Ergodic theorems for Cesàro bounded operators in L1

Francisco J Martín-Reyes,Elena De La Rosa

doi:10.1016/j.jmaa.2021.125973

Abstract

We consider a positive invertible Lamperti operator T with positive inverse. Our result concerns the averages Ak,n, 0≤k,n∈Z, and the ergodic maximal operatorMTf=supk,n≥0⁡|Ak,nf|=supk,n≥0⁡|1k+n+1∑j=−knTjf|, the ergodic Hilbert transform defined by HTf(x)=limn→∞⁡∑j=1n1j(Tjf(x)−T−jf(x)) and the ergodic power functions defined by Pr,Tf(x)=(∑n=0∞|An+1,0f(x)−An,0f(x)|r+|A0,n+1f(x)−A0,nf(x)|r)1/r, for 1<r<+∞. Several authors proved that if the averages are uniformly bounded in Lp, 1<p<∞, then these operators are bounded in Lp[23,24,27,28,21]. Regarding the case p=1, Gillespie and Torrea [12] showed that this condition is not sufficient to assure MT is of weak type (1,1). We provide two sufficient conditions that recall the assumptions in the Dunford-Schwartz theorem: if the averages are uniformly bounded in L1 and in L∞ then MT, HT and Pr,T apply L1 into weak-L1 and the corresponding sequences of functions in L1 converge a.e. and in measure. Furthermore, we reach the same conclusions by assuming a weaker condition: we replace the uniform boundedness of the averages An,n in L∞ by the assumption that, for a fixed p∈(1,∞), the averages associated with a modified operator Tp, related with T, are uniformly bounded in Lp. We end the paper showing examples of nontrivial operators satisfying the assumptions of the two main results.

Highlights

By Lp := Lp(μ), 1 ≤ p < ∞, we denote the set of measurable functions f such that X |f |p dμ < ∞
We show in this paper that if the averages are uniformly bounded in L1 and in L∞ MT is of weak type (1, 1) but, that, we show that under the same assumptions we have that HT∗, HT and Pr,T are of weak type (1, 1) and the sequences An,nf, Hnf converge a.e. and in measure for all f ∈ L1
When we work in the real line and U : R → R is a nonnegative measurable function Lp(U ) is the Lebesgue space Lp(μ), where μ is the measure with density U with respect to Lebesgue measure

Summary

Introduction

Under these assumptions, it is proved that the maximal operator MT+ is bounded in Lp and the sequence of averages Anf converges a.e. and in Lp for all f ∈ Lp. In what follows, we shall work in the setting of [24] (or [29]). We show in this paper that if the averages are uniformly bounded in L1 and in L∞ MT is of weak type (1, 1) (which resembles Dunford-Schwartz ergodic theorem) but, that, we show that under the same assumptions we have that HT∗ , HT and Pr,T are of weak type (1, 1) and the sequences An,nf , Hnf converge a.e. and in measure for all f ∈ L1. The last section is devoted to provide examples of operators which satisfy the assumptions of the theorems

Notations

Lamperti operators

Statement of the main results

Examples