Non-subelliptic estimates for the tangential Cauchy–Riemann system
Tóm tắt
We prove non-subelliptic estimates for the tangential Cauchy-Riemann system over a weakly “q-pseudoconvex” higher codimensional submanifold M of
$$\mathbb{C}^{n}$$
. Let us point out that our hypotheses do not suffice to guarantee subelliptic estimates, in general. Even more: hypoellipticity of the tangential C-R system is not in question (as shows the example by Kohn of (Trans AMS 181:273–292,1973) in case of a Levi-flat hypersurface). However our estimates suffice for existence of smooth solutions to the inhomogeneous C-R equations in certain degree. The main ingredients in our proofs are the weighted L
2 estimates by Hörmander (Acta Math 113:89–152,1965) and Kohn (Trans AMS 181:273–292,1973) of Sect. 2 and the tangential
$$\bar\partial$$
-Neumann operator by Kohn of Sect 4; for this latter we also refer to the book (Adv Math AMS Int Press 19,2001). As for the notion of q pseudoconvexity we follow closely Zampieri (Compositio Math 121:155–162,2000). The main technical result, Theorem 2.1, is a version for “perturbed” q-pseudoconvex domains of a similar result by Ahn (Global boundary regularity of the
$$\bar\partial$$
-equation on q-pseudoconvex domains, Preprint, 2003) who generalizes in turn Chen-Shaw (Adv Math AMS Int Press 19, 2001).
Tài liệu tham khảo
Ahn, H.: Global boundary regularity of the \(\bar\partial\)-equation on q-pseudoconvex domains (Preprint) (2003)
Andreotti A., Fredricks G., Nacinovich M. (1981) On the absence of a Poincaré lemma in tangential Cauchy–Riemann complexes. Ann. Scuola Norm. Sup. Pisa 8, 365–404
Baracco L., Zampieri G. (2005) Regularity at the boundary for \(\bar\partial\) on q-pseudoconvex. J. d’Anal. Math. 95, 45–61
Barrett D. (1992) Behavior of the Bergman projection on the Diederich-Fornaess worm. Acta Math. 168, 1–10
Boggess A. (1991) CR Manifolds and the Tangential Cauchy–Riemann complex. CRC Press, Boca Raton
Chen, S.C., Shaw,M.C.: Partial differential equations in several complex variables. Stud. Adv. Math. - AMS Int. Press 19 (2001)
Christ M. (1996) Global C ∞ irregularity of the \(\bar\partial\)-Neumann problem for worm domains. J. A.M.S. 9(4): 1171–1185
Derridj M. (1978) Regularité pour \(\bar\partial\) dans quelques domaines faiblement pseudo-convexes. J. Diff. Geometry 13, 559–576
Derridj M., Tartakoff D. Sur la régularité analytique lobale des solutions du problème de Neumann pour \(\bar\partial\), Sém. Goulaouic-Schwartz (1976)
Dufresnoy A. (1979) Sur l’operateur \(\bar\partial\) et les fonctions diffférentiables au sens de Whitney. Ann. de l’Inst. Fourier 29(1): 229–238
Henkin G.M. (1977) H. Lewy’s equation and analysis on pseudoconvex manifolds (Russian). I, Uspehi Mat. Nauk. 32(3): 57–118
Ho L.H. (1991) \(\bar\partial\)-problem on weakly q-convex domains. Math. Ann. 290(1): 3–18
Hörmander L. (1965) L 2 estimates and existence theorems for the \(\bar\partial\) operator, Acta Math. 113, 89–152
Hörmander L. (1966) An introduction to complex analysis in several complex variables. Van Nostrand, Princeton
Kohn J.J. (1973) Global regularity for \(\bar\partial\) on weakly pseudo-convex manifolds. Trans. AMS 181, 273–292
Kohn J.J. (1977) Methods of partial differential equations in complex analysis. Proc. Symp. Pure Math. 30, 215–237
Kohn J.J. (1979) Subellipticity of the \(\bar\partial\)-Neumann problem on pseudoconvex domains: sufficient conditions. Acta Math. 142, 79–122
Nacinovich M. (1984) Poincaré lemma for the tangential Cauchy Riemann complexes. Math. Ann. 268, 449–471
Shaw M.C.(1992) Local existence theorems with estimates for \(\bar\partial_b\) on weakly pseudoconvex boundaries. Mat. Ann. 294, 677–700
Zampieri G. (2000) q-Pseudoconvexity and regularity at the boundary for solutions of the \(\bar\partial\)-problem. Compositio Math. 121, 155–162
Zampieri G. (2000) Solvability of the \(\bar\partial\) problem with C ∞ regularity up to the boundary on wedges of \(\mathbb{C}^{n}\). Israel J. Math. 115, 321–331
Zampieri G. (2002) q=pseudoconvex hypersurfaces through higher codimensional submanifolds of \(\mathbb{C}^{n}\). J. Reine Angew Math. 544, 83–90