Convolution on L p -spaces of a locally compact group

We have recently shown that, for 2 < p < ∞, a locally compact group G is compact if and only if the convolution multiplication f * g exists for all f, g ∈ L ^p (G). Here, we study the existence of f * g for all f, g ∈ L ^p (G) in the case where 0 < p ≤ 2. Also, for 0 < p < ∞, we offer some necessary and sufficient conditions for L ^p (G) * L ^p (G) to be contained in certain function spaces on G.