Maximal operators on the infinite-dimensional torus

We study maximal operators related to bases on the infinite-dimensional torus $mathbb{T}^omega$. {For the normalized Haar measure $dx$ on $mathbb{T}^omega$ it is known that $M^{mathcal{R}_0}$, the maximal operator associated with the dyadic basis $mathcal{R}_0$, is of weak type $(1,1)$, but $M^{mathcal{R}}$, the operator associated with the natural general basis $mathcal{R}$, is not. We extend the latter result to all $q in [1,infty)$. Then we find a wide class of intermediate bases $mathcal{R}_0 subset mathcal{R} subset mathcal{R}$, for which maximal functions have controlled, but sometimes very peculiar behavior.} Precisely, for given $q_0 in [1, infty)$ we construct $mathcal{R}$ such that $M^{mathcal{R}}$ is of restricted weak type $(q,q)$ if and only if $q$ belongs to a predetermined range of the form $(q_0, infty]$ or $[q_0, infty]$. Finally, we study the weighted setting, considering the Muckenhoupt $A_p^mathcal{R}(mathbb{T}^omega)$ and reverse Holder $mathrm{RH}_r^mathcal{R}(mathbb{T}^omega)$ classes of weights associated with $mathcal{R}$. For each $p in (1, infty)$ and each $w in A_p^mathcal{R}(mathbb{T}^omega)$ we obtain that $M^{mathcal{R}}$ is not bounded on $L^q(w)$ in the whole range $q in [1,infty)$. Since we are able to show that [ bigcup_{p in (1, infty)}A_p^mathcal{R}(mathbb{T}^omega) = bigcup_{r in (1, infty)} mathrm{RH}_r^mathcal{R}(mathbb{T}^omega), ] the unboundedness result applies also to all reverse Holder weights.