On an elementary approach to the fractional Hardy inequality

Krugljak, Natan; Maligranda, Lech; Persson, Lars

doi:10.1090/S0002-9939-99-05420-9

On an elementary approach to the fractional Hardy inequality
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by Natan Krugljak, Lech Maligranda and Lars Erik Persson PDF

Proc. Amer. Math. Soc. 128 (2000), 727-734 Request permission

Abstract:

Let $H$ be the usual Hardy operator, i.e., $Hu(t)=\frac {1}{t}\int _0^tu(s) ds$. We prove that the operator $K=I-H$ is bounded and has a bounded inverse on the weighted spaces $L_p(t^{-\alpha },dt/t)$ for $\alpha >-1$ and $\alpha \not =0$. 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 $L_{p,q}$ norm is replaced by an equivalent expression. Examples show that the restrictions in the theorems are essential.

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