On the weighted $\ell^p$-space of a discrete group

Abstract

Let G be a locally compact group and 1 < p < ∞. The L p -conjecture asserts that L p (G) is closed under the convolution if and only if G is compact. For 2 < p < ∞, we have recently shown that f * g exists and belongs to L ∞ (G) for all f, g ∈ L p (G) if and only if G is compact. Here, we consider the weighted case of this result for a discrete group G and a weight function ω on G; we prove that f * g exists and belongs to ℓ ∞ (G, 1/ ω) for all f, g ∈ ℓ p (G, ω) if and only if ℓ p (G, ω) ⊆ ℓ q (G, 1/ ω), the dual of ℓ p (G, ω).