JournalspmVol. 77, No. 1pp. 1–29

The supremum-involving Hardy-type operators on Lorentz-type spaces

  • Qinxiu Sun

    Zhejiang University of Science and Technology, Hangzhou, China
  • Xiao Yu

    Shangrao Normal University, China
  • Hongliang Li

    Zhejiang International Studies University, Hangzhou, China
The supremum-involving Hardy-type operators on Lorentz-type spaces cover

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Abstract

Given measurable functions u,σu, \sigma on an interval (0,b)(0,b) and a kernel function k(x,y)k(x,y) on (0,b)2(0,b)^2 satisfying Oinarov condition, the supremum-involving Hardy-type operators

Rf(x)=supxτ<bu(τ)0τk(τ,y)σ(y)f(y)dy,x>0Rf(x)=\sup_{x\leq\tau < b}u(\tau)\int_0^\tau k(\tau,y)\sigma (y)f(y)dy, x > 0

in Orlicz-Lorentz spaces are investigated. We obtain sufficient conditions of boundedness of R:Λu0G0(w0)Λu1G1(w1)R: \Lambda_{u_0}^{G_0}(w_0)\\ \rightarrow \Lambda_{u_1}^{G_1}(w_1) and R:Λu0G0(w0)Λu1G1,(w1)R: \Lambda_{u_0}^{G_0}(w_0)\rightarrow \Lambda_{u_1}^{G_1,\infty}(w_1). Furthermore, in the case of weighted Lorentz spaces, two characterizations of the boundedness of the operator R:Λu0p0(w0)Λu1p1,q1(w1)R:\Lambda_{u_0}^{p_0}(w_0)\rightarrow\Lambda_{u_1}^{p_1,q_1}(w_1) are achieved as well as the compactness of the operator RR is characterized. It is notable that in the present paper the spaces are only required to be quasi-Banach spaces other than Banach spaces.

Cite this article

Qinxiu Sun, Xiao Yu, Hongliang Li, The supremum-involving Hardy-type operators on Lorentz-type spaces. Port. Math. 77 (2020), no. 1, pp. 1–29

DOI 10.4171/PM/2042