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Weighted inequalities for iterated convolutions

Authors:	Kenneth F Andersen

Institution:	Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Abstract:	Given a fixed exponent , $1\le p<\infty$ , and suitable nonnegative weight functions , $j=1,\dots,m$ , an optimal associated weight function $\omega _m$ is constructed for which the iterated convolution product satisfies $\begin{displaymath}\int _0^{\infty}\bigg\|\bigg\prod _{j=1}^mF_j\bigg](x)\bigg\|^p\, \dfrac{dx}{\omega _m(x)}\le \prod _{j=1}^m\int _0^{\infty}\|F_j(t)\|^p\, \dfrac{dt}{v_j(t)}\end{displaymath}$ for all complex valued measurable functions with $\int _0^{\infty}\|F_j(t)\|^p\,dt/v_j(t)<\infty$ . Here $\prod _{j=1}^2F_j](x)=F_1F_2](x)= \int _0^xF_1(t)F_2(x-t)\,dt$ and for each , $\prod _{j=1}^mF_j=\bigg\prod _{j=1}^{m-1}F_j \bigg]F_m$ . Analogous results are given when $R^+=(0,\infty)$ is replaced by and also when the convolution on is taken instead to be $\int _0^{\infty}F(t)G(x/t)\,dt/t$ . The extremal functions are also discussed.

Keywords:	Convolution weights inequalities

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