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Local solvability of the \(\overline\partial\)-equation with boundary regularity on weakly \(q\)-convex domains. (English) Zbl 1156.32303

Summary: We consider weakly \(q\)-convex domains with smooth boundary and show that the \(\overline\partial\)-equation is locally solvable with regularity up to the boundary for smooth forms of degree \((p,s)\) for \(s \geq q\).

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32F10 \(q\)-convexity, \(q\)-concavity
32F32 Analytical consequences of geometric convexity (vanishing theorems, etc.)
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[1] Andreotti, A., Grauert, H.: Théorèmes de finitude pour la cohomologie des expaces complexes. Bull. Soc. Math. Fr. 90, 193–259 (1962) · Zbl 0106.05501
[2] Andreotti, A., Hill, C.D.: E.E. Levi Convexity and the Hans Lewy problem. Part I. Ann. Sc. Norm. Super. Pisa 26, 325–363 (1972); Part II: ibid 28, 747–806 (1972) · Zbl 0256.32007
[3] Demailly, J.-P.: Sur l’identité de Bochner-Kodaira-Nakano en géométrie hermitienne. Lecture Notes in Math. 1198, Springer-Verlag, Berlin, 1986, pp. 88–97
[4] Demailly, J.-P.: Complex analytic and algebraic geometry. Available at http://www-fourier.ujf-grenoble.fr/\(\sim\)demailly/books.html
[5] Dufresnoy, A.: Sur l’opérateur et les fonctions différentiables au sens de Whitney. Ann. Inst. Fourier 29, 229–238 (1979) · Zbl 0387.32011
[6] Ho, L.: -problem on weakly q-convex domains. Math. Ann. 290, 3–18 (1991) · Zbl 0714.32006
[7] Kohn, J.J.: Global regularity for on weakly pseudo-convex manifolds. Trans. Amer. Math. 181, 272–292 (1973) · Zbl 0276.35071
[8] Michel, V.: Sur la régularité du au bord d’un domaine de dont la forme de Levi a exactement s valeurs propres strictement négatives. Math. Ann. 295, 135–161 (1993) · Zbl 0788.32010
[9] Michel, V.: Résolution locale du avec régularité Gevrey au bord d’un domaine r-convexe. Math. Z. 218, 305–317 (1995) · Zbl 0816.32013
[10] Trèves, F.: Homotopy formulas in the tangential Cauchy-Riemann complex. Memoirs Amer. Math. Society, Providence, Rhode Island, 1990 · Zbl 0707.35105
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