The maximal function on variable spaces.

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