A result on extension of C.R. functions

Makhlouf Derridj; John Erik Fornaess

Annales de l'institut Fourier (1983)

  • Volume: 33, Issue: 3, page 113-120
  • ISSN: 0373-0956

Abstract

top
Let Ω an open set in C 4 near z 0 Ω , λ a suitable holomorphic function near z 0 . If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : u = λ f , ( f is a ( 0 , 1 ) form, closed in U ( z 0 ) in U ( z 0 ) with supp ( u ) Ω U ( z 0 ) , then we deduce an extension result for C . R . functions on Ω U ( z 0 ) , as holomorphic fonctions in Ω V ( z 0 ) .

How to cite

top

Derridj, Makhlouf, and Fornaess, John Erik. "A result on extension of C.R. functions." Annales de l'institut Fourier 33.3 (1983): 113-120. <http://eudml.org/doc/74592>.

@article{Derridj1983,
abstract = {Let $\Omega $ an open set in $\{\bf C\}^4$ near $z_0\in \partial \Omega $, $\lambda $ a suitable holomorphic function near $z_0$. If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : $\bar\{\partial \}u=\lambda f$, ($f$ is a $(0,1)$ form, $\bar\{\partial \}$ closed in $U(z_0)$ in $U(z_0)$ with supp$(u)\subset \bar\{\Omega \}\cap U(z_0)$, then we deduce an extension result for $C.R.$ functions on $\partial \Omega \cap U(z_0)$, as holomorphic fonctions in $\Omega \cap V(z_0)$.},
author = {Derridj, Makhlouf, Fornaess, John Erik},
journal = {Annales de l'institut Fourier},
keywords = {extension of C. R. functions},
language = {eng},
number = {3},
pages = {113-120},
publisher = {Association des Annales de l'Institut Fourier},
title = {A result on extension of C.R. functions},
url = {http://eudml.org/doc/74592},
volume = {33},
year = {1983},
}

TY - JOUR
AU - Derridj, Makhlouf
AU - Fornaess, John Erik
TI - A result on extension of C.R. functions
JO - Annales de l'institut Fourier
PY - 1983
PB - Association des Annales de l'Institut Fourier
VL - 33
IS - 3
SP - 113
EP - 120
AB - Let $\Omega $ an open set in ${\bf C}^4$ near $z_0\in \partial \Omega $, $\lambda $ a suitable holomorphic function near $z_0$. If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : $\bar{\partial }u=\lambda f$, ($f$ is a $(0,1)$ form, $\bar{\partial }$ closed in $U(z_0)$ in $U(z_0)$ with supp$(u)\subset \bar{\Omega }\cap U(z_0)$, then we deduce an extension result for $C.R.$ functions on $\partial \Omega \cap U(z_0)$, as holomorphic fonctions in $\Omega \cap V(z_0)$.
LA - eng
KW - extension of C. R. functions
UR - http://eudml.org/doc/74592
ER -

References

top
  1. [1] A. ANDREOTTI and C.D. HILL, E.E. convexity and the H. Lewy problem. Part I : Reduction to vanishing theorems, Ann. Scuola Normale Sup. di Pisa, 26 (1972). Zbl0256.32007
  2. [2] A. ANDREOTTI and E. VESENTINI, Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Publ. IHES, vol. 24-25. Zbl0138.06604
  3. [3] M.S. BAOUENDI and F. TREVES, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. Math., 113 (1981). Zbl0491.35036MR82f:35057
  4. [4] E. BEDFORD and J.E. FORNAESS, Local extension of C.R. function from weakly pseudoconvex boundaries, Michigan Math. J., 25. Zbl0401.32007MR80c:32014
  5. [5] A. BOGGES, C.R. extendability near a point where the first leviform vanishes, Duke Math. J., 43 (3). Zbl0509.32006
  6. [6] M. DERRIDJ, Inégalités de Carleman et extension locale des fonctions holomorphes, Annali. Sci. Norm. Pisa, Serie IV vol. IX (1981). Zbl0548.32013
  7. [7] C.D. HILL and TAÏANI, Families of analytic discs in Cn with boundaries on a prescribed C.R. submanifold, Ann. Scuola Norm. Pisa, 4-5 (1978). Zbl0399.32008
  8. [8] L. HÖRMANDER, Introduction to complex analysis in several variables, van Nostrand. Zbl0138.06203
  9. [9] L. HÖRMANDER, L²-estimates and existence theorems for the ∂-operator, Acta Math., 113 (1965). Zbl0158.11002MR31 #3691
  10. [10] J.J. KOHN and H. ROSSI, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. Math., 81 (1965). Zbl0166.33802MR31 #1399
  11. [11] H. LEWY, On the local character... Ann. Math., 64 (1956). 
  12. [12] R. NIRENBERG, On the Lewy extension phenomenon, Trans. Amer. Math. Soc., 168 (1972). Zbl0241.32006MR46 #392
  13. [13] R.O. WELLS, On the local holomorphic hull... Comm. P.A.M., vol. XIX (1966). Zbl0142.33901

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.