Henkin-Ramirez formulas with weight factors

B. Berndtsson; Mats Andersson

Annales de l'institut Fourier (1982)

  • Volume: 32, Issue: 3, page 91-110
  • ISSN: 0373-0956

Abstract

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We construct a generalization of the Henkin-Ramírez (or Cauchy-Leray) kernels for the -equation. The generalization consists in multiplication by a weight factor and addition of suitable lower order terms, and is found via a representation as an “oscillating integral”. As special cases we consider weights which behave like a power of the distance to the boundary, like exp- ϕ with ϕ convex, and weights of polynomial decrease in C n . We also briefly consider kernels with singularities on subvarieties of domains in C n .

How to cite

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Berndtsson, B., and Andersson, Mats. "Henkin-Ramirez formulas with weight factors." Annales de l'institut Fourier 32.3 (1982): 91-110. <http://eudml.org/doc/74554>.

@article{Berndtsson1982,
abstract = {We construct a generalization of the Henkin-Ramírez (or Cauchy-Leray) kernels for the $\overline\{\partial \}$-equation. The generalization consists in multiplication by a weight factor and addition of suitable lower order terms, and is found via a representation as an “oscillating integral”. As special cases we consider weights which behave like a power of the distance to the boundary, like exp-$\phi $ with $\phi $ convex, and weights of polynomial decrease in $\{\bf C\}^n$. We also briefly consider kernels with singularities on subvarieties of domains in $\{\bf C\}^n$.},
author = {Berndtsson, B., Andersson, Mats},
journal = {Annales de l'institut Fourier},
keywords = {Henkin-Ramirez kernels; weight factor; delta-equation; pseudoconvex},
language = {eng},
number = {3},
pages = {91-110},
publisher = {Association des Annales de l'Institut Fourier},
title = {Henkin-Ramirez formulas with weight factors},
url = {http://eudml.org/doc/74554},
volume = {32},
year = {1982},
}

TY - JOUR
AU - Berndtsson, B.
AU - Andersson, Mats
TI - Henkin-Ramirez formulas with weight factors
JO - Annales de l'institut Fourier
PY - 1982
PB - Association des Annales de l'Institut Fourier
VL - 32
IS - 3
SP - 91
EP - 110
AB - We construct a generalization of the Henkin-Ramírez (or Cauchy-Leray) kernels for the $\overline{\partial }$-equation. The generalization consists in multiplication by a weight factor and addition of suitable lower order terms, and is found via a representation as an “oscillating integral”. As special cases we consider weights which behave like a power of the distance to the boundary, like exp-$\phi $ with $\phi $ convex, and weights of polynomial decrease in ${\bf C}^n$. We also briefly consider kernels with singularities on subvarieties of domains in ${\bf C}^n$.
LA - eng
KW - Henkin-Ramirez kernels; weight factor; delta-equation; pseudoconvex
UR - http://eudml.org/doc/74554
ER -

References

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  1. [1] B. BERNDTSSON, Integral formulas for the ∂∂-equation and zeros of bounded holomorphic functions in the unit ball, Math. Ann., 249 (1980), 163-176. Zbl0414.31007MR81m:32012
  2. [2] P. CHARPENTIER, Solutions minimales de l'équation ∂u = f dans la boule et dans le polydisque, Ann. Inst. Fourier, 30, 4 (1980), 121-153. Zbl0425.32009MR82j:32009
  3. [3] A. CUMENGE, Extension holomorphe dans des classes de Hardy, Thèse, Université Paul Sabatier de Toulouse, 1980. 
  4. [4] S.V. DAUTOV, G.M. HENKIN, Zeros of holomorphic functions of finite order and weighted estimates for solutions of the A T T -equation (russian), Math. Sb., 107 (1979), 163-174. Zbl0392.32001MR80b:32005
  5. [5] G.M. HENKIN, Integral representation of a function in a strictly pseudo-convex domain and applications to the A T T -problem (russian), Mat. Sb., 82 (1970), 300-308. Zbl0216.10402MR42 #534
  6. [6] G.M. HENKIN, Solutions with bounds of the equations of H. Lewy and Poincaré-Lelong. Construction of functions of the Nevanlinna class with given zeros in a strictly pseudoconvex domain, Soviet Math. Dokl., 16 (1976), 3-13. 
  7. [7] G.M. HENKIN, The Lewy equation and analysis on pseudoconvex manifolds, Russian Math. Surveys, 32 (1977), 59-130. Zbl0382.35038MR56 #12318
  8. [8] E. RAMIREZ DE ARELLANO, Ein Divisionsproblem und Randintegraldarstellungen in der komplexen Analysis, Math. Ann., 184 (1970), 172-187. Zbl0189.09702MR42 #4767
  9. [9] H. SKODA, Valeurs au bord pour les solutions de l’opérateur d " et caractérisation des zéros des fonctions de la classe de Nevanlinna, Bull. Soc. Math. France, 104 (1976), 225-299. Zbl0351.31007MR56 #8913
  10. [10] H. SKODA, d " -cohomologie à croissance lente dans Cn, Ann. Scient. Ec. Norm. Sup., 4 (1971), 97-120. Zbl0211.40402MR44 #4241
  11. [11] N. ØVRELID, Integral representation formulas and Lp-estimates for the A T T -equation, Math. Scand., 29 (1971), 137-160. Zbl0227.35069

Citations in EuDML Documents

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  1. Alexandre A. Shlapunov, On iterations of Green type integrals for matrix factorizations of the Laplace operator
  2. Xavier Massaneda, Interpolation by holomorphic functions in the unit ball with polynomial growth
  3. Jim Arlebrink, Zeros of bounded holomorphic functions in strictly pseudoconvex domains in 2
  4. Bo Berndtsson, Cauchy-Leray forms and vector bundles
  5. A. Yger, C. A. Berenstein, Traitement du signal et algorithmes explicites de déconvolution
  6. Miroslav Engliš, Jaak Peetre, Covariant differential operators and Green's functions
  7. Emmanuel Mazzilli, Équation de Cauchy-Riemann dans les ellipsoïdes réels de 𝐂 n
  8. C. A. Berenstein, A. Yger, Exponential polynomials and 𝒟 -modules
  9. C. Berenstein, A. Yger, The use of D-modules to study exponential polynomials
  10. Klas Diederich, Emmanuel Mazzilli, Extension and restriction of holomorphic functions

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