Hardy and Hardy–Littlewood–Pólya operators and their commutators on local fields

We introduce the Hardy and Hardy–Littlewood–Pólya operators on local fields and show that they are bounded on weighted Lebesgue spaces with power weights. Moreover, we compute the precise norms of these operators on these spaces. Further, we prove the boundedness of the commutators generated by these operators and functions with central mean oscillation on Herz spaces, and in particular, on the weighted Lebesgue spaces.