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

It is proven that if 1 <= p(.) < infinity in a bounded domain Omega subset of R-n and if p(.) epsilon EXPa(Omega) for some a > 0, then given f epsilon L-p(.)(Omega), the Hardy-Littlewood maximal function of f, Mf, is such that p(.log(Mf) epsilon EXPa/(a+1)(Omega)). Because a/(a+1) < 1, the thesis is slightly weaker than (Mf)(lambda P(.)) epsilon L-1(Omega) for some lambda > 0. The assumption that p(.) epsilon EXPa(Omega)) 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.