Abstract
Abstract 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.
Highlights
The aim of this paper is to define a new class of functions called local Morrey-Lorentz spaces Mplo,qc;λ(Rn) and to study the boundedness of the maximal operator in these spaces
We show that for Morrey-Lorentz spaces
We show that in the case λ < or λ >, the space Mplo,qc;λ(Rn) is trivial, and in the limiting case λ =, the space Mplo,qc; (Rn) is the classical Lorentz space
Summary
The aim of this paper is to define a new class of functions called local Morrey-Lorentz spaces Mplo,qc;λ(Rn) and to study the boundedness of the maximal operator in these spaces. Recall that the local Morrey-type spaces LMpθ,w were introduced and the boundedness in these spaces of the fractional integral operators and singular integral operators defined on homogeneous Lie groups were proved by Guliyev [ ] in the doctoral thesis (see, [ – ]). They are given by f LMpθ,w = w(r) f Lp(B( ,r)) Lθ ( ,∞), where w is a positive measurable function defined on ( , ∞).
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