The non-commutative neutrix product of the distributions $ln{x}_{+}$ and $x_{+}^{-s}$ is proved to exist for $s=1,2,...$ and is evaluated for $s=1,2$. The existence of the non-commutative neutrix product of the distributions $x_{+}^{-r}$ and $x_{+}^{-s}$ is then deduced for $r,s=1,2,...$ and evaluated for $r=s=1$.

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 give sufficient conditions for the support of the Fourier transform of a certain class of weighted integrable distributions to lie in the region $x_1\ge 0$ and $x_2\ge 0$.