Estimation of the continuity constants for Bogovski\uı and regularized Poincaré integral operators

Abstract

We study the dependence of the continuity constants for the regularized Poincaré and Bogovski\uı 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 $H^1(\Omega ,{\Lambda }^{\ell -1})$, $\ell \in \{1,\dots ,n\}$. 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.