An example on the limiting case of Sobolev embedding

Sobolev embedding When $p=n$

When the dimension of the domains agrees with the integrability of a function, the trading of differentiability with integrability may sometimes be problematic.

Let us consider a special case where $\Omega\subset \mathbb{R}^2$. For any $u\in H^1(\Omega):= W^{1,2}(\Omega)$, we have for a fixed $q\in [1,\infty)$,

\[\Vert u \Vert_{L^q(\Omega)}\leq C\Vert u \Vert_{W^{1,2}(\Omega)}.\]

The reason that $q\neq \infty$, is that there exists $H^1(\Omega)$-regular unbounded function like:

\[f(x) = \ln \ln \left(1 + \frac{1}{ |x| }\right).\]

It is easy to check that $f\in H^1$ but not in $L^{\infty}(\Omega)$. However when $n=1$, say $\Omega = (0,1)$ above counterexample does not work. The reason is that $f\notin W^{1,1}(\Omega)$, for functions $u\in W^{1,1}(\Omega)$, as for this space consists of the antiderivatives of Lebesgue integrable functions on $(0,1)$ which is bounded by a simple estimate: \(\sup_{x\in(0,1)} \left|f(x) - \int^1_0 f(t)\,dt \right| \leq \int^1_0 |f'(t)|\,dt.\)