Michael Langenbruch

Representation of multipliers on spaces of real analytic functions

Abstract We consider multipliers on the spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We prove representation theorems in terms of analytic functionals and in terms of holomorphic functions. We characterize Euler differential operators among multipliers. Then we characterize when such operators are surjective or have a continuous linear right inverse on the space of real analytic functions over an interval not containing zero. In particular we solve the problem when Euler differential equation of infinite order has a solution in the space of real analytic functions on an interval not containing zero.

Analytic Extension of Smooth Functions

Let F be a closed proper subset of ℝn and let ℰ* be a class of ultradifferentiable functions. We give a new proof for the following result of Schmets and Valdivia on analytic modification of smooth functions: for every function ƒ ∈ ℰ* (ℝn) there is % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Randverteilungen von Nullösungen hypoelliptischer Differentialgleichungen

{\widetilde f} \in {\cal E}_{*}(\rm R ^{n})

Surjective partial differential operators on real analytic functions defined on open convex sets

which is real analytic on ℝnF and such that ∂a ƒ ¦F = ∂a ƒ ¦F for any a ∈ ℕ0n. For bounded ultradifferentiable functions ƒ we can obtain % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Partial differential equations without solution operators in weighted spaces of (generalized) functions

{\widetilde f}

Kolmogorov diameters in solution spaces of systems of partial differential equations

by means of a continuous linear operator.

Differentiability and growth of solutions of partial differential equations

In this paper a distributional boundary value is defined for solutions f (defined on ℝn+1\ℝn) of a partially hypoelliptic differential operator (on ℝn+1)with constant coefficients. Then the following is equivalent:(1)f has a distributional boundary value.(2)f can be continued to ℝn+1 as a distribution. For hypoelliptic operators this is equivalent to:(3)f ist a locally slowly growing function. A topology is given on this function space, that makes the boundary value mapping a topological homomorphism.

Fundamental solutions with partially bounded support

Abstract: Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on a covex open set Ω⊂ℝn. Let L(Pm) denote the localizations at ∞ (in the sense of Hörmander) of the principal part Pm. Then Q(x+iτN)≠ 0 for (x,τ)∈ℝn×(ℝ\{ 0}) for any Q∈L(Pm) if N is a normal to δΩ which is noncharacteristic for Q. Under additional assumptions this implies that Pm must be locally hyperbolic.

Multiplier projections on spaces of real analytic functions in several variables

Let P(D) be a hypoelliptic pdo with constant coefficients and let E(M) (and(WM,∞b′) be the weighted spaces of C∞-functions (resp. of distributions) defined by Palamodov (resp. Gelfand/Shilov), where M is a radially symmetric weight function, eventually satisfying some mild technical conditions. Then P(D) has no continuous linear right inverse in E(M). If the index of hypoellipticity w.r.t. some x is greater than 1 and is minimal in a certain sense, then P(D) has no right inverse in (WM,∞)b′.If P(D) is elliptic however, then a continuous linear right inverse for P(D) exists in (WM,∞)b′.

Dualraum und Topologie der (lokal) langsam wachsenden Nullösungen hypoelliptischer Differentialoperatoren

Let Ep(0) denote the solutions (on 0<RN) of a system P(D) of partial differential equations with constant coefficients in a localizable analytically uniform space E (defined on 0). The relative Kolmogorov diameters of the neighbourhoods of 0 in Ep(0) are estimated from above and below, using the fundamental principle of Ehrenpreis. The diametral dimension, of Ep(0) is calculated and it is proved, that Ep(RN) and Ep(0) are nonisomorphic for (partially) bounded 0, in special cases.