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The supremum-involving Hardy-type operators on Lorentz-type spaces. (English) Zbl 1451.42035

Summary: Given measurable functions \(u, \sigma\) on an interval \((0,b)\) and a kernel function \(k(x,y)\) on \((0,b)^2\) satisfying Oinarov condition, the supremum-involving Hardy-type operators \[Rf(x)=\sup\limits_{x\leq\tau < b}u(\tau)\int_0^\tau k(\tau,y)\sigma (y)f(y)dy, \quad x > 0\] in Orlicz-Lorentz spaces are investigated. We obtain sufficient conditions of boundedness of \(R: \Lambda_{u_0}^{G_0}(w_0)\ \rightarrow \Lambda_{u_1}^{G_1}(w_1)\) and \(R: \Lambda_{u_0}^{G_0}(w_0)\rightarrow \Lambda_{u_1}^{G_1,\infty}(w_1)\). Furthermore, in the case of weighted Lorentz spaces, two characterizations of the boundedness of the operator \(R:\Lambda_{u_0}^{p_0}(w_0)\rightarrow\Lambda_{u_1}^{p_1,q_1}(w_1)\) are achieved as well as the compactness of the operator \(R\) is characterized. It is notable that in the present paper the spaces are only required to be quasi-Banach spaces other than Banach spaces.

MSC:

42B35 Function spaces arising in harmonic analysis
42B15 Multipliers for harmonic analysis in several variables
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