An inequality of p forms on Riemannian manifold

Question

Suppose $(M^n,g)$ is an Riemannian manifold with boundary, $w$ is any $p$-form in $A^p(M)$, then we have following inequality $$\|\nabla w\|^2\geq \frac{\|d w\|^2}{p+1}+\frac{\|\delta w\|^2}{n-p+2}$$ where the norm is induced norm on vector bundle by $g$, $d$ is the differential of form and $\delta$ is the codifferential. I am not familar with $p$-forms. Can any one show me some references about this inequality? I am interested in the characterization of equality. Thank you.