BOUNDEDNESS OF WEIGHTED ITERATED HARDY-TYPE OPERATORS INVOLVING SUPREMA FROM WEIGHTED LEBESGUE SPACES INTO WEIGHTED CESARO FUNCTION SPACES

In this paper the boundedness of the weighted iterated Hardy-type operators T(u,b )and T-u,T-b* involving suprema from weighted Lebesgue space L-p(nu) into weighted Cesaro function spaces 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(nu) into 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,P-b from L-P(nu) into 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,T-b involving suprema from L-P(nu) into 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 A(P)(nu) into Gamma(q)(w).