P-Functionale und Randwerte zu Hypoelliptischen Differentialoperatoren.
Darstellung von Distributionen endlicher Ordnung als Randwerte zu hypoelliptischen Differentialoperatoren.
On the Functional Dimension of Solution Spaces of Hypoelliptic Partial Differential Operators.
Bases in Spaces of Ultradifferentiable Functions with Compact Support.
Power Series Spaces and Weighted Solution Spaces of Partial Differential Equations.
Fortsetzung von Randwerten zu hypoelliptischen Differentialoperatoren und partielle Differentialgleichungen.
Bases in solution sheaves of systems of partial differential equations.
Differentiability and growth of solutions of partial differential equations.
Kolmogorov Diameters in Solution Spaces of Systems of Partial Differential Equations.
Randverteilungen von Nullösungen hypoelliptischer Differentialgleichungen.
Partial Differential Equations without Solution Operators in Weighted Spaces of (Generalized) Functions.
Dualraum und Topologie der (lokal) langsam wachsenden Nullösungen hypoelliptischer Differentialoperatoren.
Extension of ultradifferentiable functions.
Continuous linear right inverses for convolution operators in spaces of real analytic functions
We determine the convolution operators on the real analytic functions in one variable which admit a continuous linear right inverse. The characterization is given by means of a slowly decreasing condition of Ehrenpreis type and a restriction of hyperbolic type on the location of zeros of the Fourier transform μ̂(z).
Localizations of partial differential operators and surjectivity on real analytic functions
Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on an open set . Then P(D) admits shifted (generalized) elementary solutions which are real analytic on an arbitrary relatively compact open set ω ⊂ ⊂ Ω. This implies that any localization of the principal part is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for . Under additional assumptions must be locally hyperbolic.
Complemented kernels of partial differential operators in weighted spaces of (generalized) functions
Right inverses for partial differential operators on Fourier hyperfunctions
We characterize the partial differential operators P(D) admitting a continuous linear right inverse in the space of Fourier hyperfunctions by means of a dual (Ω̅)-type estimate valid for the bounded holomorphic functions on the characteristic variety near . The estimate can be transferred to plurisubharmonic functions and is equivalent to a uniform (local) Phragmén-Lindelöf-type condition.
Characterization of surjective partial differential operators on spaces of real analytic functions
Let A(Ω) denote the real analytic functions defined on an open set Ω ⊂ ℝⁿ. We show that a partial differential operator P(D) with constant coefficients is surjective on A(Ω) if and only if for any relatively compact open ω ⊂ Ω, P(D) admits (shifted) hyperfunction elementary solutions on Ω which are real analytic on ω (and if the equation P(D)f = g, g ∈ A(Ω), may be solved on ω). The latter condition is redundant if the elementary solutions are defined on conv(Ω). This extends and improves previous...
Asymptotic Fourier and Laplace transformations for hyperfunctions
We develop an elementary theory of Fourier and Laplace transformations for exponentially decreasing hyperfunctions. Since any hyperfunction can be extended to an exponentially decreasing hyperfunction, this provides simple notions of asymptotic Fourier and Laplace transformations for hyperfunctions, improving the existing models. This is used to prove criteria for the uniqueness and solvability of the abstract Cauchy problem in Fréchet spaces.
On the diametral dimension of weighted spaces of analytic germs
We prove precise estimates for the diametral dimension of certain weighted spaces of germs of holomorphic functions defined on strips near ℝ. This implies a full isomorphic classification for these spaces including the Gelfand-Shilov spaces and for α > 0. Moreover we show that the classical spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions are not isomorphic.
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