Maximal operator on variable Lebesgue spaces with radial exponent

Abstract

Consider general Lebesgue spaces with variable exponent p(.) and the Hardy-Littlewood maximal operator M. There are known sufficient conditions for p(.) which guarantee the boundedness of M on these spaces. These conditions are divided into two categories. The first one controls a local behavior of p(.) and the second one gives sufficient conditions to p(.) at infinity.We put in this paper emphasis to properties of p(.) at infinity. Certain sufficient conditions to p(.) at infinity are known to guarantee the boundedness of the maximal operator on variable Lebesgue spaces. In this paper we find a weaker condition to p(.) which still preserves the boundedness of M.Moreover, it is known that there exist some functions p(.) which have no limit at infinity for which the maximal operator is bounded. We give here a wider class of such functions p(.) with no limit which nevertheless preserves the boundedness of M.