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Sobolev estimates for the Lewy operator on weakly pseudo-convex boundaries. (English) Zbl 0588.32023
The authors prove a regularity theorem, with optimal estimates in the Sobolev norm, for the global \({\bar \partial}_ b\) problem on forms of top degree on the boundary of smooth, pseudoconvex domain. The methods are a combination of integral formulas and weighted estimates.
The paper contains a detailed account of the history of this and related problems.
Reviewer: S.Krantz

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32T99 Pseudoconvex domains
Full Text: DOI EuDML
[1] Ahern, P., Schneider, R.: Holomorphic Lipschitz functions in pseudoconvex domains. Am. J. Math.101, 543-565 (1979) · Zbl 0455.32008 · doi:10.2307/2373797
[2] Aizenberg, L.A., Dautov, Sh.A.: Differential forms orthogonal to holomorphic functions or forms, and their properties. Providence: Am. Math. Soc. 1983 · Zbl 0511.32002
[3] Andreotti, A., Hill, C.D.: E.E. Levi convexity and the Hans Lewy problem, Parts I and II. Ann. Scuola Norm Sup. Pisa26, 325-363; 747-806 (1972) · Zbl 0256.32007
[4] Boas, H.P.: The Szegö projection: Sobolev estimates in regular domains. Preprint · Zbl 0622.32006
[5] Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegö. Soc. Math. Fr. Astérisque34-35, 123-164 (1976) · Zbl 0344.32010
[6] Burns, Jr., D.M.: Global behavior of some tangential Cauchy-Riemann equations. In: Partial differential equations and geometry. Proc. Park City Conf., pp. 51-56. New York: Dekker 1979
[7] Folland, G.B., Kohn, J.J.: The Neumann problem for the Cauchy-Riemann complex. Princeton: Princeton Univ. Press 1972 · Zbl 0247.35093
[8] 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
[9] Harvey, R., Polking, J.: Fundamental solutions in complex analysis, I and II. Duke Math. J.46, 253-300; 301-340 (1979) · Zbl 0441.35043 · doi:10.1215/S0012-7094-79-04613-1
[10] Henkin, G.M.: The Lewy equation and analysis on pseudoconvex manifolds, I and II. Russ. Math. Surv.32, 59-130 (1977); Math. USSR-Sb.31, 63-94 (1977) · Zbl 0382.35038 · doi:10.1070/RM1977v032n03ABEH001628
[11] Hörmander, L.: Linear partial differential operators. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0321.35001
[12] Kerzman, N., Stein, E.M.: The Szegö kernel in terms of Cauchy-Fantappiè kernels. Duke Math. J.45, 197-224 (1978) · Zbl 0387.32009 · doi:10.1215/S0012-7094-78-04513-1
[13] Kohn, J.J.: Boundaries of complex manifolds. Proc. Conf. Complex Analysis (Minneapolis), pp. 81-94. New York: Springer 1965 · Zbl 0166.36003
[14] Kohn, J.J.: Global regularity for \(\bar \partial\) on weakly pseudoconvex manifolds. Trans. Am. Math. Soc.181, 272-292 (1973) · Zbl 0276.35071
[15] Kohn, J.J.: Estimates for \(\bar \partial _b\) on pseudo-convex CR manifolds. In: Pseudodifferential operators and applications. Providence: Am. Math. Soc. 1985 · Zbl 0571.58027
[16] 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
[17] Lewy, H.: An example of a smooth linear partial differential equation without solution. Ann. Math.66, 155-158 (1957) · Zbl 0078.08104 · doi:10.2307/1970121
[18] Phong, D.H., Stein, E.M.: Estimates for the Bergman and Szegö projections on strongly pseudoconvex domains. Duke Math. J.44, 695-704 (1977) · Zbl 0392.32014 · doi:10.1215/S0012-7094-77-04429-5
[19] Rosay, J.P.: Equation de Lewy-résolubilité global de l’équation? b u=f sur la frontière de domaines faiblement pseudo-convexes deC 2 (ouC 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. Complex Analysis (Minneapolis), pp. 242-253. New York: Springer 1965
[21] Shaw, M.-C.:L 2 estimates and existence theorems for the tangential Cauchy-Riemann complex. Invent. math.82, 133-150 (1985) · Zbl 0581.35057 · doi:10.1007/BF01394783
[22] Shaw, M.-C.: A simplification of Rosay’s theorem on global solvability of tangential Cauchy-Riemann equations. Ill. J. Math. (to appear) · Zbl 0598.32018
[23] Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton: Princeton Univ. Press 1971 · Zbl 0232.42007
[24] Kohn, J.J.: The range of the tangential Cauchy-Riemann operator. Preprint · Zbl 0609.32015
[25] Henkin, G., Leiterer, J.: Theory of functions on complex manifolds. Basel: Birkhäuser 1984
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