Weighted inequalities for iterated convolutions

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

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.