On a convolution inequality of Saitoh

Cwikel, Michael; Kerman, Ronald

doi:10.1090/S0002-9939-96-03068-7

On a convolution inequality of Saitoh
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by Michael Cwikel and Ronald Kerman PDF

Proc. Amer. Math. Soc. 124 (1996), 773-777 Request permission

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 \[ \int ^\infty _0 x^{-(m-1)(p-1)} \left |\left [\prod ^m_{j=1} *F_j\right ] (x) \right |^p dx\le [(m-1)!]^{1-p} \prod ^m_{j=1} \int ^\infty _0 |F_j(y)|^p dy\] 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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