.37, 421-430 (1988) · Zbl 0637.32016
[7] D’angelo, J.: Real hypersurfaces, order of contact, and applications. Ann. Math.115, 615-637 (1982) · Zbl 0488.32008
[8] Folland, G.B., Kohn, J.J.: The Neumann problem for the Cauchy-Riemann complex. Ann. Math. Studies,75, Princeton: Princeton University Press 1972 · Zbl 0247.35093
[9] Hörmander, L.:L 2 estimates and existence theorems for the \(\bar \partial \) operator. Acta Math.113, 89-152 (1963) · Zbl 0158.11002
[10] Kohn, J.J.: Harmonic integrals on strongly pseudoconvex manifolds I, II. Ann. Math.78, 112-148 (1963);79, 450-472 (1964) · Zbl 0161.09302
[11] Kohn, J.J.: Boundary behavior of \(\bar \partial \) on weakly pseudoconvex manifolds of dimension two. J. Differ. Geom.6, 523-542 (1972) · Zbl 0256.35060
[12] Kohn, J.J.: Subellipticity of the \(\bar \partial \) -Neumann problem on pseudo-convex domains: sufficient conditions. Acta Math.142, 79-122 (1979) · Zbl 0395.35069
[13] Kohn, J.J., Nirenberg, L.: Non-coercive boundary value problems. Commun. Pure Appl. Math.18, 443-492 (1965) · Zbl 0125.33302
[14] Sibony, N.: Un example de domaine pseudoconvexe regulier où l’équation \(\bar \partial u = f\) n’admet pas de solution bornée pourf bornée. Invent. Math.62, 235-242 (1980) · Zbl 0436.32015
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