On an elementary approach to the fractional Hardy inequality

Abstract

Let H be the usual Hardy operator, i.e., Hu(t) = 1/t ∫0t u(s)ds. We prove that the operator K = I - H is bounded and has a bounded inverse on the weighted spaces Lp(t-α, dt/t) for α > -1 and α ≠ 0. 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 Lp,q norm is replaced by an equivalent expression. Examples show that the restrictions in the theorems are essential. ©1999 American Mathematical Society.