Weights for the ergodic maximal operator and a.e. convergence of the ergodic averages for functions in Lorentz spaces (Q1313231)

Let \(T\) be an invertible measure preserving transformation in a measure space \((X, {\mathcal F}, \mu)\). Put \(T_{n,m}f = (n + m + 1)^{-1} \sum^ m_{j = - n} f \circ T^ j\), \(Mf = \sup_{n,m} T_{n,m} | f |\). For \(1 \leq q \leq p < \infty\) let \(L_{p,q} (v)\) denote the Lorentz space of functions \(f\) with \[ \| f \|_{p,q,v} : = \left( q \int^ \infty_ 0 \Bigl( \int_{\{x/ | f(x) | > y\}} vdy \Bigr)^{q/p} y^{q - 1} dy \right)^{1/q} < \infty. \] For \(q = \infty\), put \[ \| f \|_{p,q,v} = \sup_{y>0} y \Bigl( \int_{\{x/ | f(x) | > y\}} vd \mu \Bigr)^{1/p}. \] The author extends several results of \textit{F. J. Martin-Reyes} [Trans. Am. Math. Soc. 296, 61-82 (1986; Zbl 0612.28014)] to \(L_{p,q}\)-spaces. Example: Let \(1 \leq q \leq p < \infty\) and let \(u,v\) be positive measurable functions. The following statements are equivalent: (a) \(\| Mf \|_{p, \infty,u} \leq C \| f \|_{p,q,v}\), (b) \(\sup_{n,m \geq 0} \| T_{n,m} f \|_{p, \infty,u} \leq C \| f \|_{p,q,v}\), (c) \((\sum^ k_{i = 0} u \circ T^ i)/(\sum^ k_{i = 0} v^{1 - p'} \circ T^ i)^{p - 1} \leq C(k + 1)^ p\) (with \(C\) indep. of \(k\) and \(pp' = p + p')\).