References for $|u(x)|\leq C\, \|u\|_{W_{0}^{k,2}(\Omega)}\,\delta^{t} (x)$ I would like to find references for the following result:
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with smooth boundary. Then
For any $t>0$, there exist  positive constants $k$ and $C$ such that
$$|u(x)|\leq C\, \|u\|_{W_{0}^{k,2}(\Omega)}\,\delta^{t} (x),\forall x\in\Omega,\forall u\in W_{0}^{k,2}(\Omega).$$
Here $\delta(x)$ is the distance from $x$ to the boundary of $\Omega$, and $W_{0}^{k,2}(\Omega)$ is the standard Sobolev space, which is the closure of $C_{0}^{\infty}(\Omega)$ in $W^{k,2}(\Omega)$.See the notation here.
Do we have a name for this result?
 A: The statement follows from some Sobolev embedding theorem.
First, we consider the case $t=1$.
If $k$ is large enough, then $W_0^k(\Omega)$ embeds into $C^1(\Omega)$.
Then the inequality
$$
|u(x)|=|u(x)-u(y)| \leq \|u\|_{C^1(\Omega)} \|x-y\|
$$
follows for all $y$ in the boundary of $\Omega$ and $u\in C_0^\infty(\Omega)$.
Then your claimed inequality follows by taking the infimum with respect to $y$
and using the above embedding, i.e.
$$
|u(x)|\leq \|u\|_{C^1(\Omega)} \delta(x) \leq \|u\|_{W_0^k(\Omega)}\delta(x).
$$
Then you can use the density of $C_0^\infty(\Omega)$ in $W_0^k(\Omega)$
to obtain your claim.
The case $0<t<1$ is not a problem because the domain is bounded.
For $t\geq 1$, we can use a similar method.
Instead of $C^1$ we now need to embed into $C^m$ for some $m\geq 1$.
Then we can use similar estimates, but with higher derivatives
(these follow from higher-order Taylor estimates).
Note that the higher derivatives at the boundary are stilly zero
for elements in $C_0^\infty(\Omega)$, which can be used in the argumentation.
I am not aware of a name for this result, and I have never seen it in this formulation.
