Pointwise convergence up to a subsequence and Sobolev spaces

Question

Let $u_m\to u$ in $W^{k,p}(\Omega)$ for some domain $\Omega\subset\mathbb{R}^n$. I feel like, up to a subsequence, i can say that $D^\alpha u_m\to D^\alpha u$ a.e. for every $|\alpha|\leq k$.

The reason would be the following : for any $|\alpha|\leq k$, as $D^\alpha u_m\to D^\alpha u$ in $L^p(\Omega)$ we know that $D^\alpha u_m\to D^\alpha u$ a.e. up to a subsequence. Now for any other $|\beta|\leq k$, taking again a subsequence (of the subsequence of) $u_m$, we also get that $D^\beta u_m\to D^\beta u$ a.e.. Repeating this process for every multi-index of lenght at most $k$, we get the claim.

Does that work ? Thanks in advance.

Answer 1

Yes it works. If you want to write this down formally, we can proceed as follows. Relabeling the $\alpha$ such that $\lvert \alpha\rvert\leqslant k$ as $\alpha_1,\dots,\alpha_N$, it suffices to show that

Lemma If for each $\ell\in\{1,\dots,N\}$, the sequence $\left(u_{\ell,n}\right)_{n\geqslant 1}$ converges in $\mathbb L^p$ to $u_{\ell}$, then there exists an increasing map $\varphi\colon\mathbb N\to\mathbb N$ such that for each $\ell\in\{1,\dots,N\}$, the sequence $\left(u_{\ell,\varphi(n)}\right)_{n\geqslant 1}$ converges to $u_{\ell}$ almost everywhere.

We use the

Fact If $(v_n)_{n\geqslant 1}$ converges to $v$ in $\mathbb L^p$, then there exists an increasing map $\psi\colon \mathbb N\to\mathbb N$ such that $\left(v_{\varphi(n)}\right)_{n\geqslant 1}$ covnerges almost everywhere to $v$.

We can prove the Lemma by induction over $N$. The case $N=1$ is simply the previous fact. Let us to the induction step. Assume that the lemma is true for $N$ and suppose that for each $\ell\in\{1,\dots,N+1\}$, the sequence $\left(u_{\ell,n}\right)_{n\geqslant 1}$ converges in $\mathbb L^p$ to $u_{\ell}$. By the induction assumption, we know that we can find an increasing map $\varphi_1\colon\mathbb N\to\mathbb N$ such that for each $\ell\in \{1,\dots,N\}$, the sequence $\left(u_{\ell,\varphi_1(n)}\right)_{n\geqslant 1}$ converges to $u_{\ell}$ almost everywhere. Using the fact with $v_n=u_{N+1,\varphi_1(n)}$, we can find an increasing map $\varphi_2\colon\mathbb N\to\mathbb N$ such that $(v_{\varphi_2(n)})$ converges almost everywhere to $u_{N+1}$. We conclude taking $\varphi=\varphi_1\circ\varphi_2$.