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Proceedings of the American Mathematical Society

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On a convolution inequality of Saitoh

Authors: Michael Cwikel and Ronald Kerman
Journal: Proc. Amer. Math. Soc. 124 (1996), 773-777
MSC (1991): Primary 26D90
MathSciNet review: 1301493
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Abstract: Let $F_1,F_2,\dotsc,F_j,\dotsc$ be in the class $L_{{\operatorname{loc}}}(\mathbb{R}_+)$ of locally integrable functions on $\mathbb{R}_+=(0,\infty)$. Define the convolution product $\prod^m_{j=1}*F_j$ inductively by $[\prod^2_{j=1}*F_j](x) =(F_1*F_2)(x)=\int^x_0 F_1(y)F_2(x-y)\,dy$ and $\prod^m_{j=1} *F_j=[\prod^{m-1}_{j=1}*F_j]*F_m$ for $m>2$. The inequality

\begin{displaymath}\int^\infty_0 x^{-(m-1)(p-1)} \left|\left[\prod^m_{j=1} *F_j\right] (x) \right|^p\,dx\le [(m-1)!]^{1-p} \prod^m_{j=1} \int^\infty_0 |F_j(y)|^p\,dy\end{displaymath}

is obtained for each $p$, $1<p<\infty$. Further, the constant $[(m-1)!]^{1-p}$ is shown to be the best possible, and the nonzero extremal functions are determined.

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Additional Information

Michael Cwikel
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel

Ronald Kerman
Affiliation: Department of Mathematics, Brock University, St. Catharines, Ontario, Canada L2S 3A1

Keywords: Convolution, Titchmarsh theorem
Received by editor(s): November 24, 1993
Received by editor(s) in revised form: July 17, 1994
Additional Notes: The first author’s research was supported by the Fund for Promotion of Research at the Technion.
The second author’s research was supported by NSERC grant A4021.
Communicated by: Andrew M. Bruckner
Article copyright: © Copyright 1996 American Mathematical Society