I am familiar with some inequalities on convolutions, such as Young's convolution inequality which says if $\frac1r + 1 = \frac1p + \frac1q$ then $$ ||f \ast g||_r\le ||f||_p\,||g||_q $$
I am wondering if there is any known inequalities that can help me relate a triple convolution with the convolution of the outer terms, i.e. something relating $f\ast g \ast h$ and $f\ast h$. I am working in the context of a nonabelian countable group, so the convolution is not necessarily commutative here. My hope is that $$ ||f\ast g\ast h||_\infty \le C ||f\ast h||_\infty ||g||_1 $$ would hold for some $C$ depending on the underlying group.