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

Abstract EN

It is proven that if 1≤p(·)<∞ in a bounded domain Ω⊂Rn and if p(·)∈EXPa(Ω) for some a>0, then given f∈Lp(·)(Ω), the Hardy-Littlewood maximal function of f, Mf, is such that p(·)log(Mf)∈EXPa/(a+1)(Ω).

Because a/(a+1)<1, the thesis is slightly weaker than (Mf)λp(·)∈L1(Ω) for some λ>0.

The assumption that p(·)∈EXPa(Ω) for some a>0 is proven to be optimal in the framework of the Orlicz spaces to obtain p(·)log(Mf) in the same class of spaces.