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

Qinxiu Sun, Xiao Yu and Hongliang Li ()
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Qinxiu Sun: Zhejiang University of Science and Technology
Xiao Yu: Shangrao Normal University
Hongliang Li: Zhejiang International Studies University

Indian Journal of Pure and Applied Mathematics, 2020, vol. 51, issue 3, 1105-1132

Abstract: Abstract Weight criteria for the boundedness and compactness of generalized Hardy-type operators (0.1) $$Tf(x) = {u_1}(x)\int_{\left\{ {\phi (y) \le \psi (x)} \right\}} {f(y){u_2}(y){v_0}(y)d\mu (y),\;\;\;\;\;x \in X},$$ T f ( x ) = u 1 ( x ) ∫ { ϕ ( y ) ≤ ψ ( x ) } f ( y ) u 2 ( y ) v 0 ( y ) d μ ( y ) x ∈ X 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 $$T:\;\Lambda _{{v_0}}^{{G_0}}({w_0}) \to \Lambda _{{v_1}}^{{G_1}}({w_1})$$ T Λ v 0 G 0 ( w 0 ) → Λ v 1 G 1 ( w 1 ) and $$T:\;\Lambda _{{v_0}}^{{G_0}}({w_0}) \to \Lambda _{{v_1}}^{{G_1},\infty }({w_1})$$ T Λ v 0 G 0 ( w 0 ) → Λ v 1 G 1 ∞ ( 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 $$\Lambda _{{v_0}}^{{p_0}}({w_0})$$ Λ v 0 p 0 ( w 0 ) to another $$\Lambda _{{v_1}}^{{p_1},{q_1}}({w_1})$$ Λ 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.

Keywords: Hardy operator; Orlicz-Lorentz spaces; weighted Lorentz spaces; boundedness; compactness; 46E30; 46B42 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s13226-020-0453-1

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