Sharp Estimates for $p$-Adic Hardy, Hardy-Littlewood-P\'olya Operators and Commutators

In this paper we get the sharp estimates of the $p$-adic Hardy and Hardy-Littlewood-P'olya operators on $L^q(\mathrm{\mid }x{\mathrm{\mid }}_p^{\alpha }dx)$. Also, we prove that the commutators generated by the $p$-adic Hardy operators (Hardy-Littlewood-P'olya operators) and the central BMO functions are bounded on $L^q(\mathrm{\mid }x{\mathrm{\mid }}_p^{\alpha }dx)$, more generally, on Herz spaces.