Strong q-variation inequalities for analytic semigroups

By L. Christian, Quanhua Xu. 2011

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Let T : Lp --> Lp be a positive contraction, with p strictly between 1 and infinity. Assume that T is analytic, that is, there exists a constant K such that \norm{T^n-T^{n-1}} < K/n for any positive integer n. Let q strictly betweeen 2 and infinity and let v^q be the space of all complex sequences with a finite strong q-variation. We show that for any x in Lp, the sequence ([T^n(x)](\lambda))_{n\geq 0} belongs to v^q for almost every \lambda, with an estimate \norm{(T^n(x))_{n\geq 0}}_{Lp(v^q)}\leq C\norm{x}_p. If we remove the analyticity assumption, we obtain a similar estimate for the ergodic averages of T instead of the powers of T. We also obtain similar results for strongly continuous semigroups of positive contractions on Lp-spaces.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/461160ca3d9ca20e74d276225e21af7e3e3c0a6c

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Weighted variation inequalities for differential operators and singular integrals in higher dimensions (2013, background): “Bourgain’s work has been considerably improved by subsequent works and largely extended to many other operators in ergodic theory (see, for instance, [12, 14, 16]) and harmonic analysis (cf.”