Convolution of discrete measures on linear groups

Let p,q∈Rp,q∈R such that 10τ=τp>0such that for  f∈ℓp(G)f∈ℓp(G)‖f∗f‖q1/2⩽C‖f‖p1−τ(‖f‖p′)τ,where the maximum in  (∗)(∗)is taken over all abelian subgroups  G10C=Cp>0and  1>τ=τp>01>τ=τp>0such that if  f∈ℓp(SL3(Z))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  G1G1of  SL3(Z)SL3(Z)and  x∈SL3(Z)x∈SL3(Z).