Estimation of the continuity constants for Bogovski\u{\i} and regularized Poincar\'e integral operators

We study the dependence of the continuity constants for the regularized Poincar\'e and Bogovski\u{\i} integral operators acting on differential forms defined on a domain $\Omega$ of $\mathbb{R}^n$. We, in particular, study the dependence of such constants on certain geometric characteristics of the domain when these operators are considered as mappings from (a subset of) $L^2(\Omega,\Lambda^\ell)$ to $H^1(\Omega,\Lambda^{\ell-1})$, $\ell \in \{1, \ldots, n\}$. For domains $\Omega$ that are star shaped with respect to a ball $B$ we study the dependence of the constants on the ratio $diam(\Omega)/diam(B)$. A program on how to develop estimates for higher order Sobolev norms is presented. The results are extended to certain classes of unions of star shaped domains.