On the generalization of the Riesz–Zygmund theorem for the ergodic Hilbert transform

Abstract

A reverse weak type inequality for the conjugate function is generalized to the ergodic case. The inequality allows us to prove the Riesz–Zygmund theorem (if $f\geq 0$ and $f,\widetilde{f}\in L$, then $f\in L\log L$) for the ergodic Hilbert transform.