Some applications of a new integral formula for ∂ ̅ b

Moulay-Youssef Barkatou

Annales Polonici Mathematici (1998)

  • Volume: 70, Issue: 1, page 1-24
  • ISSN: 0066-2216

Abstract

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Let M be a smooth q-concave CR submanifold of codimension k in ℂ n . We solve locally the ∂ ̅ b -equation on M for (0,r)-forms, 0 ≤ r ≤ q-1 or n-k-q+1 ≤ r ≤ n-k, with sharp interior estimates in Hölder spaces. We prove the optimal regularity of the ∂ ̅ b -operator on (0,q)-forms in the same spaces. We also obtain L p estimates at top degree. We get a jump theorem for (0,r)-forms (r ≤ q-2 or r ≥ n-k-q+1) which are CR on a smooth hypersurface of M. We prove some generalizations of the Hartogs-Bochner-Henkin extension theorem on 1-concave CR manifolds.

How to cite

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Moulay-Youssef Barkatou. "Some applications of a new integral formula for $∂̅_{b}$." Annales Polonici Mathematici 70.1 (1998): 1-24. <http://eudml.org/doc/262710>.

@article{Moulay1998,
abstract = {Let M be a smooth q-concave CR submanifold of codimension k in $ℂ^n$. We solve locally the $∂̅_\{b\}$-equation on M for (0,r)-forms, 0 ≤ r ≤ q-1 or n-k-q+1 ≤ r ≤ n-k, with sharp interior estimates in Hölder spaces. We prove the optimal regularity of the $∂̅_\{b\}$-operator on (0,q)-forms in the same spaces. We also obtain $L^p$ estimates at top degree. We get a jump theorem for (0,r)-forms (r ≤ q-2 or r ≥ n-k-q+1) which are CR on a smooth hypersurface of M. We prove some generalizations of the Hartogs-Bochner-Henkin extension theorem on 1-concave CR manifolds.},
author = {Moulay-Youssef Barkatou},
journal = {Annales Polonici Mathematici},
keywords = {CR manifold; tangential Cauchy-Riemann equations; q-convexity; -convexity; -concavity; 1-concave CR manifolds; integral formulas; Hartogs extension theorem},
language = {eng},
number = {1},
pages = {1-24},
title = {Some applications of a new integral formula for $∂̅_\{b\}$},
url = {http://eudml.org/doc/262710},
volume = {70},
year = {1998},
}

TY - JOUR
AU - Moulay-Youssef Barkatou
TI - Some applications of a new integral formula for $∂̅_{b}$
JO - Annales Polonici Mathematici
PY - 1998
VL - 70
IS - 1
SP - 1
EP - 24
AB - Let M be a smooth q-concave CR submanifold of codimension k in $ℂ^n$. We solve locally the $∂̅_{b}$-equation on M for (0,r)-forms, 0 ≤ r ≤ q-1 or n-k-q+1 ≤ r ≤ n-k, with sharp interior estimates in Hölder spaces. We prove the optimal regularity of the $∂̅_{b}$-operator on (0,q)-forms in the same spaces. We also obtain $L^p$ estimates at top degree. We get a jump theorem for (0,r)-forms (r ≤ q-2 or r ≥ n-k-q+1) which are CR on a smooth hypersurface of M. We prove some generalizations of the Hartogs-Bochner-Henkin extension theorem on 1-concave CR manifolds.
LA - eng
KW - CR manifold; tangential Cauchy-Riemann equations; q-convexity; -convexity; -concavity; 1-concave CR manifolds; integral formulas; Hartogs extension theorem
UR - http://eudml.org/doc/262710
ER -

References

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  1. [1] R. A. Airapetjan and G. M. Henkin, Integral representations of differential forms on Cauchy-Riemann manifolds and the theory of CR-functions, Russian Math. Surveys 39 (1984), 41-118. 
  2. [2] R. A. Airapetjan and G. M. Henkin, Integral representations of differential forms on Cauchy-Riemann manifolds and the theory of CR-functions II, Math. USSR-Sb. 55 (1986), no. 1, 91-111. 
  3. [3] A. Andreotti, G. Fredricks and M. Nacinovich, On the absence of Poincaré lemma in tangential Cauchy-Riemann complexes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), 365-404. Zbl0482.35061
  4. [4] M. Y. Barkatou, Régularité höldérienne du ∂ ̅ b sur les hypersurfaces 1-convexes-concaves, Math. Z. 221 (1996), 549-572. 
  5. [5] M. Y. Barkatou, thesis, Grenoble, 1994. 
  6. [6] M. Y. Barkatou, Formules locales de type Martinelli-Bochner-Koppelman sur des variétés CR, Math. Nachr., 1998. 
  7. [7] M. Y. Barkatou, Optimal regularity for ∂ ̅ b on CR manifolds, J. Geom. Anal., to appear. 
  8. [8] S. Berhanu and S. Chanillo, Hölder and L p estimates for a local solution of ∂ ̅ b at top degree, J. Funct. Anal. 114 (1993), 232-256. Zbl0798.35120
  9. [9] A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, CRC Press, Boca Raton, Fla., 1991. 
  10. [10] A. Boggess and M.-C. Shaw, A kernel approach to the local solvability of the tangential Cauchy-Riemann equations, Trans. Amer. Math. Soc. 289 (1985), 643-658. Zbl0579.35062
  11. [11] L. Ehrenpreis, A new proof and an extension of Hartogs' theorem, Bull. Amer. Math. Soc. 67 (1961), 507-509. 
  12. [12] B. Fischer, Kernels of Martinelli-Bochner type on hypersurfaces, Math. Z. 223 (1996), 155-183. Zbl0864.32003
  13. [13] R. Harvey and J. Polking, Fundamental solutions in complex analysis, Parts I and II, Duke Math. J. 46 (1979), 253-300 and 301-340. Zbl0441.35043
  14. [14] G. M. Henkin, Solutions des équations de Cauchy-Riemann tangentielles sur des variétés de Cauchy-Riemann q-convexes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 27-30. Zbl0472.32014
  15. [15] G. M. Henkin, The Hans Lewy equation and analysis of pseudoconvex manifolds, Russian Math. Surveys 32 (1977), 59-130. Zbl0382.35038
  16. [16] G. M. Henkin, The method of integral representations in complex analysis, in: Several Complex Variables I, Encyclopaedia Math. Sci. 7, Springer, 1990, 19-116. 
  17. [17] G. M. Henkin, The Hartogs-Bochner effect on CR manifolds, Soviet Math. Dokl. 29 (1984), 78-82. Zbl0601.32021
  18. [18] C. Laurent-Thiébaut, Résolution du ∂ ̅ b à support compact et phénomène de Hartogs-Bochner dans les variétés CR, in: Proc. Sympos. Pure Math. 52, Amer. Math. Soc., 1991, 239-249. Zbl0742.32014
  19. [19] C. Laurent-Thiébaut and J. Leiterer, Uniform estimates for the Cauchy-Riemann equation on q-convex wedges, Ann. Inst. Fourier (Grenoble) 43 (1993), 383-436. Zbl0782.32014
  20. [20] C. Laurent-Thiébaut and J. Leiterer, Uniform estimates for the Cauchy-Riemann equation on q-concave wedges, Astérisque 217 (1993), 151-182. Zbl0796.32008
  21. [21] C. Laurent-Thiébaut and J. Leiterer, Andreotti-Grauert Theory on Hypersurfaces, Quaderni della Scuola Normale Superiore di Pisa, 1995. Zbl1161.32303
  22. [22] L. Ma and J. Michel, Local regularity for the tangential Cauchy-Riemann, J. Reine Angew. Math. 442 (1993), 63-90. Zbl0781.32022
  23. [23] J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley, 1984. 
  24. [24] P. L. Polyakov, Sharp estimates for the operator ∂ ̅ M on a q-concave CR manifold, preprint. Zbl0909.32005
  25. [25] R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Grad. Texts in Math. 108, Springer, 1986. Zbl0591.32002
  26. [26] R. M. Range and Y. T. Siu, Uniform estimates for the ∂̅-equation on domains with piecewise smooth strictly pseudoconvex boundaries, Math. Ann. 206 (1973), 325-354. Zbl0248.32015
  27. [27] M.-C. Shaw, Homotopy formulas for ∂ ̅ b in CR manifolds with mixed Levi signatures, Math. Z. 224 (1997), 113-136. 
  28. [28] E. M. Stein, Singular integrals and estimates for the Cauchy-Riemann equations, Bull. Amer. Math. Soc. 79 (1973), 440-445. Zbl0257.35040
  29. [29] F. Trèves, Homotopy formulas in the tangential Cauchy-Riemann complex, Mem. Amer. Math. Soc. 434 (1990). Zbl0707.35105

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