Convergence of the lacunary ergodic Cesàro averages

Let T be a positive linear operator with positive inverse. We consider in this paper the ergodic Cesàro-α averages A n , α f , 0 < α ⩽ 1 , and the ergodic Cesàro-α maximal operator associated to T. For Lebesgue spaces L p ( ν ) , it is known that the good range for the convergence of the Cesàro-α averages and the boundedness of the maximal operator is 1 / α < p < + ∞ . In this paper we study the convergence of A n k , α f , where { n k } is a lacunary sequence, and the boundedness of its associated ergodic maximal operator. We get positive results in the range 1 ⩽ p < 1 1 − α . We use transference arguments which leads to us to study in depth weighted inequalities of the lacunary Cesàro-α maximal operator in the setting of the integers and in the setting of the real line.