We study the dependence of the continuity constants for the regularized Poincaré and Bogovskiĭ integral operators acting on differential forms defined on a domain Ω 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) $L^2(\mathrm{\Omega },{\mathrm{\Lambda }}^{\ell })$ to $H^1(\mathrm{\Omega },{\mathrm{\Lambda }}^{\ell -1})$, $\ell \in \{1,\dots ,n\}$. For domains Ω that are star shaped with respect to a ball B we study the dependence of the constants on the ratio $\mathrm{diam}(\mathrm{\Omega })/\mathrm{diam}(B)$. 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.

我们研究正则化Poincaré和Bogovskiĭ积分算子对在Ω的域Ω上定义的微分形式起作用的连续常数的依赖性。 ${\mathbb{[R}}^{ñ}$。当这些运算符被视为（的一个子集）的映射时，我们尤其研究这些常数对域的某些几何特征的依赖性。${\mathrm{大号}}^{\mathrm{2个}}（\mathrm{\Omega }，{\mathrm{\Lambda }}^{\ell }）$ 至 $H^{\mathrm{1个}}（\mathrm{\Omega }，{\mathrm{\Lambda }}^{\ell -\mathrm{1个}}）$， $\ell \in \{\mathrm{1个}，\dots ，ñ\}$。对于相对于球B呈星形的畴Ω，我们研究了常数对比率的依赖性$\mathrm{直径}（\mathrm{\Omega }）/\mathrm{直径}（乙）$。提出了一个有关如何开发高阶Sobolev规范的估计的程序。结果扩展到星形畴的并集的某些类。