New distribution spaces associated to translation-invariant Banach spaces

We introduce and study new distribution spaces, the test function space $\mathcal{D}_E$ and its strong dual $\mathcal{D}'_{E'_{\ast}}$. These spaces generalize the Schwartz spaces $\mathcal{D}_{L^{q}}$, $\mathcal{D}'_{L^{p}}$, $\mathcal{B}'$ and their weighted versions. The construction of our new distribution spaces is based on the analysis of a suitable translation-invariant Banach space of distributions $E$ with continuous translation group, which turns out to be a convolution module over a Beurling algebra $L^{1}_{\omega}$. The Banach space $E'_{\ast}$ stands for $L_{\check{\omega}}^1\ast E'$. We also study convolution and multiplicative products on $\mathcal{D}'_{E'_{\ast}}$.