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On an elementary approach
to the fractional Hardy inequality

Authors: Natan Krugljak, Lech Maligranda and Lars Erik Persson
Journal: Proc. Amer. Math. Soc. 128 (2000), 727-734
MSC (1991): Primary 26D15; Secondary 46E30
Published electronically: September 9, 1999
MathSciNet review: 1676324
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Abstract | References | Similar Articles | Additional Information

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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Additional Information

Natan Krugljak
Affiliation: Department of Mathematics, Yaroslavl State University, Sovetskaya 14, 150 000 Yaroslavl, Russia

Lech Maligranda
Affiliation: Department of Mathematics, LuleåUniversity of Technology, S-971 87 Luleå, Sweden

Lars Erik Persson

Keywords: Inequalities, Hardy inequality, Grisvard inequality, Lorentz spaces
Received by editor(s): April 15, 1998
Published electronically: September 9, 1999
Communicated by: Frederick W. Gehring
Article copyright: © Copyright 1999 American Mathematical Society