Boundedness of the maximal operator in the local Morrey-Lorentz spaces

In this paper we define a new class of functions called local Morrey-Lorentz spaces M p , q ; λ loc ( R n ) , 0 < p , q ≤ ∞ and 0 ≤ λ ≤ 1 . These spaces generalize Lorentz spaces such that M p , q ; 0 loc ( R n ) = L p , q ( R n ) . We show that in the case λ < 0 or λ > 1 , the space M p , q ; λ loc ( R n ) is trivial, and in the limiting case λ = 1 , the space M p , q ; 1 loc ( R n ) is the classical Lorentz space Λ ∞ , t 1 p − 1 q ( R n ) . We show that for 0 < q ≤ p < ∞ and 0 < λ ≤ q p , the local Morrey-Lorentz spaces M p , q ; λ loc ( R n ) are equal to weak Lebesgue spaces W L 1 p − λ q ( R n ) . We get an embedding between local Morrey-Lorentz spaces and Lorentz-Morrey spaces. Furthermore, we obtain the boundedness of the maximal operator in the local Morrey-Lorentz spaces. MSC: 42B20, 42B25, 42B35, 47G10.