Non-solvability of the tangential ∂̅-system in manifolds with constant Levi rank

• Autorzy: Giuseppe Zampieri
• Języki publikacji: EN

Abstrakty

EN

Let M be a real-analytic submanifold of $ℂ^n$ whose "microlocal" Levi form has constant rank $s^{+}_{M} + s^{-}_{M}$ in a neighborhood of a prescribed conormal. Then local non-solvability of the tangential ∂̅-system is proved for forms of degrees $s^{-}_{M}$, $s^{+}_{M}$ (and 0).
 This phenomenon is known in the literature as "absence of the Poincaré Lemma" and was already proved in case the Levi form is non-degenerate (i.e. $s^{-}_{M} + s^{+}_{M} = n - codim M$). We owe its proof to [2] and [1] in the case of a hypersurface and of a higher-codimensional submanifold respectively. The idea of our proof, which relies on the microlocal theory of sheaves of [3], is new.

Słowa kluczowe

EN

CR manifolds; ∂̅ and $∂̅^b$ problems; tangential CR complex

Kategorie tematyczne

• 32F10: q -convexity, q -concavity

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2000
• Tom: 74
• Numer: 1
• Strony: 291-296

Daty

wydano

2000

otrzymano

1999-09-01

Twórcy

autor

Giuseppe Zampieri

• v. Miglioranza 20, Vicenza, Italy

Bibliografia

• [1] A. Andreotti, G. Fredricks and M. Nacinovich, On the absence of Poincaré lemma in tangential Cauchy-Riemann complexes, Ann. Scuola Norm. Sup. Pisa 8 (1981), 365-404.
• [2] L. Boutet de Monvel, Hypoelliptic operators with double characteristics and related pseudodifferential operators, Comm. Pure Appl. Math. 27 (1974), 585-639.
• [3] M. Kashiwara and P. Schapira, Microlocal theory of sheaves, Astérisque 128 (1985).
• [4] C. Rea, Levi-flat submanifolds and holomorphic extension of foliations, Ann. Scuola Norm. Sup. Pisa 26 (1972), 664-681.
• [5] M. Sato, M. Kashiwara and T. Kawai, Hyperfunctions and Pseudodifferential Operators, Lecture Notes in Math. 287, Springer, 1973, 265-529.
• [6] G. Zampieri, Microlocal complex foliation of ℝ-Lagrangian CR submanifolds, Tsukuba J. Math. 21 (1997), 361-366.