Abstract
We determine the cases of equality in the Riesz rearrangement inequality
\[
\int \int f(y)g(x-y)h(x)dy\; dx \le \int \int f^\ast(y)g^\ast(x-y)h^\ast(x)dy\; dx,
\]
where $f^\ast$, $g^\ast$, and $h^\ast$ are the spherically decreasing rearrangements of the functions $f$, $g$, and $h$ on $\mathbf{R}^n$. We apply our results to the weak Young inequality
