Maximal and fractional maximal operators in the Lorentz Morrey spaces and their applications to the Bochner Riesz and Schrodinger type operators

Abstract

The aim of this paper is to obtain boundedness conditions for the maximal function M f and to prove the necessary and sufficient conditions for the fractional maximal oparator M α in the Lorentz-Morrey spaces L p,q;λ (R n ) which are a new class of functions. We get our main results by using the obtained sharp rearrangement estimates. The obtained results are applied to the boundedness of particular operators such as the Bochner-Riesz operator B δ r and the Schrödingertype operators V γ (−∆ + V ) −β and V γ ∇(−∆ + V ) −β in the Lorentz-Morrey spaces L p,q;λ (R n ), where the nonnegative potential V belongs to the reverse Hölder class B ∞ (R n ).