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

 

The modular inequalities
for a class of convolution operators
on monotone functions


Author: Jim Qile Sun
Journal: Proc. Amer. Math. Soc. 125 (1997), 2293-2305
MSC (1991): Primary 26D15, 42B25
MathSciNet review: 1389538
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Abstract: This paper is devoted to the study of modular inequality

\begin{equation*}\Phi _{2}^{-1} \left ( \int _{0}^{+\infty } \Phi _{2}( a(x)K f(x)) w(x) dx \right ) \le \Phi _{1}^{-1} \left ( \int _{0}^{+\infty } \Phi _{1}( C f(x) ) v(x) dx \right ) \end{equation*}

where $\Phi _{1} \ll \Phi _{2}$ and $K$ is a class of Volterra convolution operators restricted to the monotone functions. When $\Phi _{1}(x) = x^{p}/p, \Phi _{2}(x) = x^{q}/q$ with $1 < p \le q < +\infty $ and the kernel $k(x) \equiv 1$, our results will extend those for the Hardy operator on monotone functions on Lebesgue spaces.


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Additional Information

Jim Qile Sun
Affiliation: Department of Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7
Email: jsuen@switchview.com

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03867-7
PII: S 0002-9939(97)03867-7
Received by editor(s): July 10, 1995
Received by editor(s) in revised form: January 30, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society