Some old discussion can be seen here.
Let $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given as a weight function. We further assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant $C>0$, $$ \frac{1}{|B(x,r)|}\int_{B(x,r)}\omega(y)dy\leq C\omega(x) $$ for any $x\in\mathbb R^N$ and $r>0$.
Now, given $u$, $v$ are two compact supported functions such that $u\in L^1(\mathbb R^N)$ and $v\in L^1_\omega(\mathbb R^N)$, i.e., $\int_{\mathbb R^N}|v(x)|\omega(x)dx<\infty$.
My question: does $u\ast v$, i.e. the convolution, belongs to $L^1_\omega(\mathbb R^N)$ as well?
I understand if I assume that $\omega$ is locally bounded then this question is trivial. But I am interested to see if we could remove the locally bounded assumption, or replace it with a weaker condition.