Let $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given. We 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 $r>0$.

We define the weighted $L^1_\omega$ space by, for function $u$, $$ \int_{\mathbb R^N}|u(x)|\omega(x)dx<\infty $$

It is well known that $L^1(\mathbb R^N)\ast L^1(\mathbb R^N)\subset L^1(\mathbb R^N)$, i.e., the convolution for two $L^1$ function is still $L^1$.

Now my question, for what condition do I need for $\omega$ that $$ L^1_\omega(\mathbb R^N)\ast L^1(\mathbb R^N)\subset L^1_\omega(\mathbb R^N)? $$

I feel I may need some growth condition on $\omega$ but I am not sure. I also search online. There are many papers dealing with weighted $L^p$ space but they are just interested in the case $p>1$ and $\omega=|x|^\alpha$ for some $\alpha>0$. But I am really just interested in the case $p=1$...

Thank you!

PS: Of course making $\omega$ be bounded above will work but it makes this theorem not so useful. I wish to keep $\omega$ has the ability to blow up to infinity still.