Is convolution integrable in $L_1(\mathbb{R}^n)$? Let $|h(y)|\in L_1(\mathbb{R}^n)$, i.e. $\int\limits_{\mathbb{R}^n}|h(y)|\,dy<+\infty$.
Consider the function $F(x)=\int\limits_{\mathbb{R}^n}|h(x-y)|\,dy$. It is known that $F(x)$ be bounded and uniformly continuous. I try to find condtitions when $F(x)\in L_1(\mathbb{R}^n)$, i.e.
$$
\int\limits_{\mathbb{R}^n}F(x)\,dx=
\int\limits_{\mathbb{R}^n}\biggl(\int\limits_{\mathbb{R}^n}|h(x-y)|\,dy\biggr)\,dx<+\infty.
$$
But I can't even find a specific example for that my assumption holds. Is this possible? Because I think $F(x)$ also be an increasing function, so its integral will always divergents.

No. Indeed, $F(x)\equiv \|h\|_{L_1(\mathbb{R}^n)}$, so that
$$
\int\limits_{\mathbb{R}^n}F(x)\,dx=\|h\|_{L_1(\mathbb{R}^n)}\int
\limits_{\mathbb{R}^n}\,dy=\infty.
$$
Thanks for comments to @Chris and @RyszardSzwarc!
