Uniform estimates with weights for the \(\overline{\partial}\)-equation. (English) Zbl 0923.32014

The paper gives some \(L^\infty\) variants of Hörmander’s weighted \(L^2\) estimates for the \(\overline\partial\)-equation on strongly pseudoconvex domains. In more detail, in the context of the unit ball \(B\) of \(\mathbf C^n\), a \(\overline\partial\)-closed (0,1) form \(f\) in \(B\), a plurisubharmonic function \(\phi\) on \(B\), and the solution \(u\) of \(\overline\partial u=f\) which is of minimal norm in \(L^2(e^{-\phi}(1-| z| ^2)^N)\), for \(N>(n+1)^2\), estimates are given for \(\sup | u| e^{-\widetilde\phi/2}(1-| z| ^2)^{N/2}\) in terms of \(\sup (| f| +| \partial f|) e^{-\phi/2}(1-| z| ^2)^{N/2}\), where \(\widetilde\phi\) is a certain regularization of \(\phi\) and the norms \(| f| ,| \partial f| \) are taken with respect to the metric \(\partial\overline\partial\phi\) or to the Bergman metric on \(B\) (in the latter case, \(| \partial f| \) can even be omitted). In addition to strongly pseudoconvex domains, a variant is also established for the case of the entire complex space \(\mathbf C^n\). The proofs use a refined \(L^2\)-estimate for the canonical solution \(u\). Finally, a relation of these results to the corona problem in the ball is discussed.
Reviewer: M.Engliš (Praha)


32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
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[1] Andersson, M., and Carlsson, H. Hp-estimates for holomorphic division formulas.Pacific J. Math. 173 (2) (1996), 307–335. · Zbl 0857.32006
[2] Berndtsson, B. \(\bar \partial _b \) and Carleson type estimates.Complex Analysis, II (C. A. Berenstein, ed.), Lecture Notes in Math., vol. 1276, pp. 42–54. Springer-Verlag, New York, 1987.
[3] Berndtsson, B. Weighted estimates for the \(\bar \partial - equation\) in domains in \(\mathbb{C}\).Duke Math. J. (1992), 239–255. · Zbl 0774.35048
[4] Berndtsson, B. A smoothly bounded pseudoconvex domain in \(\mathbb{C}\)2 where Lestimates for \(\bar \partial \) don’t hold.Arkiv. för Matemalik 31 (1993), 209–218. · Zbl 0827.32018
[5] Berndtsson, B. Some recent results on estimates for the \(\bar \partial - equation\) . InContributions to Complex Analysis and Analytic Geometry (H. Skoda and J-M. Trepreau, eds.), pp. 27–42. Vieweg, Braunschweig, 1994. · Zbl 0823.32007
[6] Berndtsson, B. \(\bar \partial \) and Schrödinger operators.Math. Z. 221 (3) (1996), 401–413. · Zbl 0855.35104
[7] Carleson, L. Interpolation by bounded analytic functions and the Corona problem.Ann. of Math. 76 (1962), 547–559. · Zbl 0112.29702
[8] Christ, M. On the \(\bar \partial - equation\) in \(\mathbb{C}\)1 with weights.J. Geom. Anal. 1 (1991), 193–230. · Zbl 0737.35011
[9] Dautov, S. A., and Henkin, G. M. Zeros of holomorphic functions of finite order and weighted estimates for solutions of the \(\bar \partial - equation\) .Mat. Sb. 107 (1978), 163–174. · Zbl 0392.32001
[10] Fornaess, J., and Sibony, N. Lp-estimates for \(\bar \partial \) .Proc. Symp. Pure Math. A M S52.3 (1990), 129–163. · Zbl 0827.32019
[11] Fornaess, J., and Sibony, N. Pseudoconvex domains in \(\mathbb{C}\)2, where the Corona Theorem and Lp-estimates for \(\bar \partial \) don’t hold.Complex Analysis and Geometry, Univ. Ser. Math., pp. 209–222. Plenum, New York, 1993. · Zbl 0812.32007
[12] Gamelin, T.Uniform Algebras. Prentice Hall, Englewood Cliffs, NJ, 1966.
[13] Gamelin, T. Wolff’s proof of the Corona Theorem.Israel J. Math. 37 (1980), 113–119. · Zbl 0466.46050
[14] Henkin, G. M., and Leiterer, J.Theory of Functions on Complex Manifolds. Akademie-Verlag, Berlin, 1984.
[15] Hörmander, L. Generators for some rings of analytic functions.B.A.M. S. 73, 943–949 (1967). · Zbl 0172.41701
[16] Hörmander, L. L2-estimates and existence theorems for the \(\bar \partial - operator\) .Acta Math. 113 (1965), 89–152. · Zbl 0158.11002
[17] Kohn, J., and Folland, G.The Neumann Problem for the Cauchy-Riemann complex. Annals of Mathematical Studies. Princeton University Press, Princeton, NJ, 1972. · Zbl 0247.35093
[18] Michel, V. Private communication.
[19] Rudin, W.Function Theory in the Unit Ball of \(\mathbb{C}\)n. Springer-Verlag, Berlin, 1980. · Zbl 0495.32001
[20] Sibony, N. Prolongement analytique des fonctions holomorph bornées et métrique de Caratheodory.Inv. Math. 29 (1975), 205–230. · Zbl 0333.32011
[21] Sibony, N. Un example de domaine pseudoconvexe regulier ou l’equation \(\bar \partial u = f\) n’admet pas de solution bornée pourf bornée.Inv. Math. 62 (1980), 235–242. · Zbl 0436.32015
[22] Sibony, N. Problème de la Couronne pour des domaines pseudoconvex.Ann. of Math. 126 (1987), 675–689. · Zbl 0636.32007
[23] Varopoulos, N. Th. BMO functions and the \(\bar \partial - equation\) .Pacific J. Math. 71 (1977), 221–273. · Zbl 0371.35035
[24] Wu, H. The Bochner technique.Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, vols. 1, 2, 3 (Beijing, 1980), pp. 929–1071. Science Press, Beijing, 1982.
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