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

Research paper by Canay Aykol, Vagif S Guliyev, Ayhan Serbetci

Indexed on: 26 Jul '13Published on: 26 Jul '13Published in: Journal of Inequalities and Applications



Abstract

In this paper we define a new class of functions called local Morrey-Lorentz spaces Mp,q;λloc(Rn)Open image in new window, 0<p,q≤∞Open image in new window and 0≤λ≤1Open image in new window. These spaces generalize Lorentz spaces such that Mp,q;0loc(Rn)=Lp,q(Rn)Open image in new window. We show that in the case λ<0Open image in new window or λ>1Open image in new window, the space Mp,q;λloc(Rn)Open image in new window is trivial, and in the limiting case λ=1Open image in new window, the space Mp,q;1loc(Rn)Open image in new window is the classical Lorentz space Λ∞,t1p−1q(Rn)Open image in new window. We show that for 0<q≤p<∞Open image in new window and 0<λ≤qpOpen image in new window, the local Morrey-Lorentz spaces Mp,q;λloc(Rn)Open image in new window are equal to weak Lebesgue spaces WL1p−λq(Rn)Open image in new window. 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.