Boundedness of weighted iterated Hardy-type operators involving suprema from weighted Lebesgue spaces into weighted Cesàro function spaces

Abstract

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 \(\Gamma^q(w)\).