Let $\{f_{n}\}_{n\in \mathbb{N}}$ be a sequence of functions living in the Sobolev space $H^{1}(\mathbb{R}^{d})$, for some dimension $d \ge 1$. Suppose $f_{n} \rightharpoonup f$ in $H^{1}(\mathbb{R}^{d})$, that is, $f_{n}$ converges weakly to $f$ in $H^{1}(\mathbb{R}^{d})$, meaning that: $$\langle f_{n},g\rangle_{H^{1}}:= \langle f_{n},g\rangle_{L^{2}} + \langle \nabla f_{n},\nabla g\rangle_{L^{2}} \to \langle f,g\rangle_{L^{2}} + \langle \nabla f, \nabla g\rangle_{L^{2}}$$ for every $g \in H^{1}(\mathbb{R}^{d})$. Note that this in particular implies that $f_{n}$ converges weakly to $f$ in $L^{2}(\mathbb{R}^{d})$.
Question: Suppose $h \in L^{p}(\mathbb{R}^{d})$, with $p > \max\{d/2,1\}$ (this condition allows us to use Sobolev inequalities, if necessary). Is it true that: $$\int_{\mathbb{R}^{d}}|f_{n}(x)-f(x)|^{2}|h(x)|dx \to 0$$ as $n \to \infty$? I mean, of course this is true if $p=2$, but is it true for an arbitrary $p > \max\{d/2,1\}$ as well? I was trying to use Hölder's inequality, but I couldn't get anywhere.