Maximal theorems and square functions for analytic operators on Lp-spaces

TL;DR

This paper proves maximal inequalities and square function bounds for analytic contraction operators on Lp-spaces, including noncommutative cases, with applications to R-boundedness and semigroups.

Contribution

It establishes new boundedness results for Littlewood-Paley square functions and maximal inequalities for analytic operators, extending to noncommutative Lp-spaces and semigroup contexts.

Findings

Boundedness of Littlewood-Paley square functions for positive analytic contractions.

Maximal inequalities for operators of the form (n+1)^m |T^n(T-I)^m|.

Extension of results to noncommutative Lp-spaces and applications to R-boundedness.

Abstract

Let T : Lp --> Lp be a contraction, with p strictly between 1 and infinity, and assume that T is analytic, that is, there exists a constant K such that n\norm{T^n-T^{n-1}} < K for any positive integer n. Under the assumption that T is positive (or contractively regular), we establish the boundedness of various Littlewood-Paley square functions associated with T. As a consequence we show maximal inequalities of the form $\norm{\sup_{n\geq 0}\, (n+1)^m\bigl |T^n(T-I)^m(x) \bigr |}_p\,\lesssim\, \norm{x}_p$, for any nonnegative integer m. We prove similar results in the context of noncommutative Lp-spaces. We also give analogs of these maximal inequalities for bounded analytic semigroups, as well as applications to R-boundedness properties.