Closed Range Estimates for \(\bar{\partial }_b\) on CR Manifolds of Hypersurface Type

  • Joel Coacalle
  • Andrew RaichEmail author


The purpose of this paper is to establish sufficient conditions for closed range estimates on (0, q)-forms, for some fixedq, \(1 \le q \le n-1\), for \(\bar{\partial }_b\) in both \(L^2\) and \(L^2\)-Sobolev spaces in embedded, not necessarily pseudoconvex CR manifolds of hypersurface type. The condition, named weak Y(q), is both more general than previously established sufficient conditions and easier to check. Applications of our estimates include estimates for the Szegö projection as well as an argument that the harmonic forms have the same regularity as the complex Green operator. We use a microlocal argument and carefully construct a norm that is well suited for a microlocal decomposition of form. We do not require that the CR manifold is the boundary of a domain. Finally, we provide an example that demonstrates that weak Y(q) is an easier condition to verify than earlier, less general conditions.


Weak Z(qWeak Y(qTangential Cauchy–Riemann operator \(\bar{\partial }_b\) Closed range Microlocal analysis 

Mathematics Subject Classification

Primary 32W10 Secondary 32F17 32V20 35A27 35N15 



  1. 1.
    Baracco, L.: The range of the tangential Cauchy–Riemann system to a CR embedded manifold. Invent. Math. 190(2), 505–510 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Boas, H., Shaw, M.-C.: Sobolev estimates for the Lewy operator on weakly pseudoconvex boundaries. Math. Ann. 274, 221–231 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Biard, S., Straube, E.: \(L^2\)-Sobolev theory for the complex Green operator. Int. J. Math. 28(9), 1740006–31 (2017)CrossRefzbMATHGoogle Scholar
  4. 4.
    Folland, G.B., Kohn, J.J.: The Neumann Problem for the Cauchy–Riemann Complex. Annals of Mathematics Studies. Princeton University Press, Princeton (1972)zbMATHGoogle Scholar
  5. 5.
    Hörmander, L.: \({L}^{2}\) estimates and existence theorems for the \(\bar{\partial }\) operator. Acta Math. 113, 89–152 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Harrington, P.S., Peloso, M.M., Raich, A.S.: Regularity equivalence of the Szegö projection and the complex Green operator. Proc. Am. Math. Soc. 143(1), 353–367 (2015). arXiv:1305.0188 CrossRefzbMATHGoogle Scholar
  7. 7.
    Harrington, P., Raich, A.: Strong closed range estimates for the Cauchy–Riemann operator. SubmittedGoogle Scholar
  8. 8.
    Harrington, P.S., Raich, A.: Closed range of \(\bar{\partial }\) on unbounded domains in \(\mathbb{C}^n\). J. Anal. Math. 138(1), 185–208 (2019)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Harrington, P., Raich, A.: Regularity results for \(\bar{\partial }_b\) on CR-manifolds of hypersurface type. Commun. Partial Differ. Equ. 36(1), 134–161 (2011)CrossRefzbMATHGoogle Scholar
  10. 10.
    Harrington, P., Raich, A.: Closed range for \(\bar{\partial }\) and \(\bar{\partial }_b\) on bounded hypersurfaces in Stein manifolds. Ann. Inst. Fourier 65(4), 1711–1754 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Harrington, P.S., Raich, A.: Closed range of \(\bar{\partial }\) in \(L^2\)-Sobolev spaces on unbounded domains in \(\mathbb{C} ^n\). J. Math. Anal. Appl. 459(2), 1040–1461 (2018). arXiv:1704.07507 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kohn, J.J.: Global regularity for \(\bar{\partial }\) on weakly pseudo-convex manifolds. Trans. Am. Math. Soc. 181, 273–292 (1973)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kohn, J.J.: The range of the tangential Cauchy–Riemann operator. Duke Math. J. 53, 525–545 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Khanh, T.V., Raich, A.: The Kohn-Laplace equation on abstract CR manifolds: global regularity. Trans. Am. Math. Soc. (to appear). arXiv:1612.07445
  15. 15.
    Kohn, J.J., Rossi, H.: On the extension of holomorphic functions from the boundary of a complex manifold. Ann. Math. 81, 451–472 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nicoara, A.: Global regularity for \(\bar{\partial }_b\) on weakly pseudoconvex CR manifolds. Adv. Math. 199, 356–447 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Raich, A.: Compactness of the complex Green operator on CR-manifolds of hypersurface type. Math. Ann. 348(1), 81–117 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Raich, A., Straube, E.: Compactness of the complex Green operator. Math. Res. Lett. 15(4), 761–778 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Shaw, M.-C.: \({L}^2\)-estimates and existence theorems for the tangential Cauchy–Riemann complex. Invent. Math. 82, 133–150 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Straube, E.: Lectures on the \({\cal{L}}^2\)-Sobolev Theory of the \(\bar{\partial }\)-Neumann Problem. ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich (2010)zbMATHGoogle Scholar

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© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de São CarlosSão CarlosBrazil
  2. 2.SCEN 327FayettevilleUSA

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