Let $M \subset \mathbb R^n$ be a submanifold of euclidean space whose boundary is locally a Lipschitz graph. Let $\omega \in L^2\Lambda^k(M)$ be a differential form with square-integrable coefficients.

The Poincaré constant for $k = 0$ can be estimated by $1/2 \cdot \operatorname{diam}(M)$ in case the domain is convex and bounded. I have not seen a formal statement in literature that provides a comparable statement for $k > 0$.

Could you provide me with a reference? I need this only for citation and an estimate in terms of the diameter of the domain would be completely sufficient. Thank you.