Let be the usual Hardy operator, i.e., . We prove that the operator is bounded and has a bounded inverse on the weighted spaces for -1$" type="#_x005F_x0000_t75">and . Moreover, by using these inequalities we derive a somewhat generalized form of some well-known fractional Hardy type inequalities and also of a result due to Bennett-DeVore-Sharpley, where the usual Lorentz norm is replaced by an equivalent expression. Examples show that the restrictions in the theorems are essential.
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机译：设为通常的Hardy运算符，即。我们证明了该算子是有界的，并且在-1 $“ type =” #_ x005F_x0000_t75“>和的加权空间上有界逆。而且，通过使用这些不等式，我们得出了一些著名的分数Hardy类型的某种广义形式不等式，也归因于Bennett-DeVore-Sharpley的结果，其中通常的Lorentz范数被等价表达式代替，实例表明定理中的限制是必不可少的。
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