Weighted inequalities for iterated convolutions

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