A Local Estimate for the Maximal Function in Lebesgue Spaces with EXP-Type Exponents

Abstract

It is proven that if 1 ≤ (⋅) < ∞ in a bounded domain Ω ⊂ R and if (⋅) ∈ EXP (Ω) for some > 0, then given ∈ (⋅) (Ω), the Hardy-Littlewood maximal function of , , is such that (⋅)log( ) ∈ EXP /( +1) (Ω). Because /( + 1) < 1, the thesis is slightly weaker than ( ) (⋅) ∈ 1 (Ω) for some > 0. The assumption that (⋅) ∈ EXP (Ω) for some > 0 is proven to be optimal in the framework of the Orlicz spaces to obtain (⋅)log( ) in the same class of spaces.