An example on the limiting case of Sobolev embedding

Sobolev embedding When $p=n$

Here are some notes after reading some MathOverflow questions¹²³.

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 Poincare-type estimate:

\[\sup_{x\in(0,1)} \left|f(x) - \int^1_0 f(t)\,dt \right| \leq \int^1_0 |f'(t)|\,dt.\]

1. Giuseppe (https://mathoverflow.net/users/23223/giuseppe), Example for the Sobolev embedding theorem when $p=n$., URL (version: 2012-04-26): https://mathoverflow.net/q/95235 ↩

2. Chris Gerig (https://mathoverflow.net/users/12310/chris-gerig), What goes wrong for the Sobolev embeddings at $k=n/p$?, URL (version: 2013-03-11): https://mathoverflow.net/q/124028 ↩

3. JumpJump https://mathoverflow.net/users/62560/jumpjump, The Hölder inequality for fractional order Sobolev seminorm?, URL (version: 2017-04-13): https://mathoverflow.net/q/254344 ↩