# Boundedness of weighted iterated Hardy-type operators involving suprema from weighted Lebesgue spaces into weighted Ces\`{a}ro function spaces

## TL;DR

This paper characterizes the boundedness of weighted Hardy-type operators involving suprema from weighted Lebesgue spaces to weighted Cesàro spaces, including applications to fractional maximal functions.

## Contribution

It provides new characterizations of the boundedness of iterated Hardy-type operators involving suprema between weighted Lebesgue and Cesàro spaces, extending previous results.

## Key findings

- Characterization of boundedness of $T_{u,b}$ and $T_{u,b}^*$ operators.
- Boundedness criteria for the supremal operator $R_u$ on monotone functions.
- Norm calculation of the fractional maximal function $M_{eta}$ between specific function spaces.

## Abstract

In this paper 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 Ces\`{a}ro 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 $\Gamma^q(w)$.