Is $H_0^1$ a redundant assumption in the 2D Agmon inequality?

The Wikipedia article 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?

• There is probably a regularity assumption on $\Omega$ in the lecture notes, right? Zero traces are very convenient 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 in the proofs. 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. – Hannes Nov 27 '18 at 16:15
• @Hannes: Thanks for your comment! Indeed in Agmon's notes, he assumes that the domain has "nice" boundary. I overlooked this. I would accept your comment as an answer. – Jack Nov 27 '18 at 17:08

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.