Let $\Omega \subset \mathbb{R}^n$ be a bounded open subset with a $C^1$- boundary.
Is the embedding
\begin{align}
H^1(\Omega) \subset L^6(\Omega)
\end{align}
continuous for $n=1,2,3$?
I think it's clear for $n=3$, since we can apply the Sobolev embedding theorem to obtain that $\frac{1}{p^*}= \frac{1}{p}- \frac{1}{n}=\frac{1}{2}-\frac{1}{3} = \frac{1}{6}$. Hence, the embedding is continuous for $n=3$.
But how can I argue for the cases $n=1$ and $n=2$?