Endpoint estimates for n-dimensional Hardy operators and their commutators

In this paper, the sharp bound for the weak-type (1, 1) inequality for the n-dimensional Hardy operator is obtained. Moreover, the precise norms of generalized Hardy operators on the type of Campanato spaces are worked out. As applications, the corresponding norms of the Riemann-Liouville integral operator and the n-dimensional Hardy operator are deduced. It is also proved that the n-dimensional Hardy operator maps from the Hardy space into the Lebesgue space. The endpoint estimate for the commutator generated by the Hardy operator and the (central) BMO function is also discussed.