I'm trying to solve the following exercise:
Let $p,q \in [1, \infty]$ be conjugate indices and consider the sequences $(u_n) \subseteq L^p(\mathbb{R}^d)$, $(v_n) \subseteq L^q(\mathbb{R}^d)$. Assume that $u_n \rightarrow u$ strongly and $v_n \rightharpoonup v$ (where $u \in L^p$ and $v \in L^q$). Given $R>0$, show that for every $r \in [1,\infty]$ $$(u_n*v_n)|_{B_R} \rightharpoonup (u*v)|_{B_R}$$ in $L^r(B_R)$ (weak* convergence when $r=\infty$), where $B_R=B(0,R)$.
I've tried to use the following characterization of weak convergence: for every $f \in L^{r'}$, $|\int_{B_R} (u_n*v_n - u*v) f \rightarrow 0|$, but I'm stucked and maybe this isn't the best way.
Can someone give me a hint please?