Startseite Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces
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Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces

  • Hongliang Li EMAIL logo , Qinxiu Sun und Xiao Yu
Veröffentlicht/Copyright: 10. Januar 2018

Abstract

Given measurable functions ϕ, ψ on + and a kernel function k(x,y)0 satisfying the Oinarov condition, we study the Hardy operator

Kf(x)=ψ(x)0xk(x,y)ϕ(y)f(y)𝑑y,x>0,

between Orlicz–Lorentz spaces ΛXG(w), where f is a measurable function on +. We obtain sufficient conditions of boundedness of K:Λu0G0(w0)Λu1G1(w1) and K:Λu0G0(w0)Λu1G1,(w1). We also look into boundedness and compactness of K:Λu0p0(w0)Λu1p1,q1(w1) between weighted Lorentz spaces. The function spaces considered here are quasi-Banach spaces rather than Banach spaces. Specializing the weights and the Orlicz functions, we restore the existing results as well as we achieve new results in the new and old settings.

MSC 2010: 46E30; 46B42

Communicated by Christopher D. Sogge


Award Identifier / Grant number: 11401530

Award Identifier / Grant number: 11461033

Award Identifier / Grant number: 11561057

Award Identifier / Grant number: LQ13A010018

Award Identifier / Grant number: 20151BAB211002

Funding statement: Supported by National Natural Science Foundation of China (11401530, 11461033, 11561057), Natural Science Foundation of Zhejiang Province of China (LQ13A010018) and Natural Science Foundation of Jiangxi Province of China (20151BAB211002).

Acknowledgements

The first author expresses his deep gratitude to Professor Anna Kamińska for fruitful discussions. The paper was written while the first author was visiting the Department of Mathematics of the University of Memphis. He expresses his thanks to the university for the generous hospitality given to him during his visit.

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Received: 2017-07-03
Published Online: 2018-01-10
Published in Print: 2018-07-01

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