Multiparameter pointwise ergodic theorems for Markov operators on $L_\infty$

Let $P_1,\dots,P_d$ be commuting Markov operators on $L_{\infty}(X,\Cal F, \mu)$, where $(X, \Cal F,\mu)$ is a probability measure space. Assuming that each $P_i$ is either conservative or invertible, we prove that for every $f$ in $L_p(X, \Cal F,\mu)$ with $1\le p < \infty$ the averages
$$
A_nf=(n+1)^{-d}\sum_{0\le n_i\le n}P_1^{n_1}P_2^{n_2}\dots P_d^{n_d}f\quad (n\ge 0)
$$
converge almost everywhere if and only if there exists an invariant and equivalent finite measure $\lambda$ for which the Radon-Nikodym derivative $v=d\lambda/d\mu$ is in the dual space $L_{p'}(X,\Cal F,\mu)$. Next we study the case in which there exists $p_1$, with $1\le p_1\le\infty$, such that for every $f$ in $L_p(X,\Cal F,\mu)$ the limit function belongs to $L_{p_1}(X,\Cal F,\mu)$. We give necessary and sufficient conditions for this problem.