Maximal Operators on the Infinite-Dimensional Torus

We study maximal operators related to bases on the infinite-dimensional torus 𝕋^ω . For the normalized Haar measure dx on 𝕋^ω it is known that M^ℛ_0 , the maximal operator associated with the dyadic basis ℛ_0 , is of weak type (1, 1), but M^ℛ , the operator associated with the natural general basis ℛ , is not. We extend the latter result to all q ∈ [1,∞ ) . Then we find a wide class of intermediate bases ℛ_0 ⊂ℛ' ⊂ℛ , for which maximal functions have controlled, but sometimes very peculiar behavior. Precisely, for given q_0 ∈ [1, ∞ ) we construct ℛ' such that M^ℛ' is of restricted weak type ( q , q ) if and only if q belongs to a predetermined range of the form (q_0, ∞ ] or [q_0, ∞ ] . Finally, we study the weighted setting, considering the Muckenhoupt A_p^ℛ(𝕋^ω ) and reverse Hölder RH_r^ℛ(𝕋^ω ) classes of weights associated with ℛ . For each p ∈ (1, ∞ ) and each w ∈ A_p^ℛ(𝕋^ω ) we obtain that M^ℛ is not bounded on L^q(w) in the whole range q ∈ [1,∞ ) . Since we are able to show that ⋃ _p ∈ (1, ∞ )A_p^ℛ(𝕋^ω ) = ⋃ _r ∈ (1, ∞ )RH_r^ℛ(𝕋^ω ), the unboundedness result applies also to all reverse Hölder weights.