If a function belongs to $W^{k,p}(\Omega)$ for every $p$, does it belong to $W^{k,\infty}(\Omega)$?

Question

If a function belongs to $W^{k,p}(\Omega)$ for every $p$, does it belong to $W^{k,\infty}(\Omega)$?

Under what assumptions on the domain is this true?

Answer 1

The answer is no, and there are no assumptions on the domain that can help you with this. For example, in one dimension the function $\log x$ is in $L^p((0, 1))$ for all $p<\infty$, but is not in $L^\infty((0, 1))$. Integrating it $k$ times yields an example in $W^{k, p}((0, 1))$; or simpler yet, consider $x^k \log x$.

There are interesting function spaces in between $\bigcap_{p<\infty} W^{k, p}$ and $W^{k, \infty}$:

• Orlicz-Sobolev spaces $W^{k, \Phi}$: replace the integrability of $p$th power of $|D^k u|$ with the integrability of $\Phi(|D^ku|)$ for some Orlicz function $\Phi$ such as $\Phi(t) = e^t$.
• BMO-Sobolev spaces, defined by the requirement that $D^k$ has bounded mean oscillation.
• Grand Sobolev spaces: require a bound on $\|D^k\|_p$ of the form $O(p)$ as $p\to\infty$.