On the weighted lp- space of a discrete group

Abstract

Let G be a locally compact group and 1 < p < ∞. The L p-conjecture asserts that LP(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 ε LP(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/ωÃ) for all f, g ε lp(G,ω) if and only if lp(G,ω) Ç lq(G, 1/ωÃ), the dual of lP(G, ωÃ).