### Characterization of the $\omega $ hypoelliptic convolution operators on ultradistributions.

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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).

Let ${\u2107}_{\left(\omega \right)}\left(\Omega \right)$ denote the non-quasianalytic class of Beurling type on an open set Ω in ${\mathbb{R}}^{n}$. For $\mu \in {\u2107}_{\left(\omega \right)}^{\text{'}}\left({\mathbb{R}}^{n}\right)$ the surjectivity of the convolution operator ${T}_{\mu}:{\u2107}_{\left(\omega \right)}\left({\Omega}_{1}\right)\to {\u2107}_{\left(\omega \right)}\left({\Omega}_{2}\right)$ is characterized by various conditions, e.g. in terms of a convexity property of the pair $({\Omega}_{1},{\Omega}_{2})$ 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 ${S}_{\mu}:{D}_{\omega}^{\text{'}}\left({\Omega}_{1}\right)\to {D}_{\omega}^{\text{'}}\left({\Omega}_{2}\right)$ between ultradistributions of Roumieu type whenever $\mu \in {\u2107}_{\omega}^{\text{'}}\left({\mathbb{R}}^{n}\right)$. These...

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.

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.

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.