Convolution of discrete measures on linear groups

AbstractLet p,q∈R such that 1<p<2 and 2p=1+1q. Define(∗)‖f‖p′=maxx,G1(∑y∈G1|f(xy)|p)1/p, where G1 is taken in some class of subgroups specified later. We prove the following two theorems about convolutions.Theorem 2Let G=SL2(C) equipped with the discrete topology. Then there is a constant τ=τp>0 such that for f∈ℓp(G)‖f∗f‖q1/2⩽C‖f‖p1−τ(‖f‖p′)τ, where the maximum in (∗) is taken over all abelian subgroups G1<G and x∈G.Theorem 3There is a constant C=Cp>0 and 1>τ=τp>0 such that if f∈ℓp(SL3(Z)), then‖f∗f‖q1/2⩽C‖f‖p1−τ(‖f‖p′)τ, where the maximum in (∗) is taken over all nilpotent subgroups G1 of SL3(Z) and x∈SL3(Z)