An answer to a question on the convolution of functions

摘要

Let G be a locally compact group. Recently, G?a?b and Strobin [2] asked when f*g exists for all f,g ∈ L ~p(G) and also: is the set {(f,g) ∈ L ~p(G) × L ~p(G): f*g ∈ L ~p(G)}σ-c-lower porous (in particular, meager) for p ∈ (1,2]? In this paper, we answer these questions. In particular, we prove that if 1 < p < ∞, 1 ≤ q < ∞, and G is a non-unimodular locally compact group, then the set {(f,g) ∈ L ~p(G) × L ~q(G): f*g is not λ -a. e. finite on G} is a residual set in L ~p(G) × L ~q(G).