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

Abstract

In this paper the boundedness of the weighted iterated Hardy-type operators $T_{u,b}$ and $T_{u,b}^{\ast }$ involving suprema from weighted Lebesgue space $L_p(v)$ into weighted Cesàro function spaces ${\mathrm{Ces}}_q(w,a)$ are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator $R_u$ from $L^p(v)$ into ${\mathrm{Ces}}_q(w,a)$ 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 ${\mathrm{Ces}}_q(w,a)$ 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 ${\mathrm{Ces}}_q(w,a)$ 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 ${\Gamma }^q(w)$.

26 pages. arXiv admin note: text overlap with arXiv:1109.5400 by other authors