On the L p -conjecture for locally compact groups

摘要

Let G be a locally compact group. For 1 < p < ∞, it is well-known that f * g exists and belongs to L p (G) for all f, g $in$ L p (G) if and only if G is compact. Here, for 2 < p < ∞, we show that f * g exists for all f, g $in$ L p (G) if and only if G is compact. We also show that this result does not remain true for 1 < p ≤ 2.