Multiparameter ergodic Cesàro-α averages

Abstract

Let (X,F,ν) be a σ-finite measure space. Associated with k Lamperti operators on Lp(ν), T1,…,Tk, nˉ=(n1,…,nk)∈Nk and αˉ=(α1,…,αk) with 0<αj≤1, we define the ergodic Cesàro-αˉ averages Rnˉ,αˉf=1∏kj=1Aαjnj∑ik=0nk⋯∑i1=0n1∏j=1kAαj−1nj−ijTikk⋯Ti11f. For these averages we prove the almost everywhere convergence on X and the convergence in the Lp(ν) norm, when n1,…,nk→∞ independently, for all f∈Lp(dν) with p>1/α∗ where α∗=min1≤j≤kαj. In the limit case p=1/α∗, we prove that the averages Rnˉ,αˉf converge almost everywhere on X for all f in the Orlicz–Lorentz space Λ(1/α∗,φm−1) with φm(t)=t(1+log+t)m. To obtain the result in the limit case we need to study inequalities for the composition of operators Ti that are of restricted weak type (pi,pi). As another application of these inequalities we also study the strong Cesàro-αˉ continuity of functions.