Let $(M,g)$ be a smooth, compact, oriented Riemannian manifold with smooth, oriented boundary $\partial M$. Further, let $\Omega^p(M)$ and $\Omega^p(\partial M)$ be the spaces of smooth differential $p$-forms over $M$, respectively $\partial M$, and let $$i_p: \Omega^p(M) \to \Omega^p(\partial M)$$ be the pullback of the smooth boundary inclusion $i: \partial M \hookrightarrow M$.

A version of the Trace theorem then asserts that for each $p \in \mathbb N$, $i_p$ extends to a bounded linear map between Sobolev (Hilbert-)spaces $$T_p: H^1(\Omega^p(M)) \to H^0(\Omega^p(\partial M)) = L^2(\Omega^p(\partial M)).$$ I want to better understand the space $\ker(T_p)$ of traceless $p$-forms.

Using coordinate charts, partitions of unity and assuming, without loss of generality, the Riemannian metric to be a product near the boundary, I can prove via classic methods that

$$\Omega^0(M) \cap \ker(T_0) \; \text{is $H^1$-dense in} \; \ker(T_0)$$ Among other things, I use the fact that $T_0f$ is simply the restriction of $f$ to $\partial M$. Of course, this does not hold in general for $p \geq 1$, which is also why my proof annoyingly breaks down in this case.

Question: Is it still true that for $p \geq 1$, $\Omega^p(M) \cap \ker(T_p)$ is $H^1$-dense in $\ker(T_p)$ ?


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