Skip to main content
SpringerLink
Log in

L p and Lipschitz estimates for the\(\bar \partial \)-equation and the\(\bar \partial \)-Neumann problem

  • 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

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Agmon, S., Doublis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Commun. Pure Appl. Math.12, 623–727 (1959)

    Google Scholar 

  2. Alt, W.: Hölderabschätzungen für Ableiten von Lösungen der Gleichung\(\bar \partial \) u=f bei streng pseudokonvexen Rand. Manuscr. Math.13, 381–414 (1974)

    Google Scholar 

  3. Aronszajn, N.: Boundary values of functions with finite Dirichlet integral. Conference on Partial Differential Equations, Studies in Eigenvalue Problems Lawrence, Kansas 1956.

  4. Ash, M.E.: The basic estimate of the\(\bar \partial \)-Neumann problem in the non-Kählerian case. Am. J. Math.86, 247–254 (1964)

    Google Scholar 

  5. Babič, V.M., Slobodeckii, L.N.: On the boundedness of the Dirichlet integral. Dokl. Akad. Nauk106, 604–606 (1956)

    Google Scholar 

  6. Beals, R.: Weighted distribution spaces and pseudodifferential operators. J. Anal. Math.39, 131–187 (1981)

    Google Scholar 

  7. Beals, R., Greiner, P.C.: Pseudodifferential operators associated to hyperplane bundles. Sem. Mat. Torino 7-40 (1983)

  8. Beals, R., Greiner, P.C., Stanton, N.K.: The heat equation on aCR manifold. J. Differ. Geom.20, 343–387 (1984)

    Google Scholar 

  9. Beals, R., Stanton, N.K.: The heat equation for the\(\bar \partial \)-Neumann problem. I.12 (1987) (to appear)

  10. Browder, F.E.: A priori estimates for solutions of elliptic boundary value problems. I. Proc. Koninkl. Ned. Akad. Wetenschap.22, 145–159 (1959); II, 160–169 (1959), III,23, 404–410 (1961)

    Google Scholar 

  11. Calderón, A.P.: Lebesgue spaces of differentiable functions and distributions. Proc. Symp. Pure Math. Vol. 4, Providence, Amer. Math. Soc., 33–49 (1961)

    Google Scholar 

  12. Calderón, A.P.: boundary value problems for elliptic equations. Joint Soviet-American Symposium on Partial Differential Equations. Novosibirsk, 303–304 (1963)

  13. Calderón, A.P., Vaillancourt, R.: A class of bounded pseudodifferential operators. Proc. Nat. Acad. Sci. USA69, 1185–1187 (1972)

    Google Scholar 

  14. Chang, D.-C.: Dissertation, Princeton University, 1987

  15. Coifman, R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes Math. 242. Berlin, Heidelberg, New York: Springer 1971

    Google Scholar 

  16. Ehrenpreis, L.: Some applications of the theory of distributions to several complex variables. Seminar on Analytic Functions I, 65–79, Institute for Advanced Study, Princeton, 1957

    Google Scholar 

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

    Google Scholar 

  18. Gagliardo, E.: Caratterizzazioni delle trace sulla frontera relative ad alcune classi di funzioni inn variabili. Rend. Sem. Mat. Univ. Padova27, 284–305 (1957)

    Google Scholar 

  19. Grauert, H., Lieb, I.: Das Ramirezsche Integral und die Gleichung\(\bar \partial \) f=α im Bereich der beschränkten Formen. Rice Univ. Stud.56, 29–50 (1970)

    Google Scholar 

  20. Greiner, P.C., Stein, E.M.: A parametrix for the\(\bar \partial \)-Neumann problem. Proc. 1974 Montreal Conference. Montreal: Univ. of Montreal Press 1975

    Google Scholar 

  21. Greiner, P.C., Stein, E.M.: Estimates for the\(\bar \partial \)-Neumann problem. Princeton: Princeton Univ. Press 1977

    Google Scholar 

  22. Henkin, G.M.: Integral representations of functions holomorphic in strictly pseudoconvex domains and applications to the\(\bar \partial \)-problem. Mat. Sb.82 (124), 300–308 (1970); Math. USSR Sb.11, 273–281 (1970)

    Google Scholar 

  23. Henkin, G.M., Romanov, A.V.: Exact Hölder estimates of solutions of the\(\bar \partial \) equation. Izv. Akad. Nauk SSSR35, 1171–1183 (1971); Math. USSR Izv.5, 1180–1192 (1971)

    Google Scholar 

  24. Hörmander, L.:L 2 estimates and existence theorems for the\(\bar \partial \) operator. Acta Math.113, 89–152 (1965)

    Google Scholar 

  25. Hörmander, L.: Pseudo-differential operators and non-elliptic boundary problems. Ann. Math.83, 129–209 (1966)

    Google Scholar 

  26. Kerzman, N.: Hölder andL p estimates for solutions of\(\bar \partial \) u=f on strongly pseudoconvex domains. Commun. Pure Appl. Math.24, 301–379 (1971)

    Google Scholar 

  27. Kohn, J.J.: Harmonic integrals on strongly pseudoconvex manifolds, I. Ann. Math.78, 112–148 (1963); II,79, 450–472 (1964)

    Google Scholar 

  28. Košelev, A.I.: A priori estimates inL p and general solutions of elliptic equations and systems. Uspehi Mat. Nauk13, 29–88 (1958). Am. Math. Soc. Transl. (2),20, 105–171 (1962)

    Google Scholar 

  29. Krantz, S.G.: Optimal Lipschitz andL p regularity for the equation\(\bar \partial \) u=f on strongly pseudoconvex domains. Math. Ann.219, 233–260 (1976)

    Google Scholar 

  30. Lieb, I.: Die Cauchy-Riemannschen Differentialgleichungen auf streng pseudokonvexen Gebieten. Math. Ann.190, 6–44 (1970)

    Google Scholar 

  31. Lieb, I., Range, R.M.: Lösungsoperatoren für den Cauchy-Riemann-Komplex mitC k-Abschätzungen. Math. Ann.253, 145–164 (1980)

    Google Scholar 

  32. Lieb, I., Range, R.M.: On integral representations and a priori Lipschitz estimates for the canonical solution of the\(\bar \partial \)-equation. Math. Ann.265, 221–251 (1983)

    Google Scholar 

  33. Lieb, I., Range, R.M.: Integral representations and estimates in the theory of the\(\bar \partial \)-Neumann problem. Ann. Math.123, 265–301 (1986)

    Google Scholar 

  34. Lieb, I., Range, R.M.: Estimates for a class of integral operators and applications to the\(\bar \partial \) u=f-Neumann problem. Invent. Math.85, 415–438 (1987)

    Google Scholar 

  35. Morrey, Jr., C.B.: The analytic embedding of abstract real analytic manifolds. Ann. Math. (2),68, 150–201 (1958)

    Google Scholar 

  36. Nagel, A., Stein, E.M.: Lectures on pseudo-differential operators: regularity theorems and applications to non-elliptic problems. Princeton: Princeton Univ. Press 1979

    Google Scholar 

  37. Øvrelid, N.: Integral representation formulas andL p estimates for the\(\bar \partial \)-equation. Math. Scand.29, 137–160 (1971)

    Google Scholar 

  38. Øvrelid, N.: Pseudodifferential operators and the\(\bar \partial \)-equation. Lect. Notes Math. 512, 185–192. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  39. Phong, D.H., Stein, E.M.: Estimates for the Bergman and Szegö projections on strongly pseudoconvex domains. Duke Math. J.44, 695–704 (1977)

    Google Scholar 

  40. Phong, D.H., Stein, E.M.: Singular integrals related to the Radon transform and boundary value problems. Proc. Nat. Acad. Sci. USA80, 7697–7701 (1983)

    Google Scholar 

  41. Rothschild, L.P., Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math.137, 247–320 (1976)

    Google Scholar 

  42. Schauder, J.: Über lineare Differentialgleichungen zweiter Ordnung. Math. Z.38, 509–530 (1934)

    Google Scholar 

  43. Schauder, J.: Numerische Abschätzungen in elliptischen linearen Differentialgleichungen. Studia Math.5, 34–42 (1934)

    Google Scholar 

  44. Siu, Y.T.: The\(\bar \partial \)-problem with uniform bounds on derivatives. Math. Ann.207, 163–176 (1974)

    Google Scholar 

  45. Slobodeckii, L.N.: Spaces of S. L. Sobolev of fractional order and their application to boundary problems for partial differential equations. Dokl. Akad. Nauk SSSR118, 243–246 (1958)

    Google Scholar 

  46. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton: Princeton Univ. Press 1970

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Yale University, 06520-2155, New Haven, CT, USA

    Richard Beals

  2. Department of Mathematics, University of Toronto, M5S 1A1, Toronto, Ontario, Canada

    Peter C. Greiner

  3. Department of Mathematics, University of Notre Dame, 46556, Notre Dame, IN, USA

    Nancy K. Stanton

Authors
  1. Richard Beals
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter C. Greiner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nancy K. Stanton
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research partially supported by NSF grant DMS-8402637

Research partially supported by the National Research Council of Canada

Research partially supported by NSF grant DMS-8200442-01

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beals, R., Greiner, P.C. & Stanton, N.K. L p and Lipschitz estimates for the\(\bar \partial \)-equation and the\(\bar \partial \)-Neumann problem. Math. Ann. 277, 185–196 (1987). https://doi.org/10.1007/BF01457358

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01457358

Keywords

Search

Navigation