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Hardy-Type Operators in Lorentz-Type Spaces Defined on Measure Spaces

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Abstract

Weight criteria for the boundedness and compactness of generalized Hardy-type operators

$$Tf(x) = {u_1}(x)\int_{\left\{ {\phi (y) \le \psi (x)} \right\}} {f(y){u_2}(y){v_0}(y)d\mu (y),\;\;\;\;\;x \in X},$$
((0.1))

in Orlicz-Lorentz spaces defined on measure spaces is investigated where the functions ϕ, ψ, u1, u2, v0 are positive measurable functions. Some sufficient conditions of boundedness of \(T:\;\Lambda _{{v_0}}^{{G_0}}({w_0}) \to \Lambda _{{v_1}}^{{G_1}}({w_1})\) and \(T:\;\Lambda _{{v_0}}^{{G_0}}({w_0}) \to \Lambda _{{v_1}}^{{G_1},\infty }({w_1})\) are obtained on Orlicz-Lorentz spaces. Furthermore, we achieve sufficient and necessary conditions for T to be bounded and compact from a weighted Lorentz space \(\Lambda _{{v_0}}^{{p_0}}({w_0})\) to another \(\Lambda _{{v_1}}^{{p_1},{q_1}}({w_1})\). It is notable that the function spaces concerned here are quasi-Banach spaces instead of Banach spaces.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, 310023, China

    Qinxiu Sun

  2. Department of Mathematics, Shangrao Normal University, Shangrao, 334001, China

    Xiao Yu

  3. Department of Mathematics, Zhejiang International Studies University, Hangzhou, 310012, China

    Hongliang Li

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  1. Qinxiu Sun
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  2. Xiao Yu
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  3. Hongliang Li
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Correspondence to Hongliang Li.

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Supported by Natural Science Foundation of Zhejiang Province of China (LY19A010001), National Natural Science Foundation of China (11961056), Natural Science Foundation of Jiangxi Province of China (20151BAB211002).

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Sun, Q., Yu, X. & Li, H. Hardy-Type Operators in Lorentz-Type Spaces Defined on Measure Spaces. Indian J Pure Appl Math 51, 1105–1132 (2020). https://doi.org/10.1007/s13226-020-0453-1

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