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Strong \(q\)-variation inequalities for analytic semigroups. (Inégalités de \(q\)-variation forte pour les semi-groupes analytiques.) (English. French summary) Zbl 1269.47011

The aim of this paper is to exhibit a large class of operators \(T\) on \(L^{p}(\Omega)\) for a fixed \(1 < p < \infty\) with the following property: for any \(2 < q < \infty\), there exists a constant \(C>0\) which may depend on \(q\) and \(T\) such that, for any \(x\in L^{p}(\Omega)\), the sequence \(\big([T^{n} (x)](\lambda)\big)_{n\geqslant 0}\) belongs to \(v^{q}\) for almost every \(\lambda \in \Omega \), with an estimate \(\|(T^{n}(x))_{n\geqslant 0}\|_{L^{p}(v^{q})}\leqslant C\|x\|_{p}\). The authors show that this holds true provided that \(T\) is a positive contraction and \(T\) is analytic, in the sense that \(\sup_{n\geq1}n\|T^n-T^{n-1}\|<\infty\). When the analyticity assumption is removed, then, for a positive contraction \(T\) on \(L^{p}(\Omega)\) with \(1 < p < \infty\), they obtain the estimate \(\|(M_{n}(T)x)_{n\geqslant 0}\|_{L^{p}(v^{q})}\leqslant C\|x\|_{p}\), \(x\in L^{p}(\Omega)\), \(1 < p < \infty\), where \(M_{n}(T)=(n+1)^{-1}\sum_{k=0}^n T^{k}\) denotes the ergodic average of \(T\). The authors also obtain similar results for strongly continuous semigroups. The paper concludes with some examples and applications.

MSC:

47A35 Ergodic theory of linear operators
37A99 Ergodic theory
47B38 Linear operators on function spaces (general)
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