Uniformly bounded operator family and pointwise convergence

Question

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.

Answer 1

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.

Answer 2

Suppose $p > 1$ and $\|Q_n\| \to 1$. Using Fatou's lemma for a lower bound and the norm for an upper bound, $\lim_n \|Q_n u\|_p = \|u\|_p$, and similarly $\lim_n \|(u + Q_n u)/2\|_p = \|u\|_p$. By Clarkson's inequalities, $\lim_n \|Q_n u - u\|_p = 0$, i.e. $Q_n \to \text{id}$ in the strong operator topology.

This still leaves the case $p=1$ open.