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Continuous linear right inverses for convolution operators in spaces of real analytic functions

Studia Mathematica

We determine the convolution operators ${T}_{\mu }:=\mu *$ 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).

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

Studia Mathematica

Let ${ℇ}_{\left(\omega \right)}\left(\Omega \right)$ denote the non-quasianalytic class of Beurling type on an open set Ω in ${ℝ}^{n}$. For $\mu \in {ℇ}_{\left(\omega \right)}^{\text{'}}\left({ℝ}^{n}\right)$ the surjectivity of the convolution operator ${T}_{\mu }:{ℇ}_{\left(\omega \right)}\left({\Omega }_{1}\right)\to {ℇ}_{\left(\omega \right)}\left({\Omega }_{2}\right)$ is characterized by various conditions, e.g. in terms of a convexity property of the pair $\left({\Omega }_{1},{\Omega }_{2}\right)$ 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 ${S}_{\mu }:{D}_{\omega }^{\text{'}}\left({\Omega }_{1}\right)\to {D}_{\omega }^{\text{'}}\left({\Omega }_{2}\right)$ between ultradistributions of Roumieu type whenever $\mu \in {ℇ}_{\omega }^{\text{'}}\left({ℝ}^{n}\right)$. These...

Opérateurs pseudo-différentiels définis en un point

Annales Polonici Mathematici

We introduce the notion of pseudo-differential operators defined at a point and we establish a canonical one-to-one correspondence between such an operator and its symbol. We also prove the invertibility theorem for special type operators.

Optimal regularity for the pseudo infinity Laplacian

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we find the optimal regularity for viscosity solutions of the pseudo infinity Laplacian. We prove that the solutions are locally Lipschitz and show an example that proves that this result is optimal. We also show existence and uniqueness for the Dirichlet problem.

Partial differential operators of infinite order with constant coefficients on the space of analytic functions on the polydisc

Studia Mathematica

Real Analytic Curves in Fréchet Spaces and Their Duals.

Monatshefte für Mathematik

Strongly nonlinear problem of infinite order with ${L}^{1}$ data.

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

Syzygies of modules and applications to propagation of regularity phenomena.

Publicacions Matemàtiques

Propagation of regularity is considered for solutions of rectangular systems of infinite order partial differential equations (resp. convolution equations) in spaces of hyperfunctions (resp. C∞ functions and distributions). Known resulys of this kind are recovered as particular cases, when finite order partial differential equations are considered.

The existence and the continuation of holomorphic solutions for convolution equations in tube domains

Bulletin de la Société Mathématique de France

The micro-support of the complex defined by a convolution operator in tube domains

Banach Center Publications

Variational inequalities of strongly nonlinear elliptic operators of infinite order.

International Journal of Mathematics and Mathematical Sciences

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