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

This paper aims to investigate the boundedness of the p -adic analog of the weighted Hardy–Cesàro operator U ψ , s : f → ∫ Z p ⋆ f ( s ( t ) ⋅ ) ψ ( t ) d t on weighted Lebesgue spaces and weighted BMO spaces. In each case, we obtain the corresponding operator norms | U ψ , 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 ψ so that the commutator of U ψ , s is bounded on L ω r ( Q p n ) with symbols in BMO ω ( Q p n ) .