An example on the limiting case of Sobolev embedding
Sobolev embedding When $p=n$
Here are some notes after reading some MathOverflow questions^{1}^{2}^{3}.
A counterexample when $p=n=2$
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 $\Omega\subset \mathbb{R}^2$ including the origin. 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 $L^{\infty}(\Omega)$.
Same counterexample for $p=n=1$?
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 $W^{1,1}(\Omega)$ space consists of the antiderivatives of Lebesgue integrable functions on $(0,1)$, which is bounded by a simple Poincaretype estimate:
\[\sup_{x\in(0,1)} \leftf(x)  \int^1_0 f(t)\,dt \right \leq \int^1_0 f'(t)\,dt.\]
Giuseppe (https://mathoverflow.net/users/23223/giuseppe), Example for the Sobolev embedding theorem when $p=n$., URL (version: 20120426): https://mathoverflow.net/q/95235 ↩

Chris Gerig (https://mathoverflow.net/users/12310/chrisgerig), What goes wrong for the Sobolev embeddings at $k=n/p$?, URL (version: 20130311): https://mathoverflow.net/q/124028 ↩

JumpJump https://mathoverflow.net/users/62560/jumpjump, The Hölder inequality for fractional order Sobolev seminorm?, URL (version: 20170413): https://mathoverflow.net/q/254344 ↩
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