Maximal operators on the infinite-dimensional torus

Abstract

We study maximal operators related to bases on the infinite-dimensional torus Tω. For the normalized Haar measure dx on Tω it is known that MR0, the maximal operator associated with the dyadic basis R, is of weak type (1, 1), but MR, 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⊂ R′⊂ R, for which maximal functions have controlled, but sometimes very peculiar behavior. Precisely, for given q∈ [1 , ∞) we construct R′ such that MR′ is of restricted weak type (q, q) if and only if q belongs to a predetermined range of the form (q, ∞] or [q, ∞]. Finally, we study the weighted setting, considering the Muckenhoupt ApR(Tω) and reverse Hölder RHrR(Tω) classes of weights associated with R. For each p∈ (1 , ∞) and each w∈ApR(Tω) we obtain that MR is not bounded on Lq(w) in the whole range q∈ [1 , ∞). Since we are able to show that ⋃p∈(1,∞)ApR(Tω)=⋃r∈(1,∞)RHrR(Tω),the unboundedness result applies also to all reverse Hölder weights.