POROSITY OF CERTAIN SUBSETS OF LEBESGUE SPACES ON LOCALLY COMPACT GROUPS

Let G be a locally compact group. In this paper, we show that if G is a nondiscrete locally compact group, p epsilon (0, 1) and q epsilon (0 + infinity], then {(f, g) epsilon L-p(G) x L-q(G) : f * g is finite lambda- a.e.} is a set of first category in L-p(G) x L-q(G). We also show that if G is a nondiscrete locally compact group and p, q, r epsilon [1, + infinity] such that 1/p + 1/q > 1 + 1/r, then {(f,g) epsilon L-p(G) x L-q(G) : f * g epsilon L-r(G)}, is a set of first category in L-p(G) x L-q(G). Consequently, for p, q epsilon [1 + infinity) and r epsilon [1, + infinity] with 1/p + 1/q > 1 + 1/r, G is discrete if and only if L-p(G) * L-q(G) subset of L-r(G); this answers a question raised by Saeki ['The L-p-conjecture and Young's inequality', Illinois J. Math. 34 (1990), 615-627].