Let $1 \leq p < \infty$ be fixed and let $\Omega \subseteq \mathbb{R}^n$ be open. Let $(Q\_n)\_{n \in \mathbb{N}}$ be a uniformly bounded family of operators on $L^p(\Omega)$, i.e. there exists $C>0$ such that $\|Q\_n\| \leq C$ for all $n \in \mathbb{N}$.


Now suppose that for all $u \in L^p(\Omega)$, we have $Q\_n u \longrightarrow u$ pointwise almost everywhere. Does this imply that $Q\_n \longrightarrow \mathrm{id}$ in the strong operator topology, i.e. $Q\_n u \longrightarrow u$ in $L^p(\Omega)$ for each $u \in L^p(\Omega)$?


I am looking for a proof or a counterexample. Does the answer depend on the choice of $\Omega$?


**Edit:** What if we additionally have $\|Q\_n\| \longrightarrow 1$, or even $\|Q\_n\| \leq 1+ \frac{C}{n}$ for some $C>0$? I forgot to ask about this additional condition in my first post. I am aware that this probably still does not fix the situation, but I didn't manage to construct an example from the answer given that satisfies this additional requirement.



Uniformly bounded operator family and pointwise convergence
Let us consider the case $p=1$. Let $u \in L^1(\mathbb{R})$ be a positive function.


Define $f\_n := \chi\_{[n,n+1]}$ and $Q\_nu := u + f\_n \star u$, where $\star$ denotes the convolution.


Notice that, for any such $u$, we have $\|Q\_nu - u\|\_{L^1(\mathbb{R}^n)} = \|u\|\_{L^1} > 0$ and that the sequence $(Q\_n)$ satisfies : 


* $ \|Q\_nu\|\_{L^1} \leq 2 \|u\|\_{L^1} $
* $ (Q\_nu - u)$ is a sequence of continuous functions which vanishes pointwise when $n$ goes to $+ \infty$ (if you further assume that $u$ is compactly supported, then for any $x \in \mathbb{R}, (Q\_nu - u)(x) \equiv 0$ for $n$ big enough)


For bounded $\Omega$, replacing $f\_n$ by (something like) $g\_n := n\chi\_{[0,\frac{1}{n}]}$ probably works, provided that you mind about the boundary.


Anyhow, your claim is morally equivalent to a stronger version of the dominated convergence theorem (without domination), so it was bound to fail. 


Hope this was clear enough, don't hesitate to ask for details.


