Let $\Omega \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, $k \in \mathbb{N}$, and $1 \leq p < \infty$.

1. Subcritical case ($kp < n$): $W^{k,p}(\Omega) \hookrightarrow L^{p^*}(\Omega)$ continuously, where $\frac{1}{p^*} = \frac{1}{p} - \frac{k}{n}$

2. Supercritical case ($kp > n$): $W^{k,p}(\Omega) \hookrightarrow C^{m,\alpha}(\overline{\Omega})$ continuously, where $m = \lfloor k - n/p \rfloor$ and $\alpha = k - n/p - m$

3. Critical case ($kp = n$): $W^{k,p}(\Omega) \hookrightarrow L^q(\Omega)$ for all $q < \infty$, but not $L^\infty$

Moreover, for $\Omega$ bounded, embeddings to strictly smaller exponents/spaces are compact (Rellich-Kondrachov).

For connected $\Omega$ with finite measure and $1 \leq p < \infty$: $\inf_{c \in \mathbb{R}}\|u - c\|_{L^p(\Omega)} \leq C(\Omega, p)\|\nabla u\|_{L^p(\Omega)}$ for all $u \in W^{1,p}(\Omega)$.

For $u \in W^{1,p}_0(\Omega)$, this gives: $\|u\|_{L^p} \leq C\|\nabla u\|_{L^p}$

For $p > n$, there exists $C = C(n,p)$ such that for all $u \in W^{1,p}(\mathbb{R}^n)$: $|u(x) - u(y)| \leq C|x - y|^{1-n/p}\|Du\|_{L^p(B(x,2|x-y|))}$

This quantifies the Hölder continuity implied by Sobolev embedding when $p > n$.

For $u \in H^1_0(\Omega)$ where $0 \in \Omega \subset \mathbb{R}^n$ with $n \geq 3$: $\int_\Omega \frac{u^2}{|x|^2}\,dx \leq \frac{4}{(n-2)^2}\int_\Omega |\nabla u|^2\,dx$

The constant is sharp. This controls singular weights and appears in problems with inverse-square potentials.