Necessary and sufficient conditions for the boundedness of the maximal operator from lebesgue spaces to morrey-type spaces

It is proved that the boundedness of the maximal operator M from a Lebesgue space Lp1 (Rn) to a general local Morrey-type space LMpθ ,w(Rn) is equivalent to the boundedness of the embedding operator from Lp1 (Rn) to LMpθ ,w(Rn) and in its turn to the boundedness of the Hardy operator from L p1 p2 (0,8) to the weighted Lebesgue space L θ p2 ,v(0,8) 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 Lp1 (Rn) to LMp2θ ,w(Rn) for any 0> <8, 0> p2 ≤ p1 <8, p1 ≥1. These conditions with p1 = p2 =1 are necessary and sufficient for the boundedness of M from L1(R n) to the weak local Morreytype space WLM1 ,w(Rn) .