Maximal operators on the infinite-dimensional torus

Abstract

We study maximal operators related to bases on the infinitedimensional 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 R p (T ω ) and reverse Hölder RH R r (T ω ) classes of weights associated with R. For each p ∈ (1, ∞) and each w ∈ A R p (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 thatthe unboundedness result applies also to all reverse Hölder weights.

On the Doubling Condition in the Infinite-Dimensional Setting

“…• divergence of Fourier series of certain smooth functions [7]; • no Lebesgue differentiation theorem for natural differentiation bases [8,10]; • unboundedness of maximal operators [10,11]; • problems with introducing a satisfactory theory of weights [11].…”