Strong q-variation inequalities for analytic semigroups

Abstract

Let T : Lp(ω) → Lp(ω) be a positive contraction, with 1 < p < 1. Assume that T is analytic, that is, there exists a constant K ≥ 0 such that ||Tn -Tn-1|| ≤ K/n for any integer n ≥ 1. Let 2 < q < ∞ and let vq be the space of all complex sequences with a finite strong q-variation. We show that for any x 2 Lp(ω), the sequence ([Tn(x)] (λ) ) n≥0 belongs to vq for almost every λ 2 , with an estimate ||(Tn(x))n≥0||Lp(vq) ≤ C||x||p. If we remove the analyticity assumption, we obtain an estimate ||(Mn(T)x)n≥0||Lp(vq) ≤ C||x||p, where Mn(T) = (n + 1)-1σn k=0 Tk denotes the ergodic average of T. We also obtain similar results for strongly continuous semigroups (Tt)t≥0 of positive contractions on Lp-spaces.