Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions
HTML articles powered by AMS MathViewer
- by Natan Kruglyak and Eric Setterqvist
- Proc. Amer. Math. Soc. 136 (2008), 2505-2513
- DOI: https://doi.org/10.1090/S0002-9939-08-09200-9
- Published electronically: March 7, 2008
- PDF | Request permission
Abstract:
It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions $f$ in $L^{p}$, then we have the sharp estimate \begin{equation*} \left \| (I-H)f\right \| _{L^p}\leq \frac {1}{(p-1)^{\frac {1}{p}}}\left \| f\right \| _{L^p} \end{equation*} for $p=2,3,4,....$ In other words, \begin{equation*} \left \| f^{**}-f^* \right \| _{L^p}\leq \frac {1}{(p-1)^{\frac {1}{p}}} \left \| f\right \| _{L^p} \end{equation*} for each $f \in L^p$ and each integer $p\ge 2$.
It is also shown, via a connection between the operator $I-H$ and Laguerre functions, that \begin{equation*} \|(1-\alpha ) I+\alpha (I-H)\|_{L^2\to L^2}=\|I-\alpha H\|_{L^2\to L^2}=1 \end{equation*} for all $\alpha \in [0,1]$.
References
- Natan Krugljak, Lech Maligranda, and Lars Erik Persson, On an elementary approach to the fractional Hardy inequality, Proc. Amer. Math. Soc. 128 (2000), no. 3, 727–734. MR 1676324, DOI 10.1090/S0002-9939-99-05420-9
- Arlen Brown, P. R. Halmos, and A. L. Shields, Cesàro operators, Acta Sci. Math. (Szeged) 26 (1965), 125–137. MR 187085
- N. Kaiblinger, L. Maligranda, and L.-E. Persson, Norms in weighted $L^2$-spaces and Hardy operators, Function spaces (Poznań, 1998) Lecture Notes in Pure and Appl. Math., vol. 213, Dekker, New York, 2000, pp. 205–216. MR 1772126
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
- Colin Bennett, Ronald A. DeVore, and Robert Sharpley, Weak-$L^{\infty }$ and BMO, Ann. of Math. (2) 113 (1981), no. 3, 601–611. MR 621018, DOI 10.2307/2006999
- G. Sansone, Orthogonal functions, Dover Publications, Inc., New York, 1991. Translated from the Italian by Ainsley H. Diamond; With a foreword by Einar Hille; Reprint of the 1959 edition. MR 1118381
Bibliographic Information
- Natan Kruglyak
- Affiliation: Department of Mathematics, Luleå University of Technology, SE-971 87, Luleå, Sweden
- Email: natan@ltu.se
- Eric Setterqvist
- Affiliation: Global Sun Engineering AB, Aurorum Science Park 2, SE-97775 Luleå, Sweden
- Email: eric.setterquist@globalsunengineering.com
- Received by editor(s): February 9, 2006
- Received by editor(s) in revised form: January 26, 2007, and March 30, 2007
- Published electronically: March 7, 2008
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 2505-2513
- MSC (2000): Primary 26D10, 46E30
- DOI: https://doi.org/10.1090/S0002-9939-08-09200-9
- MathSciNet review: 2390520