Convolution of L infinity and L1 functions
= Convolution of L infinity and L1 functions
{title2=$L^\infty*L^1\subset C_b$}
For $f\in L^\infty(\mathbb R^n)$ and $g\in L^1(\mathbb R^n)$,
$$
(f*g)(x)=\int_{\mathbb R^n}f(y)g(x-y)\,dy
$$
is bounded by $\|f\|_\infty\|g\|_1$. Continuity follows from
$$
|(f*g)(x+h)-(f*g)(x)|
\le\|f\|_\infty\|\tau_hg-g\|_1
$$
and continuity of translation in $L^1$.