Almost everywhere convergence and boundedness of Cesàro-$\alpha$ ergodic averages in $L_{p,q}$-spaces

Let $(X, \mu )$ be a $\sigma$-finite measure space and let $\tau$ be an ergodic invertible measure preserving transformation. We study the a.e. convergence of the Cesàro-$\alpha$ ergodic averages associated with $\tau$ and the boundedness of the corresponding maximal operator in the setting of $L_{p,q}(w\,d\mu)$ spaces.