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Weighted inequalities for iterated convolutions

Author: Kenneth F. Andersen
Journal: Proc. Amer. Math. Soc. 127 (1999), 2643-2651
MSC (1991): Primary 26D15, 44A35, 42A85; Secondary 26D10
Published electronically: May 4, 1999
MathSciNet review: 1657742
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Abstract: Given a fixed exponent $p$, $1\le p<\infty$, and suitable nonnegative weight functions $v_j$, $j=1,\dots,m$, an optimal associated weight function $\omega _m$ is constructed for which the iterated convolution product satisfies

\begin{displaymath}\int _0^{\infty}\bigg|\bigg[\prod _{j=1}^m*F_j\bigg](x)\bigg|^p\, \dfrac{dx}{\omega _m(x)}\le \prod _{j=1}^m\int _0^{\infty}|F_j(t)|^p\, \dfrac{dt}{v_j(t)}\end{displaymath}

for all complex valued measurable functions $F_j$ with $\int _0^{\infty}|F_j(t)|^p\,dt/v_j(t)<\infty$. Here $[\prod _{j=1}^2*F_j](x)=[F_1*F_2](x)= \int _0^xF_1(t)F_2(x-t)\,dt$ and for each $m>2$, $\prod _{j=1}^m*F_j=\bigg[\prod _{j=1}^{m-1}*F_j \bigg]*F_m$. Analogous results are given when $R^+=(0,\infty)$ is replaced by $R^n$ and also when the convolution $F_1*F_2$ on $R^+$ is taken instead to be $\int _0^{\infty}F(t)G(x/t)\,dt/t$. The extremal functions are also discussed.

References [Enhancements On Off] (What's this?)

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Additional Information

Kenneth F. Andersen
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Keywords: Convolution, weights, inequalities
Received by editor(s): June 24, 1997
Published electronically: May 4, 1999
Additional Notes: This research was supported in part by NSERC research grant #OGP-8185.
Communicated by: Frederick W. Gehring
Article copyright: © Copyright 1999 American Mathematical Society

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