Weighted L p -conjecture for locally compact groups

Porosity and the L p-conjecture

“…Finally, we will generalize the second result from [1] (cf. Remark from [1] and note that any discrete topological group is unimodular).…”

“…During the next 30 years this conjecture was established in special cases, and, finally, in 1990 Saeki [4] proved the L p -conjecture in its general form. In [1] the conjecture was strengthened for p > 2. Namely, it was shown that if G is not compact, then for every compact, symmetric neighbourhood of the neutral element K, there exist functions f, g ∈ L p such that for any x ∈ K, f g(x) = ∞.…”

“…Remark from [1] and note that any discrete topological group is unimodular). This result is another strengthening of the L p -conjecture for p > 2 and unimodular groups.…”

Convolution on L p-spaces of a locally compact group

“…In the recent work [1], we proved that for 2 < p < ∞, the compactness of G is equivalent to the existence of f * g for all f, g ∈ L p (G); see [20], as well. We also showed that f * g exists for all f, g ∈ L p (G) if 1 < p < 2 and G is discrete, or p = 2 and G is unimodular.…”

“…It is well-known that L 1 (G) is always closed under the convolution. Saeki [22] proved that, for 1 < p < ∞, the space L p (G) is closed under the convolution if and only if G is compact; see also Burnham [4], Crombez [6]- [7], Duggal [8], Gaudet [10], Johnson [12], Kunze [13], Lohoue [14], Milnes [15], Rajagopalan [16]- [19], Rickert [20]- [21], Urbanik [23], Zelazko [24]- [27] for some special cases, Kinani and Benazzouz [9] and also Abtahi, Nasr Isfahani and Rejali [2]- [3] for the more general case of weighted L p -spaces.…”