The maximal function on variable L-p spaces

We give continuity conditions on the exponent function p(x) which are sufficient for the Hardy-Littlewood maximal operator to be bounded on the variable Lebesgue space L-p(x)(Omega), where Omega is any open subset of R-n. Further, our conditions are necessary on R. Our result extends the recent work of Pick and Ruzitka [20], Diening [3] and Nekvinda [19]. We also show that under much weaker assumptions on p(x), the maximal operator satisfies a weak-type modular inequality.