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1998 | 70 | 1 | 1-24
Tytuł artykułu

Some applications of a new integral formula for $∂̅_{b}$

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Kategorie tematyczne
Rocznik
Tom
70
Numer
1
Strony
1-24
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-12-28
poprawiono
1998-08-31
Twórcy
  • Département de Mathématiques Université de Poitiers 40 Avenue du Recteur Pineau 86022 Poitiers Cedex, France
Bibliografia
  • [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] 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] 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.
  • [4] M. Y. Barkatou, Régularité höldérienne du $∂̅_b$ sur les hypersurfaces 1-convexes-concaves, Math. Z. 221 (1996), 549-572.
  • [5] M. Y. Barkatou, thesis, Grenoble, 1994.
  • [6] M. Y. Barkatou, Formules locales de type Martinelli-Bochner-Koppelman sur des variétés CR, Math. Nachr., 1998.
  • [7] M. Y. Barkatou, Optimal regularity for $∂̅_b$ on CR manifolds, J. Geom. Anal., to appear.
  • [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.
  • [9] A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, CRC Press, Boca Raton, Fla., 1991.
  • [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.
  • [11] L. Ehrenpreis, A new proof and an extension of Hartogs' theorem, Bull. Amer. Math. Soc. 67 (1961), 507-509.
  • [12] B. Fischer, Kernels of Martinelli-Bochner type on hypersurfaces, Math. Z. 223 (1996), 155-183.
  • [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.
  • [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.
  • [15] G. M. Henkin, The Hans Lewy equation and analysis of pseudoconvex manifolds, Russian Math. Surveys 32 (1977), 59-130.
  • [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] G. M. Henkin, The Hartogs-Bochner effect on CR manifolds, Soviet Math. Dokl. 29 (1984), 78-82.
  • [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.
  • [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.
  • [20] C. Laurent-Thiébaut and J. Leiterer, Uniform estimates for the Cauchy-Riemann equation on q-concave wedges, Astérisque 217 (1993), 151-182.
  • [21] C. Laurent-Thiébaut and J. Leiterer, Andreotti-Grauert Theory on Hypersurfaces, Quaderni della Scuola Normale Superiore di Pisa, 1995.
  • [22] L. Ma and J. Michel, Local regularity for the tangential Cauchy-Riemann, J. Reine Angew. Math. 442 (1993), 63-90.
  • [23] J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley, 1984.
  • [24] P. L. Polyakov, Sharp estimates for the operator $∂̅_M$ on a q-concave CR manifold, preprint.
  • [25] R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Grad. Texts in Math. 108, Springer, 1986.
  • [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.
  • [27] M.-C. Shaw, Homotopy formulas for $∂̅_b$ in CR manifolds with mixed Levi signatures, Math. Z. 224 (1997), 113-136.
  • [28] E. M. Stein, Singular integrals and estimates for the Cauchy-Riemann equations, Bull. Amer. Math. Soc. 79 (1973), 440-445.
  • [29] F. Trèves, Homotopy formulas in the tangential Cauchy-Riemann complex, Mem. Amer. Math. Soc. 434 (1990).
Typ dokumentu
Bibliografia
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Identyfikator YADDA
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