Search

WOS: 000384638700002 In this paper we proved the boundedness of the Hardy- Littlewood maximal operator M, the Calderon- Zygmund operators T and the maximal Calderon- Zygmund operators T on the local Morrey- Lorentz spaces M-p,q ,lambda(loc) (R-n). Finally, we give some applications of these results. Science Development Foundation under Republic of AzerbaijanScience Development Foundation (SDF) - Azerbaijan [EIF-2013-9(15)-46/10/1]; Presidium Azerbaijan National Academy of ScienceAzerbaijan National Academy of Sciences (ANAS) The research of V.S. Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan, Grant [EIF-2013-9(15)-46/10/1] and by the grant of Presidium Azerbaijan National Academy of Science 2015.

WOS: 000371347600006 In this paper, we investigate the boundedness of the Hilbert transform H in the local Morrey-Lorentz spaces [GRAPHICS] , [GRAPHICS] , [GRAPHICS] . We prove that the operator H is bounded in [GRAPHICS] under the condition [GRAPHICS] , [GRAPHICS] . In the limiting case [GRAPHICS] , [GRAPHICS] , we prove that the operator H is bounded from the space [GRAPHICS] to the weak local Morrey-Lorentz space [GRAPHICS] . Also we show that for the limiting case [GRAPHICS] , [GRAPHICS] , the modified Hilbert transform [GRAPHICS] is bounded from the space [GRAPHICS] to the bounded mean oscillation space. Science Development Foundation under Republic of AzerbaijanScience Development Foundation (SDF) - Azerbaijan [EIF-2013-9(15)-46/10/1]; Presidium Azerbaijan National Academy of ScienceAzerbaijan National Academy of Sciences (ANAS) The research of V.S. Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF-2013-9(15)-46/10/1 and by the grant of Presidium Azerbaijan National Academy of Science 2015.

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.