Convolution of Discrete Measures on Linear Groups

Let p, q ∈ R such that 1 < p < 2 and 2 p = 1 + 1q . Define ‖f‖p = max x,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 2. Let G = SL2(C) equipped with the discrete topology. Then there is a constant τ = τp > 0 such that for f ∈ `(G) ‖f ∗ f‖1/2 q ≤ C‖f‖1−τ p (‖f‖p) , where the maximum in (∗) is taken over all abelian subgroups G1 < G and x ∈ G. Theorem 3. There is a constant C = Cp > 0 and 1 > τ = τp > 0 such that if f ∈ `pSL3(Z) ) , then ‖f ∗ f‖1/2 q ≤ C‖f‖1−τ p (‖f‖p) where the maximum in (∗) is taken over all nilpotent subgroups G1 of SL3(Z) and x ∈ SL3(Z). This paper is a continuation of our earlier work [C] on product theorems in the groups SL2 and SL3.We show here how they may be applied to obtain nontrivial convolution estimates of discrete measures on SL2(C) and SL3(Z) (see Theorem 2 in 1partially supported by NSF. 2000 Mathematics Subject Classification. 05A99, 15A99, 05C25; 20G40, 20D60, 11B75 .