Abstract
The aim of this paper is to study the behaviour of the ergodic Hilbert transform associated to a flow which is Cesàro bounded in the space of integrable functions. In particular we see that if the flow is Cesàro bounded in this space, and f and its ergodic Hilbert transform are integrable functions then the ergodic Hilbert transform is not only defined as an a.e. limit or a limit in measure, but it is also defined as a limit in the norm of the space of integrable functions. In order to prove this result, we show that the ergodic Hilbert transform and the ergodic maximal operator are of weak type (1, 1) if the flow is Cesàro bounded in the space of integrable functions. It is also shown that the ergodic Hilbert transform and the ergodic maximal operator are of strong type (p, p), with p greater than one and finite, if the flow is Cesàro bounded in the space of measurable functions with integrable pth-power. The last section of the paper is devoted to providing nontrivial examples of Cesàro bounded flows. The proofs use ideas of the theory of Muckenhoupt's weights.
