Convergence of the integration of three functions

Question

I'm considering a convergence problem of the integration of three functions say $f_{k},g_{k},h_{k}$.
What I have is on a bounded smooth domain $\Omega$, $f_{k}\to f$ strongly in $L^{2}$, $g_{k}\rightharpoonup g$ weakly in $L^{2}$ and $h_{k}\to h$ a.e. in $\Omega$ and $|h_{k}|,|h|\leq C$ for all $x\in\Omega$. Besides, I also have $\lVert f_{k}\rVert_{2},\lVert g_{k}\rVert_{2}\leq C$. Then my question is whether \begin{align} \int_{\Omega}f_{k}g_{k}h_{k}dx\to\int_{\Omega}fghdx. \end{align} My try is \begin{align*} &|\int_{\Omega}f_{k}g_{k}h_{k}-\int_{\Omega}fghdx|\\ \leq&|\int_{\Omega}f(g_{k}-g)hdx|+|\int_{\Omega}(f_{k}-f)g_{k}h_{k}dx|+|\int_{\Omega}fg_{k}(h_{k}-h)dx| \end{align*} The first term goes to zero by weak convergence, the second by strong convergence and the last by dominated convergence theorem.
Does my thought make sense?

Answer 1

Yes, your argument makes sense, but you should be a little clearer. The first term using weak convergence part makes sense. For the second term sure, it vanishes by strong convergence, but you should use Cauchy-Schwarz to be a little more explicit about where you’re using the bounds you’re given. Finally, for the last term, yes it does vanish by dominated convergence, but again, it’s not a literal direct application because what is the dominating function for $fg_k(h_k-h)$? For this last part, I’d suggest proceeding as follows: using Cauchy-Schwarz, \begin{align} \left|\int_{\Omega}fg_k(h_k-h)\,dx\right|&\leq\|g_k\|_{L^2}\|f(h_k-h)\|_{L^2}\leq \left(\sup_{j\geq 1}\|g_j\|_{L^2}\right)\cdot\|f(h_k-h)\|_{L^2}. \end{align} Recall that since $\{g_k\}$ converges weakly, it is norm-bounded. So, we only have to deal with $\|f(h_k-h)\|_{L^2}$. Here, you can see that $f(h_k-h)\to 0$ pointwise a.e, and $|f(h_k-h)|^2\leq \tilde{C}|f|^2\in L^1$, for some other large constant $\tilde{C}$ (due to your uniform bounds on $h_k,h$). Hence, by dominated convergence, this term vanishes too.