Multiparameter ergodic Cesàro-α averages

A.L. BERNARDIS; R. CRESCIMBENI; C. FERRARI FREIRE

COLLOQUIUM MATHEMATICUM

Polish Academy of Sciences

Año: 2015 vol. 140 p. 15 - 29

0010-1354

Let (X, F, ν) be a σ-finite measure space. Associated with k Lamperti operators on Lp(ν), T1,?, Tk, n = (n1,?,nk) ∈ Nk and α = (α1,?, αk) with 0 < αj ≤ 1, we define the ergodic Cesàro-α averages (Equation Presented). 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≤k αj. In the limit case p = 1/α*, we prove that the averages Rn,αf converge almost everywhere on X for all f in the Orlicz-Lorentz space Λ(1/α*,ϕm?i) 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-α continuity of functions.