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On the range of convolution operators on non-quasianalytic ultradifferentiable functions

Jóse Bonet; Antonio Galbis; R. Meise

Studia Mathematica (1997)

  • Volume: 126, Issue: 2, page 171-198
  • ISSN: 0039-3223

Abstract

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Let ℇ ( ω ) ( Ω ) denote the non-quasianalytic class of Beurling type on an open set Ω in ℝ n . For μ ∈ ℇ ( ω ) ' ( ℝ n ) the surjectivity of the convolution operator T μ : ℇ ( ω ) ( Ω 1 ) → ℇ ( ω ) ( Ω 2 ) is characterized by various conditions, e.g. in terms of a convexity property of the pair ( Ω 1 , Ω 2 ) and the existence of a fundamental solution for μ or equivalently by a slowly decreasing condition for the Fourier-Laplace transform of μ. Similar conditions characterize the surjectivity of a convolution operator S μ : D ω ' ( Ω 1 ) → D ω ' ( Ω 2 ) between ultradistributions of Roumieu type whenever μ ∈ ℇ ω ' ( ℝ n ) . These results extend classical work of Hörmander on convolution operators between spaces of C ∞ -functions and more recent one of Ciorănescu and Braun, Meise and Vogt.

How to cite

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Bonet, Jóse, Galbis, Antonio, and Meise, R.. "On the range of convolution operators on non-quasianalytic ultradifferentiable functions." Studia Mathematica 126.2 (1997): 171-198. <http://eudml.org/doc/216450>.

@article{Bonet1997,
abstract = {Let $ℇ_\{(ω)\}(Ω)$ denote the non-quasianalytic class of Beurling type on an open set Ω in $ℝ^n$. For $μ ∈ ℇ^\{\prime \}_\{(ω)\}(ℝ^n)$ the surjectivity of the convolution operator $T_μ: ℇ_\{(ω)\}(Ω_1) → ℇ_\{(ω)\}(Ω_2)$ is characterized by various conditions, e.g. in terms of a convexity property of the pair $(Ω_1, Ω_2)$ and the existence of a fundamental solution for μ or equivalently by a slowly decreasing condition for the Fourier-Laplace transform of μ. Similar conditions characterize the surjectivity of a convolution operator $S_μ: D^\{\prime \}_\{\{ω\}\}(Ω_1) → D^\{\prime \}_\{\{ω\}\}(Ω_2)$ between ultradistributions of Roumieu type whenever $μ ∈ ℇ^\{\prime \}_\{\{ω\}\}(ℝ^n)$. These results extend classical work of Hörmander on convolution operators between spaces of $C^∞$-functions and more recent one of Ciorănescu and Braun, Meise and Vogt.},
author = {Bonet, Jóse, Galbis, Antonio, Meise, R.},
journal = {Studia Mathematica},
keywords = {non-quasianalytic class of Beurling type; surjectivity of the convolution operator; ultradistributions of Roumieu},
language = {eng},
number = {2},
pages = {171-198},
title = {On the range of convolution operators on non-quasianalytic ultradifferentiable functions},
url = {http://eudml.org/doc/216450},
volume = {126},
year = {1997},
}

TY - JOUR
AU - Bonet, Jóse
AU - Galbis, Antonio
AU - Meise, R.
TI - On the range of convolution operators on non-quasianalytic ultradifferentiable functions
JO - Studia Mathematica
PY - 1997
VL - 126
IS - 2
SP - 171
EP - 198
AB - Let $ℇ_{(ω)}(Ω)$ denote the non-quasianalytic class of Beurling type on an open set Ω in $ℝ^n$. For $μ ∈ ℇ^{\prime }_{(ω)}(ℝ^n)$ the surjectivity of the convolution operator $T_μ: ℇ_{(ω)}(Ω_1) → ℇ_{(ω)}(Ω_2)$ is characterized by various conditions, e.g. in terms of a convexity property of the pair $(Ω_1, Ω_2)$ and the existence of a fundamental solution for μ or equivalently by a slowly decreasing condition for the Fourier-Laplace transform of μ. Similar conditions characterize the surjectivity of a convolution operator $S_μ: D^{\prime }_{{ω}}(Ω_1) → D^{\prime }_{{ω}}(Ω_2)$ between ultradistributions of Roumieu type whenever $μ ∈ ℇ^{\prime }_{{ω}}(ℝ^n)$. These results extend classical work of Hörmander on convolution operators between spaces of $C^∞$-functions and more recent one of Ciorănescu and Braun, Meise and Vogt.
LA - eng
KW - non-quasianalytic class of Beurling type; surjectivity of the convolution operator; ultradistributions of Roumieu
UR - http://eudml.org/doc/216450
ER -

References

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