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;lambda(loc)(R-n), 0 < p,q <= infinity and 0 <= lambda <= 1. These spaces generalize Lorentz spaces such that M-p,q;0(loc) (R-n) = L-p,L-q(R-n). We show that in the case lambda < 0 or lambda > 1, the space M-p,q;lambda(loc) (R-n) is trivial, and in the limiting case lambda = 1, the space M-p,q;1(loc) (R-n) is the classical Lorentz space Lambda (infinity,t1/p - 1/q) (R-n). We show that for 0 < q <= p < infinity and 0 < lambda <= q/p, the local Morrey-Lorentz spaces M-p,q;lambda(loc) (R-n) are equal to weak Lebesgue spaces WL1/p-lambda/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.