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We study the dependence of the continuity constants for the regularizedPoincar\'e and Bogovski\u{\i} integral operators acting on differential formsdefined on a domain $\Omega$ of $\mathbb{R}^n$. We, in particular, study thedependence of such constants on certain geometric characteristics of the domainwhen 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 orderSobolev norms is presented. The results are extended to certain classes ofunions of star shaped domains.

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