Fall 2012 - Problem 1

Solution.

The condition on the $L^p$ norms imply that $\sup_n \norm{f_n}_{L^p} < \infty$. Let $1 < q < \infty$ be such that $\frac{1}{p} + \frac{1}{q} = 1$ so that we may view $L^p\p{\R^3}$ is the dual of $L^q\p{\R^3}$.

Let $\set{f_{n_k}}_k$ be a weakly convergent subsequence, i.e., $f_{n_k} \to f$ weakly. By assumption, $f_{n_k} \to g$ pointwise as well. We claim that $f = g$: if that were not the case, then by Hahn-Banach, there exists $h \in L^q\p{\R^3}$ such that $\int \p{f - g}h > 0$. But by Fatou's lemma and weak convergence,

a contradiction. Hence, any weakly convergent subsequence of $\set{f_{n_k}}_k$ converges weakly to the same limit. Now let $\set{f_{n_k}}_k$ be any subsequence. This sequence is bounded in $\p{L^q\p{\R^3}}^*$, so by Banach-Alaoglu, it admits a weakly convergent subsequence, and by the argument above, this subsequence converges weakly to $f$. Hence, any subsequence of $\set{f_n}_n$ admits a further subsequence which converges weakly to $f$, and this implies that $f_n \to f$ weakly.

Indeed, if that were not the case, then there exists $h \in L^q\p{\R^3}$ such that $\limsup_{n\to\infty} \int f_nh$ or $\liminf_{n\to\infty} \int f_nh$ is not equal to $\int fh$. Without loss of generality, suppose $\limsup_{n\to\infty} \int f_nh \neq \int fh$ and so there exists a subsequence $\set{f_{n_k}}_k$ such that

But this means that no further subsequence of $\set{f_{n_k}}_k$ converges weakly to $f$, which is impossible. This completes the proof.