Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

On an elementary approach to the fractional Hardy inequality

Author(s): Natan Krugljak; Lech Maligranda; Lars Erik Persson
Journal: Proc. Amer. Math. Soc. 128 (2000), 727-734.
MSC (1991): Primary 26D15; Secondary 46E30
Posted: September 9, 1999
MathSciNet review: 1676324
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Abstract: Let be the usual Hardy operator, i.e., $Hu(t)=\frac{1}{t}\int _0^tu(s)\,ds$ . We prove that the operator 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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