Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

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 PDF
Proc. Amer. Math. Soc. 136 (2008), 2505-2513 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
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 26D10, 46E30
  • Retrieve articles in all journals with MSC (2000): 26D10, 46E30
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
  • 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