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

Jóse BonetAntonio GalbisR. Meise — 1997

Studia Mathematica

Let ( ω ) ( Ω ) denote the non-quasianalytic class of Beurling type on an open set Ω in n . For μ ( ω ) ' ( n ) the surjectivity of the convolution operator T μ : ( ω ) ( Ω 1 ) ( 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 ω ' ( n ) . These...

Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse

B. A. TaylorR. MeiseDietmar Vogt — 1990

Annales de l'institut Fourier

Solving a problem of L. Schwartz, those constant coefficient partial differential operators P ( D ) are characterized that admit a continuous linear right inverse on ( Ω ) or 𝒟 ' ( Ω ) , Ω an open set in R n . For bounded Ω with C 1 -boundary these properties are equivalent to P ( D ) being very hyperbolic. For Ω = R n they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial P .

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