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1
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Localizations of partial differential operators and surjectivity on real analytic functions

100%
Langenbruch M.
Studia Mathematica
|
2000
|
tom 140
|
nr 1
15-40
EN
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 $ Ω ⊂ ℝ^n$. 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 $P_{m,Θ}$ of the principal part $P_m$ is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for $P_{m,Θ}$. Under additional assumptions $P_m$ must be locally hyperbolic.
2
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Characterization of surjective convolution operators on Sato's hyperfunctions

100%
Langenbruch M.
Banach Center Publications
|
2010
|
tom 88
|
nr 1
185-193
EN
Let $μ ∈ 𝓐 (ℝ^{d})'$ be an analytic functional and let $T_μ$ be the corresponding convolution operator on Sato's space $𝓑 (ℝ^{d})$ of hyperfunctions. We show that $T_μ$ is surjective iff $T_μ$ admits an elementary solution in $𝓑 (ℝ^{d})$ iff the Fourier transform μ̂ satisfies Kawai's slowly decreasing condition (S). We also show that there are $0 ≠ μ ∈ 𝓐 (ℝ^{d})'$ such that $T_μ$ is not surjective on $𝓑 (ℝ^{d})$.
3
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Continuous linear right inverses for convolution operators in spaces of real analytic functions

100%
Langenbruch M.
Studia Mathematica
|
1994
|
tom 110
|
nr 1
65-82
EN
We determine the convolution operators $T_μ := μ*$ 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).
4
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Right inverses for partial differential operators on Fourier hyperfunctions

100%
Langenbruch M.
Studia Mathematica
|
2007
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tom 183
|
nr 3
273-299
EN
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 $V_{P}$ near $ℝ^{d}$. The estimate can be transferred to plurisubharmonic functions and is equivalent to a uniform (local) Phragmén-Lindelöf-type condition.
5
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Bases in spaces of analytic germs

100%
Langenbruch M.
Annales Polonici Mathematici
|
2012
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tom 106
|
nr 1
223-243
EN
We prove precise decomposition results and logarithmically convex estimates in certain weighted spaces of holomorphic germs near ℝ. These imply that the spaces have a basis and are tamely isomorphic to the dual of a power series space of finite type which can be calculated in many situations. Our results apply to the Gelfand-Shilov spaces $S¹_{α}$ and $S₁^{α}$ for α > 0 and to the spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions.
6
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Characterization of surjective partial differential operators on spaces of real analytic functions

100%
Langenbruch M.
Studia Mathematica
|
2004
|
tom 162
|
nr 1
53-96
EN
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 results of Andersson, Kawai, Kaneko and Zampieri. For convex Ω, a different characterization of surjective operators P(D) on A(Ω) was given by Hörmander using a Phragmén-Lindelöf type condition, which cannot be extended to the case of noncovex Ω. The paper is based on a surjectivity criterion for exact sequences of projective (DFS)-spectra which improves earlier results of Braun and Vogt, and Frerick and Wengenroth.
7
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On the diametral dimension of weighted spaces of analytic germs

100%
Langenbruch M.
Studia Mathematica
|
2016
|
tom 233
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nr 1
85-100
EN
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 $S¹_{α}$ and $S₁^{α}$ for α > 0. Moreover we show that the classical spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions are not isomorphic.
8
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Asymptotic Fourier and Laplace transformations for hyperfunctions

100%
Langenbruch M.
Studia Mathematica
|
2011
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tom 205
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nr 1
41-69
EN
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.
9
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Algebra of multipliers on the space of real analytic functions of one variable

64%
Domański P., Langenbruch M.
Studia Mathematica
|
2012
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tom 212
|
nr 2
155-171
EN
We consider the topological algebra of (Taylor) multipliers on spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We describe multiplicative functionals and algebra homomorphisms on that algebra as well as idempotents in it. We show that it is never a Q-algebra and never locally m-convex. In particular, we show that Taylor multiplier sequences cease to be so after most permutations.
10
Content available remote

Complemented kernels of partial differential operators in weighted spaces of (generalized) functions

63%
Langenbruch M.
Studia Mathematica
|
1988
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tom 89
|
nr 1
37-61
11
Content available remote

A note on composition operators on spaces of real analytic functions

52%
Domański P., Goliński M., Langenbruch M.
Annales Polonici Mathematici
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2012
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tom 103
|
nr 2
209-216
EN
We characterize composition operators on spaces of real analytic functions which are open onto their images. We give an example of a semiproper map φ such that the associated composition operator is not open onto its image.
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