The Wikipedia [article](https://en.wikipedia.org/wiki/Agmon%27s_inequality) on Agomon's inequality states the following: 



> 
> Let $u\in H^2(\Omega)\cap H^1\_0(\Omega)$ where $\Omega\subset\mathbb{R}^2$. Then Agmon's inequality in 2D states that there exists a constant $C$ such that
>  $$
> \displaystyle \|u\|\_{L^\infty(\Omega)}\leq C \|u\|\_{L^2(\Omega)}^{1/2} \|u\|\_{H^2(\Omega)}^{1/2}.
> $$
> 
> 
> 


In Agmon's lecture notes, the general version is as follows:



> 
> Lemma 13.2. Let $m>n/2$ and let $u\in H\_m(\Omega)$.Then there exists a constant $\gamma\_s$, depending only on $\Omega$ and $m$, such that, after modification of $u$ on a set of measure zero, 
>  $$
> |u(x)|\leq\gamma\_s \|u\|\_{m}^{n/2m}\|u\|\_{0}^{1-(n/2m)},\quad x\in\overline{\Omega}. 
> $$ 
> 
> 
> 


In Agmon's notation, $n$ denotes the dimensional of the Euclidean space and $\|\cdot\|\_k=\|\cdot\|\_{H\_k}$.


**Question**: Is the assumption $H\_0^1$ redundant in the Wikipedia article? 



Is $H_0^1$ a redundant assumption in the 2D Agmon inequality?
There is probably a regularity assumption on $\Omega$ in the lecture notes, right? 


Zero traces are very convenient in such proofs because then $\Omega$ may be very irregular and one may rely on results for the full space $\mathbb{R}^n$ by considering the zero extension of the respective functions. If one wants the result thus obtained also for non-zero trace functions, one usually needs some boundary regularity for $\partial\Omega$ to have suitable extensions to the full space at hand, but then the proofs work very similarly. I would expect something like that here.


