In this paper we aim to extend a result of Hörmander's, that \mathcal{B}p,kloc(W)\subset\mathcal{C}m(W) if \frac{(1+\left|\cdot\right|)m}{k}\in Lp\prime, to the setting of vector valued local Hörmander-Beurling spaces, as well as to show that the space \bigcapj=1\infty\mathcal{B}pj,kjloc (W, E) (1\leq pj\leq\infty, kj=ejw, j=1,2,\dots) is topologically isomorphic to \mathcal{E}\omega(W, E ). Moreover, it is well known that the union of Sobolev spaces \mathcal{H}sloc(W) (=\mathcal{B}2,(1+|\cdot|2)s/2loc(W)) coincides with the space \mathcal{D}\prime\,F(W) of finite order distributions on W. We show that this is also verified in the context of vector valued Beurling ultradistributions.

In this paper we aim to extend a result of Hörmander's, that \mathcal{B}p,kloc(W)\subset\mathcal{C}m(W) if \frac{(1+\left|\cdot\right|)m}{k}\in Lp\prime, to the setting of vector valued local Hörmander-Beurling spaces, as well as to show that the space \bigcapj=1\infty\mathcal{B}pj,kjloc (W, E) (1\leq pj\leq\infty, kj=ejw, j=1,2,\dots) is topologically isomorphic to \mathcal{E}\omega(W, E ). Moreover, it is well known that the union of Sobolev spaces \mathcal{H}sloc(W) (=\mathcal{B}2,(1+|\cdot|2)s/2loc(W)) coincides with the space \mathcal{D}\prime\,F(W) of finite order distributions on W. We show that this is also verified in the context of vector valued Beurling ultradistributions.