Estimation of the continuity constants for Bogovskiu{i} and regularized Poincare integral operators

We study the dependence of the continuity constants for the regularized Poincare and Bogovskiu{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.