H^∞Functional Calculus and Maximal Inequalities for Semigroups of Contractions on Vector-ValuedL_p-Spaces

Abstract

Let {Tt} t>0 be a strongly continuous semigroup of positive contractions on Lp(X, µ) with 1 < p < ∞. Let E be a UMD Banach lattice of measurable functions on another measure space (Ω, ν). For f ∈ Lp(X; E) define M(f )(x, ω) = sup t>0 1 t t 0

Vector valued q-variation for differential operators and semigroups I

“…[7] and [27]). Secondly, inequality (1.3) allows us to conclude the convergence (1.2) and the speed of the convergence, see Section 5 for the statements; Thirdly, this result can be regarded as a Banach latticevalued version of Corollary 4.5 in [16] by Le Merdy and Xu; Finally, this result immediately implies the main result-Theorem 2 of [34] by Xu in the radial case.…”

“…Among the works, we would like to mention the results obtained recently by (Ma/Torrea/Xu [19]) Le Merdy/Xu [16], where (weighted) variational inequalities for (differential operator) semigroups were established, as well as Xu [34] where vector-valued H ∞ functional calculus for analytic semigorups on vector-valued L p spaces was built. The results or idea in these papers will be exploited in the present paper.…”

“…The estimate (1.5) will be shown through approximation arguments based on the semigroup property and a continuous version of (1.5) established by Xu in [34] using vector-valued H ∞ functional calculus. We also explain shortly how one can adapt the arguments in [15] to deduce inequality (1.5) for any single analytic operator (replacing T t by T ) from Xu's vector-valued H ∞ functional calculus.…”

Dimension-free estimates for the vector-valued variational operators

“…In [43], Xu establishes UMD lattice-valued maximal functions for analytic semigroups. Here we state the result for the Possion semigroup.…”

Noncommutative harmonic analysis on semigroups

“…In this section, motivated by Proposition 7 in [31], we apply the maximal inequalities proved in the previous section to the pointwise ergodic convergence. To this end we need an appropriate analogue for the noncommutative setting of the usual almost everywhere convergence.…”