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\(L^ 2\) estimates and existence theorems for the tangential Cauchy-Riemann complex. (English) Zbl 0581.35057
The author proves that if \(\alpha\) is a \({\bar \partial}\)-closed (p,q) form on the boundary \(\partial \Omega\) of a smoothly bounded pseudoconvex domain \(\Omega \subseteq {\mathbb{C}}^ n\), \(0<q<n-2\), then there is a (p,q- 1) form u on \(\partial \Omega\) such that \({\bar \partial}_ bu=\alpha\) and u satisfies Sobolev estimates.
The starting point for this work is an observation of Rosay that this result with \(C^{\infty}\) estimates follows from earlier work of Kohn and Kohn/Rossi. The author uses these ideas, together with a two-sided extension theorem for forms \(\alpha\), to obtain the more precise Sobolev estimates. These results and techniques should prove useful in later work.
Reviewer: S.G.Krantz

MSC:
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
32T99 Pseudoconvex domains
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References:
[1] Andreotti, A., Hill, C.D.: Levi, E.E. Convexity and the Hans Lewy problem, I and II. Ann. Sc. Norm. Super. Pisa26, 325-363, 747-806 (1971) · Zbl 0256.32007
[2] Boutet de Monvel, L.: Integration des equations de Cauchy-Riemann induites formelles. Séminaire Goulaouic-Lions-Schwartz. Exposé IX (1974-1975)
[3] Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegö. Soc. Math. France Astérisque34-35, 123-164 (1976) · Zbl 0344.32010
[4] Burns, D.M.: Global behavior of some tangential Cauchy-Riemann equations. Proc. Conf. Park City, Utah, 1977. Lect. Notes Pure Appl. Math. 48, pp. 51-56. New York: Dekker 1979
[5] Folland, G.B., Kohn, J.J.: The Neumann problem for the Cauchy-Riemann complex. Ann. Math. Stud. Princeton, NJ: Princeton Univ. Press 1972 · Zbl 0247.35093
[6] Folland, G.B., Stein, E.M.: Estimates for the \(\bar \partial _b\) -complex and analysis on the Heisenberg group. Commun. Pure Appl. Math.27, 429-522 (1974) · Zbl 0293.35012 · doi:10.1002/cpa.3160270403
[7] Friedrichs, K.: The identity of weak and strong extensions of differential operators. Trans. Am. Math. Soc.55, 132-151 (1944) · Zbl 0061.26201
[8] Harvey, R., Polking, J.: Fundamental solutions in complex analysis I, II. Duke Math. J.47, 253-300, 301-340 (1979) · Zbl 0441.35043 · doi:10.1215/S0012-7094-79-04613-1
[9] Henkin, G.M.: Solutions with bounds of the equations of H. Lewy and Poincaré-Lelong. In: Construction of functions of Nevanlinna class with given zeros in a strictly pseudo-convex domain. Dokl. Akad. Nauk SSSR224, 771-774 (1975) [English Transl.: Sov. Math., Dokl.16, 1310-1314 (1976)]
[10] Henkin, G.M.: The Lewy equation and analysis of pseudo-convex manifolds. Usp. Mat. Nauk32, 57-118 (1977) [English Transl.: Russ. Math. Surv.32, 59-130 (1977)] · Zbl 0358.35057
[11] Hörmander, L.: Linear partial differential operators. New York: Springer 1963 · Zbl 0108.09301
[12] Hörmander, L.:L 2 estimates and existence for the \(\bar \partial\) operators. Acta. Math.113, 89-152 (1965) · Zbl 0158.11002 · doi:10.1007/BF02391775
[13] Hörmander, L.: An introduction to complex analysis in several variables. Princeton: Van Nostrand 1966 · Zbl 0138.06203
[14] Kohn, J.J.: Boundaries of complex manifolds. Proc. Conf. on complex manifolds, Minneapolis. New York: Springer 1965 · Zbl 0166.36003
[15] Kohn, J.J.: Global regularity for \(\bar \partial\) on weakly pseudo-convex manifolds. Trans. Am. Math. Soc.181, 272-292 (1973) · Zbl 0276.35071
[16] Kohn, J.J., Nirenberg, L.: Noncoercive boundary value problems. Commun. Pure. Appl. Math.18, 443-492 (1965) · Zbl 0125.33302 · doi:10.1002/cpa.3160180305
[17] Kohn, J.J., Rossi, H.: On the extension of holomorphic functions from the boundary of a complex manifold. Ann. Math.81, 451-472 (1965) · Zbl 0166.33802 · doi:10.2307/1970624
[18] Kuranishi, M.: Strongly pseudo-convexCR structures over small balls, I, II. Ann. Math.115, 451-500 (1982) ibid Ann. Math.116, 1-64 (1982) · Zbl 0505.32018 · doi:10.2307/2007010
[19] Rosay, J.P.: Equation de Lewy-résolubilite globale de l’équation ? b u =f sur la frontiére de domaines faiblement pseudo-convexes de ?2 (ou ? n ). Duke Math. J.49, 121-128 (1982) · Zbl 0536.35022 · doi:10.1215/S0012-7094-82-04908-0
[20] Rossi, H.: Attaching analytic spaces to an analytic space along a pseudo-concave boundary. Proc. Conf. on Complex Analysis, Minneapolis, pp. 242-253 (1964)
[21] Shaw, M.-C.: A simplification of Rosay’s theorem on global solvability of tangential Cauchy-Riemann equations. Illinois J. Math. (to appear) · Zbl 0598.32018
[22] Shaw, M.-C.: Global solvability and regularity for \(\bar \partial\) on an annulus between two weakly pseudo-convex domains. Trans. Am. Math. Society (to appear) · Zbl 0594.35010
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