The pp-adic weighted Hardy–Cesàro operator and an application to discrete Hardy inequalities

This paper aims to investigate the boundedness of the pp-adic analog of the weighted Hardy–Cesàro operator Uψ,s:f→∫Zp⋆f(s(t)⋅)ψ(t)dt on weighted Lebesgue spaces and weighted BMO   spaces. In each case, we obtain the corresponding operator norms |Uψ,s||Uψ,s|. In particular, these results have a surprising relevance to discrete Hardy inequalities on the real field. We prove a reverse BMO  –Hardy inequality and give a necessary condition on ψψ so that the commutator of Uψ,sUψ,s is bounded on Lωr(Qpn) with symbols in BMOω(Qpn).