The ∂ ¯ -Neumann operator and commutators of the Bergman projection and multiplication operators

Friedrich Haslinger

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 4, page 1247-1256
  • ISSN: 0011-4642

Abstract

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We prove that compactness of the canonical solution operator to ∂ ¯ restricted to ( 0 , 1 ) -forms with holomorphic coefficients is equivalent to compactness of the commutator [ 𝒫 , M ¯ ] defined on the whole L ( 0 , 1 ) 2 ( Ω ) , where M ¯ is the multiplication by z ¯ and 𝒫 is the orthogonal projection of L ( 0 , 1 ) 2 ( Ω ) to the subspace of ( 0 , 1 ) forms with holomorphic coefficients. Further we derive a formula for the ∂ ¯ -Neumann operator restricted to ( 0 , 1 ) forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by z and z ¯ .

How to cite

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Haslinger, Friedrich. "The $\bar{\partial }$-Neumann operator and commutators of the Bergman projection and multiplication operators." Czechoslovak Mathematical Journal 58.4 (2008): 1247-1256. <http://eudml.org/doc/37901>.

@article{Haslinger2008,
abstract = {We prove that compactness of the canonical solution operator to $\bar\{\partial \}$ restricted to $(0,1)$-forms with holomorphic coefficients is equivalent to compactness of the commutator $[\mathcal \{P\},\bar\{M\}]$ defined on the whole $L^2_\{(0,1)\}(\Omega ),$ where $\bar\{M\}$ is the multiplication by $\bar\{z\}$ and $\mathcal \{P\} $ is the orthogonal projection of $L^2_\{(0,1)\}(\Omega )$ to the subspace of $(0,1)$ forms with holomorphic coefficients. Further we derive a formula for the $\bar\{\partial \}$-Neumann operator restricted to $(0,1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by $z $ and $\bar\{z\}$.},
author = {Haslinger, Friedrich},
journal = {Czechoslovak Mathematical Journal},
keywords = {$\bar\{\partial \}$-equation; $\bar\{\partial \}$-Neumann operator; compactness; -equation; compactness},
language = {eng},
number = {4},
pages = {1247-1256},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The $\bar\{\partial \}$-Neumann operator and commutators of the Bergman projection and multiplication operators},
url = {http://eudml.org/doc/37901},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Haslinger, Friedrich
TI - The $\bar{\partial }$-Neumann operator and commutators of the Bergman projection and multiplication operators
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 1247
EP - 1256
AB - We prove that compactness of the canonical solution operator to $\bar{\partial }$ restricted to $(0,1)$-forms with holomorphic coefficients is equivalent to compactness of the commutator $[\mathcal {P},\bar{M}]$ defined on the whole $L^2_{(0,1)}(\Omega ),$ where $\bar{M}$ is the multiplication by $\bar{z}$ and $\mathcal {P} $ is the orthogonal projection of $L^2_{(0,1)}(\Omega )$ to the subspace of $(0,1)$ forms with holomorphic coefficients. Further we derive a formula for the $\bar{\partial }$-Neumann operator restricted to $(0,1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by $z $ and $\bar{z}$.
LA - eng
KW - $\bar{\partial }$-equation; $\bar{\partial }$-Neumann operator; compactness; -equation; compactness
UR - http://eudml.org/doc/37901
ER -

References

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