Sharp bound for the ergodic maximal operator associated to Cesàro bounded operators.

We consider positive invertible Lamperti operators T f ( x ) = h ( x ) Φ f ( x ) such that Φ has no periodic part. Let A n , T be the sequence of averages of T and M T the ergodic maximal operator. It is obvious that if M T is bounded on some L p , 1 < p < ∞ , then sup ⁡ ‖ A n , T ‖ L p ( ν ) ≤ ‖ M T ‖ L p ( ν ) < ∞ . It is known that the converse is true. In this paper we search the sharp dependence of the norm ‖ M T ‖ L p ( ν ) with respect to sup n ⁡ ‖ A n , T ‖ L p ( ν ) < ∞ . We prove that ‖ M T ‖ L p ( ν ) ≤ C ( p ) ( sup n ∈ N ⁡ ‖ A n , T ‖ L p ( d ν ) ) p ′ , where p ′ = p / ( p − 1 ) is the conjugate exponent and C ( p ) depends only on p . Furthermore, the exponent p ′ is sharp. Our results are closely related to Buckley's theorem about sharp bounds for the Hardy–Littlewood maximal function. [ABSTRACT FROM AUTHOR]