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Boundedness of weighted iterated Hardy-type operators involving suprema from weighted Lebesgue spaces into weighted Cesàro function spaces. (English) Zbl 07229051
Summary: The boundedness of the weighted iterated Hardy-type operators $$T_{u,b}$$ and $$T_{u,b}^*$$ involving suprema from weighted Lebesgue space $$L_p(v)$$ into weighted Cesàro function spaces $${\operatorname{Ces}}_q(w,a)$$ are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator $$R_u$$ from $$L^p(v)$$ into $${\operatorname{Ces}}_q(w,a)$$ on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator $$P_{u,b }$$ from $$L^p(v)$$ into $${\operatorname{Ces}}_q(w,a)$$ on the cone of monotone non-increasing functions. Under additional condition on $$u$$ and $$b$$, we are able to characterize the boundedness of weighted iterated Hardy-type operator $$T_{u,b}$$ involving suprema from $$L^p(v)$$ into $${\operatorname{Ces}}_q(w,a)$$ on the cone of monotone non-increasing functions. At the end of the paper, as an application of obtained results, we calculate the norm of the fractional maximal function $$M_{\gamma}$$ from $$\Lambda^p(v)$$ into $$\Gamma^q(w)$$.
##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26D10 Inequalities involving derivatives and differential and integral operators 42B25 Maximal functions, Littlewood-Paley theory 42B35 Function spaces arising in harmonic analysis
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