Localized operators on weighted Herz spaces

We introduce the notion of localized operators. We extend the boundedness of the localized operators from the weighted Lebesgue spaces to the weighted Herz spaces. The localized operators include the Hardy operator, the Riemann–Liouville fractional integrals, the general Hardy-type operators, the geometric mean operator, and the one-sided maximal function. Therefore, this paper extends the mapping properties of the the Hardy operator, the Riemann–Liouville fractional integrals, the general Hardy-type operators, the geometric mean operator, and the one-sided maximal function to the weighted Herz spaces.