Skip to main content
Log in

L p estimates for local solutions of\(\bar \partial _b \) on strongly pseudo-convex CR manifolds

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Akahori, T.: A new approach to the local embedding theorem of CR structures forn≧4, Mem. Am. Math. Soc., No. 336, Providence, I.R., 1987

  2. Andreotti, A., Hill, D.C.: E.E. Levi convexity and the Hans Lewy problem. I, II, Ann.. Sc. Norm. Super Pisa26, 325–363, 747–806 (1972)

    Google Scholar 

  3. Boggess, A.: Kernels for the tangential Cauchy-Riemann equations. Trans. Am. Math. Soc.262, 1–49 (1980)

    Google Scholar 

  4. Boggess, A., Shaw, M.-C.: A kernel approach to the local solvability of the tangential Cauchy-Riemann equations. Trans. Am. Math. Soc.289, 643–659 (1985)

    Google Scholar 

  5. Boutet de Monvel, L.: Integration des équations de Cauchy-Riemann induites formelles. Seminaire Goulaouic-Lions-Schwartz, Exposé IX (1974–1975)

  6. Folland, G.B., Kohn, J.J.: The Neumann problem for the Cauchy-Riemann complex. Princeton: Princeton University Press 1972

    Google Scholar 

  7. 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).

    Google Scholar 

  8. Friedrichs, K.: The identity of weak and strong extension of differential operators. Trans. Am. Math. Soc.55, 132–151 (1944)

    Google Scholar 

  9. Greiner, P.C., Kohn, J.J., Stein, E.M.: Necessary and sufficient conditions for the solvability of the Lewy equation. Proc. Nat. Acad. Sci.72, 3287–3289 (1975)

    Google Scholar 

  10. Harvey, R., Polking, J.: Fundamental solutions in complex analysis, I, II. Duke Math. J.46, 253–340 (1979)

    Google Scholar 

  11. Henkin, G.M.: The Lewy equation and analysis on pseudo-convex manifolds. Russ. Math. Surv.32, 59–130 (1977) (from Usp. Mat. Nauk,32, 57–118 (1977))

    Google Scholar 

  12. Henkin, G.M., Leiterer, J.: Theory of functions on complex manífolds. Boston: Birkhäuser 1984

    Google Scholar 

  13. Hörmander, L.: Linear partial differential operators. Springer New York: Academic Press 1963

    Google Scholar 

  14. Kohn, J.J.: Boundaries of complex manifolds, Proc. Conf. Complex Analysis (Minneapolis 1964), pp 81–94. Berlin Heidelberg New York: Springer 1965

    Google Scholar 

  15. Kohn, J.J., Nirenberg, L.: Non-coercive boundary problems. Commun. Pure Appl. Math.18, 443–492 (1965)

    Google Scholar 

  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)

    Google Scholar 

  17. Kuranishi, M.: Strongly pseudo-convex CR structures over small balls, Part I, An a-priori estimate. Ann. Math.115, 451–500 (1982); III,116, 249–330 (1982)

    Google Scholar 

  18. Lewy, H.: On the local character of the solution of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables. Ann. Math.64, 514–522 (1956)

    Google Scholar 

  19. Phong, D.H. Stein, E.M.: Hilbert integrals, singular integrals and Rodon transforms II. Invent. Math.86, 75–113 (1986)

    Google Scholar 

  20. Range, R.M., Siu, Y.-T.: Uniform estimates for the\(\bar \partial \)-equation on domains with piecewise smooth strictly pseudo-convex boundaries. Math. Ann.206, 325–354 (1974)

    Google Scholar 

  21. Rosay, J.P.: Some applications of Cauchy-Fantappie Forms to (Local) problems in\(\bar \partial _b \), Ann. Sc. Norm. Super Pisa Ser. 4,13, 225–243 (1986)

    Google Scholar 

  22. Shaw, M.-C.: The range of the tangential Cauchy-Riemann operator over a small ball. (to appear in Jour. of Diff. Eq.)

  23. Skoda, H.: Valeurs au bord pour les solutions de l'opérateur\(\bar \partial \) et caractérisation des zéros des fonctions de la classe de Nevanlinna. Bull. Soc. Math. Fr.104, 225–229 (1976)

    Google Scholar 

  24. Treves, F.: Poincaré lemma in analytic complex with nondegenerate Levi form. Commun. Partial Differ. Equations7, 1467–1482 (1982)

    Google Scholar 

  25. Treves, F.: Homotopy formulas in the tangential Cauchy-Riemann complex. preprint

  26. Webster, S.M.: On the local solution of the tangential Cauchy-Riemann equations. Ann. Inst. H. Poincaré6, 167–182 (1989)

    Google Scholar 

  27. Webster, S.M.: On the proof of Kuranishi's embedding theorem. Ann. Inst. H. Poincaré6, 183–207 (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Research supported in part by NSF grant DMS 89-01455 and by NSF Visiting Professorships for Women award at the University of Wisconsin-Madison

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shaw, MC. L p estimates for local solutions of\(\bar \partial _b \) on strongly pseudo-convex CR manifolds. Math. Ann. 288, 35–62 (1990). https://doi.org/10.1007/BF01444520

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01444520

Keywords

Navigation