Compactness in translation invariant Banach spaces of distributions and compact multipliers

It is shown that for a comprehensive family of translation invariant Banach spaces (B, ∥ ∥_B) of (classes of) measurable functions or distributions on a locally compact group (including most of the spaces of interest in harmonic analysis) the following compactness criterion generalizing the well-known results due to Kolmogorov-Riesz-Weil concerning compact sets in L^p(G), 1 ? p < ∞, holds true: A closed subset M ? B is compact in B if and only if it satisfies the following conditions: (a) sup_{? ? M} ∥?∥_B < ∞; (b) $\text{? ? > 0 ?k ∈ K(G):∥k1???∥}{}_{\mathrm{B}}\text{? ?}$ for all $\text{?∈M}$; (c) $\text{?? > 0 ?h∈K(G):∥h???∥}{}_{\mathrm{B}}\text{? ?}$ for all $\text{?∈M}$. Among various applications a characterization of the space of all compact multipliers between suitable pairs of such spaces can be derived.