  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.
