Boundedness of the maximal operator in the local Morrey-Lorentz spaces (original) (raw)

In this paper we define a new class of functions called local Morrey-Lorentz spaces M loc p,q;λ (R n), 0 < p, q ≤ ∞ and 0 ≤ λ ≤ 1. These spaces generalize Lorentz spaces such that M loc p,q;0 (R n) = L p,q (R n). We show that in the case λ < 0 or λ > 1, the space M loc p,q;λ (R n) is trivial, and in the limiting case λ = 1, the space M loc p,q;1 (R n) is the classical Lorentz space ∞,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 loc p,q;λ (R n) are equal to weak Lebesgue spaces WL 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.