# Traceless sobolev forms on compact manifolds with boundary

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 $$\ker(T_p)$$ ?