Ergodic theorems for Cesaro bounded operators in L1

We consider a positive invertible Lamperti operator T with positive inverse. Our result concerns the averages A(k,n), 0 <= k, n is an element of Z, and the ergodic maximal operator M(T)f = sup(k,n >= 0) vertical bar A(k,n)f vertical bar = sup(k,n >= 0) vertical bar 1/k + n + 1 Sigma(n)(j=-k) T(j)f vertical bar, the ergodic Hilbert transform defined by H-T f (x) = lim(n ->infinity)Sigma(n)(j=1) 1/j(T-j f(x) - T-j f(x)) and the ergodic power functions defined by P-r,P-T f (x) = (Sigma(infinity)(n=0)vertical bar A(n+1,0)f (x) -A(n,0)f (x)vertical bar(r)+vertical bar A(0,n+1)f (x) -A(0,n)f (x)vertical bar(r))(1/r), for 1 < r < +infinity. Several authors proved that if the averages are uniformly bounded in L-p, 1 < p < infinity, then these operators are bounded in L-p [23,24,27,28,21]. Regarding the case p = 1, Gillespie and Torrea [12] showed that this condition is not sufficient to assure M-T 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 L-1 and in L-infinity then M-T, H-T and P-r,P-T apply L-1 into weak-L-1 and the corresponding sequences of functions in L-1 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 A(n,n) in L-infinity by the assumption that, for a fixed p is an element of (1, infinity), the averages associated with a modified operator T-p, related with T, are uniformly bounded in L-p. We end the paper showing examples of nontrivial operators satisfying the assumptions of the two main results. (C) 2022 The Author(s). Published by Elsevier Inc.