Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces

Abstract

It is proved that the boundedness of the maximal operator M from a Lebesgue space L-p1 (R-n) to a general local Morrey-type space LMp2 theta,w(R-n) is equivalent to the boundedness of the embedding operator from L-p1(R-n) to LMp2 theta,(w)(R-n) and in its turn to the boundedness of the Hardy operator from L-p1/p2 (0,infinity) to the weighted Lebesgue space L-theta/p2,L-v (0,infinity) for a certain weight function v determined by the functional parameter w. This allows obtaining necessary and sufficient conditions on the function w ensuring the boundedness of M from L-p1 (R-n) to LMp2 theta,w(R-n) for any 0 1. These conditions with p(1) = p(2) = 1 are necessary and sufficient for the boundedness of M from L-1(R-n) to the weak local Morreytype space WLM1 theta,w(R-n).