Abstractcharacterization of the spaces dual to weighted Lorentz spaces are given by means of reverse Hölder inequalities (Theorems 2.1, 2.2). This principle of duality is then applied to characterize weight functions for which the identity operator, the Hardy-Littlewood maximal operator and the Hilbert transform are bounded on weighted Lorentz spaces.
Abstract. It is proved that the boundedness of the maximal operator M from a Lebesgue space L p 1 (R n ) to a general local Morrey-type space LM p 2 θ ,w (R n ) is equivalent to the boundedness of the embedding operator from L p 1 (R n ) to LM p 2 θ ,w (R n ) and in its turn to the boundedness of the Hardy operator from L p 1 p 2 (0,∞) to the weighted Lebesgue space L θ These conditions with p 1 = p 2 = 1 are necessary and sufficient for the boundedness of M from L 1 (R n ) to the weak local Morreytype space W LM 1θ ,w (R n ) .Mathematics subject classification (2010): 42B20, 42B25, 42B35.
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