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.References
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Additional Information
- Natan Krugljak
- Affiliation: Department of Mathematics, Yaroslavl State University, Sovetskaya 14, 150 000 Yaroslavl, Russia
- Email: natan@univ.uniyar.ac.ru
- Lech Maligranda
- Affiliation: Department of Mathematics, LuleåUniversity of Technology, S-971 87 Luleå, Sweden
- MR Author ID: 118770
- Email: lech@sm.luth.se
- Lars Erik Persson
- Email: larserik@sm.luth.se
- Received by editor(s): April 15, 1998
- Published electronically: September 9, 1999
- Communicated by: Frederick W. Gehring
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 727-734
- MSC (1991): Primary 26D15; Secondary 46E30
- DOI: https://doi.org/10.1090/S0002-9939-99-05420-9
- MathSciNet review: 1676324