Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Boundary Hölder and $L^p$ estimates for local solutions of the tangential Cauchy-Riemann equation

Authors: Christine Laurent-Thiébaut and Mei-Chi Shaw
Journal: Trans. Amer. Math. Soc. 357 (2005), 151-177
MSC (1991): Primary 32F20, 32F10, 32F40
Published electronically: July 22, 2004
MathSciNet review: 2098090
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the local solvability of the tangential Cauchy-Riemann equation on an open neighborhood $\omega$ of a point $z_0\in M$ when $M$ is a generic $q$-concave $CR$ manifold of real codimension $k$ in $\mathbb{C} ^n$, where $1\le k\le n-1$. Our method is to first derive a homotopy formula for $\overline\partial_b$ in $\omega$ when $\omega$ is the intersection of $M$ with a strongly pseudoconvex domain. The homotopy formula gives a local solution operator for any $\overline\partial_b$-closed form on $\omega$ without shrinking. We obtain Hölder and $L^p$ estimates up to the boundary for the solution operator.

RÉSUMÉ. Nous étudions la résolubilité locale de l'opérateur de Cauchy- Riemann tangentiel sur un voisinage $\omega$ d'un point $z_0$d'une sous-variété $CR$ générique $q$-concave $M$ de codimension quelconque de $\mathbb C^n$. Nous construisons une formule d'homotopie pour le $\overline\partial_b$ sur $\omega$, lorsque $\omega$ est l'intersection de $M$ et d'un domaine strictement pseudoconvexe. Nous obtenons ainsi un opérateur de résolution pour toute forme $\overline\partial_b$-fermée sur $\omega$. Nous en déduisons des estimations $L^p$ et des estimations hölderiennes jusqu'au bord pour la solution de l'équation de Cauchy-Riemann tangentielle sur $\omega$.

References [Enhancements On Off] (What's this?)

  • 1. R. A. Airapetjan and G. M. Henkin, Integral representation of differential forms on Cauchy-Riemann manifolds and the theory of CR function, Russian Math. Survey 39 (1984), 41-118. MR 0747791 (86b:32003)
  • 2. A. Andreotti and C. D. Hill, Convexity and the H. Levi problem. Part I : Reduction to the vanishing theorems, Ann. Scuola Norm. Sup. Pisa 26 (1972), 325-363. MR 0460725 (57:718)
  • 3. M. Y. Barkatou, Formules locales de type Bochner-Martinelli-Koppelman sur des variétés CR, Math. Nachr. 196 (1998), 5-41. MR 1657982 (99k:32001)
  • 4. M. Y. Barkatou and C. Laurent-Thiébaut, Estimations optimales pour l'opérateur de Cauchy-Riemann tangentiel, Prépublication de l'Institut Fourier 593 (2003), 1-46.
  • 5. A. Boggess and M.-C. Shaw, A kernel approach to local solvability of the tangential Cauchy-Riemann equations, Trans. Amer. Math. Soc. 289 (1985), 643-659. MR 0784007 (86g:32028)
  • 6. S.-C. Chen and M.-C. Shaw, Partial differential equations in several complex variables, Studies in Advanced Math., vol. 19, AMS-International Press, 2001. MR 1800297 (2001m:32071)
  • 7. G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Ann. Math. Studies, vol. 75, Princeton University Press, Princeton, N.J., 1972. MR 0461588 (57:1573)
  • 8. G. M. Henkin, The Hans Lewy equation and analysis on pseudoconvex manifolds, Math. USSR Sbornik 31 (1977), 59-130. MR 0454067 (56:12318)
  • 9. -, Solution des équations de Cauchy-Riemann tangentielles sur des variétés Cauchy-Riemann $q$-concaves, Comptes Rendus Acad. Sciences 293 (1981), 27-30. MR 0610140 (82b:32031)
  • 10. G. M. Henkin and J. Leiterer, Andreotti-Grauert theory by integral formulas, Progress in Math., vol. 74, Birkhaüser, 1988. MR 0986248 (90h:32002b)
  • 11. C. Laurent-Thiébaut and J. Leiterer, On the Hartogs-Bochner extension phenomenon for differential forms, Math. Ann. 284 (1989), 103-119. MR 0995385 (90c:32026)
  • 12. -, Uniform estimates for the Cauchy-Riemann equation on q-convex wedges, Ann. Inst. Fourier 43 (1993), 383-436. MR 1220276 (95a:32025)
  • 13. -, Andreotti-Grauert theory on real hypersurfaces, Quaderni, Scuola Normale Superiore, Pisa, 1995.
  • 14. M. Nacinovich, Poincaré lemma for tangential Cauchy Rieman complexes, Math. Ann. 268 (1984), 449-471. MR 0753407 (86e:32025)
  • 15. I. Naruki, Localisation principle for differential complexes and its applications, Pub. Res. Inst. Math. Sci. Kyoto Univ. 8 (1972), 43-110. MR 0321144 (47:9677)
  • 16. P. L. Polyakov, Sharp estimates for operator $\overline \partial_{M}$ on a q-concave CR manifold, J. Geom. Anal. 6 (1996), 233-276. MR 1469123 (98h:32025)
  • 17. R. M. Range and Y. T. Siu, Uniform estimates for the $\overline\partial$ on domains with piecewise smooth strictly pseudoconvex boundaries, Math. Ann. 206 (1974), 325-354. MR 0338450 (49:3214)
  • 18. M.-C. Shaw, ${L}^p$ estimates for local solutions of $\overline\partial_b$ on strongly pseudoconvex CR manifolds, Math. Ann. 288 (1990), 35-62. MR 1070923 (92b:32028)
  • 19. -, Homotopy formulas for $\overline\partial_b$ in CR manifolds with mixed Levi signatures, Math. Zeit. 224 (1997), 113-135. MR 1427707 (98f:32019)
  • 20. M.-C. Shaw and L. Wang, Hölder and ${L}^p$ estimates for $\square_b$ on CR manifolds of arbitrary codimension, (Preprint).
  • 21. F. Treves, Homotopy formulas in the tangential Cauchy-Riemann complex, Memoirs of the Amer. Math. Soc., Providence, Rhode Island, 1990. MR 1028234 (90m:32012)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 32F20, 32F10, 32F40

Retrieve articles in all journals with MSC (1991): 32F20, 32F10, 32F40

Additional Information

Christine Laurent-Thiébaut
Affiliation: Université de Grenoble, Institut Fourier, UMR 5582 CNRS/UJF, BP 74, 38402 St Martin d’Hères Cedex, France

Mei-Chi Shaw
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Keywords: CR manifolds, H\"older estimates, $L^p$-estimates, tangential Cauchy Riemann equation
Received by editor(s): May 28, 2003
Published electronically: July 22, 2004
Additional Notes: The second author was supported by NSF grant DMS01-00492
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society