Let $U$ be a bounded domain in the Euclid space $(\mathbb{R}^d,g)$.
$g_{\wedge^p}$ denotes the fiber metric of $\wedge^pT^{\ast}\mathbb{R}^d$ derived from $g$.
$A^p$ denotes the set of p- forms on $\mathbb{R}^d$.
\begin{eqnarray} \langle\omega,\eta\rangle_{L^2_{p}(U)}:=\int_U g_{\wedge^p}(\omega,\eta)dx\ \ \ (\omega,\eta\in A^p). \end{eqnarray}
$d:A^p \to A^{p+1}$ denotes the exterior derivative on $A^p$. $\delta:A^p\to A^{p-1}$ denotes the formal adjoint operator of $d$ w.r.t. the $L^2$ metric on p-forms.
\begin{eqnarray} \langle\omega,\eta\rangle_{H^1_{p}(U)}:=\langle\omega,\eta\rangle_{L^2_{p}(U)}+\langle d\omega,d\eta\rangle_{L^2_{p+1}(U)}+\langle\delta\omega,\delta\eta\rangle_{L^2_{p-1}(U)}\\ \langle\omega,\eta\rangle_{W(U)}:=\int_U \sum_{i_1<\cdots<i_p}\left[\omega_{i_1,\cdots,i_p}\eta_{i_1,\cdots,i_p}+\sum_{j=1}^d\dfrac{\partial\omega_{i_1,\cdots,i_p}}{\partial x^j}\dfrac{\partial\eta_{i_1,\cdots,i_p}}{\partial x^j}\right]dx\\ where\ \ \ \omega=\sum_{i_1<\cdots<i_p}\omega_{i_1,\cdots,i_p}dx^{i_1}\wedge\cdots\wedge dx^{i_p}\\ \eta=\sum_{i_1<\cdots<i_p}\eta_{i_1,\cdots,i_p}dx^{i_1}\wedge\cdots\wedge dx^{i_p}\ \ \ (\omega,\eta\in A^p). \end{eqnarray}
Then the question is following: Is it true that \begin{eqnarray} \exists C>0\ \ \ s.t. \forall\omega\in A^p\ \ \ ||\omega||_{W(U)}\leq C||\omega||_{H^1_{p}(U)} ? \end{eqnarray}
Since I cannot understand the following part in Jost, Riemannian Geometry and Geometric Anasysis 6th ed. , I ask this question.
