Fix an integer k ≥ 2. Let G be an amenable group and (M, dμ) a measure space. For 0 ≤ j ≤ k, let 0 < p_j ≤ ∞, and assume that p_0 = p is given by 1/(p_0)=1/(p_1)+···+1/(p_k). Assume that for any 0 ≤ j ≤ k and any u ε G, R_u~j is a bounded map from the Banach space L~(p_j)(M) into itself. We denote by‖R_u~j‖_(op) the operator norm of R_u~j: L~(p_j) (M) → L~(p_j) (M). We say that R_u~j is strongly continuous if for any sequence u_n → u in the topology of G, we have ‖R_(u_n)~j-R_u~j‖_(op) → 0.
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