Decay stuff

Sections 3, 4 and 5 give results under Lp assumptions without any further assumptions on the decay rates. Section 3 deals with the case r ∈ Lp, p > 2. Section 4 generalises to p > 1. Section 5 covers the case p = 1 for mean zero observables. There is still the general p = 1 case to consider! The optimal result in the stretched exponential case is given in Section 6. Section 7 gives an alternative proof for the case of polynomial decrease (but it is already covered by the results in Section 3 and 4). 1 Preliminaries Formally, recall that T = (I −R)−1 and T ′ = (I −R′)−1. Hence T = (I + A′)−1T ′, A′ = T ′(R′ −R) T ′ = (I − Ã′)−1T, Ã′ = T (R′ −R). Multiplying throughout by (1− z), we obtain B = (I + A′)−1B′, B′ = (I − Ã′)−1B. In addition, A′ = B′(1− z)−1(R′ −R) = B′C ′, Ã′ = B(1− z)−1(R′ −R) = BC ′, where C ′ = (1− z)−1(R′ −R). To make these formulas rigorous, first note that (I − R)−1 and (I − R′)−1 are well defined on D. In particular, the definitions B = ((I − R)/(1 − z))−1 and B′ = ((I −R′)/(1− z))−1 make sense on D, as do the definitions of A′ and Ã′.