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 $({\Omega }_1,{\Omega }_2)$ 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 $\mu \in {ℇ}_{\omega }^{\text{'}}\left({ℝ}^n\right)$. 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.