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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions


Authors: Natan Kruglyak and Eric Setterqvist
Journal: Proc. Amer. Math. Soc. 136 (2008), 2505-2513
MSC (2000): Primary 26D10, 46E30
Published electronically: March 7, 2008
MathSciNet review: 2390520
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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

$\displaystyle \left\Vert (I-H)f\right\Vert _{L^p}\leq \frac{1}{(p-1)^{\frac{1}{p}}}\left\Vert f\right\Vert _{L^p} $

for $ p=2,3,4,....$ In other words,

$\displaystyle \left\Vert f^{**}-f^* \right\Vert _{L^p}\leq \frac{1}{(p-1)^{\frac{1}{p}}} \left\Vert f\right\Vert _{L^p} $

for each $ f \in L^p$ and each integer $ p\ge2$.

It is also shown, via a connection between the operator $ I-H$ and Laguerre functions, that

$\displaystyle \Vert(1-\alpha) I+\alpha (I-H)\Vert _{L^2\to L^2}=\Vert I-\alpha H\Vert _{L^2\to L^2}=1 $

for all $ \alpha \in [0,1]$.


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Additional 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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09200-9
PII: S 0002-9939(08)09200-9
Keywords: The Hardy operator, cone of decreasing functions, sharp estimates
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
Article copyright: © Copyright 2008 American Mathematical Society