A converse of Bernstein's inequality for locally compact groups

Abstract

Let G be a Hausdorff locally compact abelian group, Γ its character group. We shall prove that, if S is a translation-invariant subspace of Lp (G) (p ∈ [1, ∞]),for each a ∈ G and , then is relatively compact (where Σ(f) denotes the spectrum of f). We also obtain a similar result when G is a Hausdorff compact (not necessarily abelian) group. These results can be considered as a converse of Bernstein's inequality for locally compact groups.