Convolution on L p -spaces of a locally compact group

Convolution on Lp -spaces of a locally compact group

F. Abtahi

• Department of Mathematics, University of Isfahan, Isfahan, Iran.

R. Nasr-Isfahani

• Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran.

A. Rejali

• Department of Mathematics, University of Isfahan, Isfahan, Iran

Abstract:

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 allf, 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.

Keywords: Convolution, locally compact group, L p –space.

MSC: Primary: 43A15, Secondary: 43A20.