Maximal operators on the infinite-dimensional torus

Abstract

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 ${\mathcal{R}}_0\subset {\mathcal{R}}^{\prime }\subset \mathcal{R}$, for which maximal functions have controlled, but sometimes very peculiar behavior.} Precisely, for given $q_0\in [1,\infty )$ we construct ${\mathcal{R}}^{\prime }$ such that $M^{{\mathcal{R}}^{\prime }}$ 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 Hölder ${\mathrm{RH}}_r^{\mathcal{R}}({\mathbb{T}}^{\omega })$ 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 \[ \bigcup_{p \in (1, \infty)}A_p^\mathcal{R}(\mathbb{T}^ω) = \bigcup_{r \in (1, \infty)} \mathrm{RH}_r^\mathcal{R}(\mathbb{T}^ω), \] the unboundedness result applies also to all reverse Hölder weights.