An elementary proof of the continuity from $L_0^2(\Omega)$ to   $H^1_0(\Omega)^n$ of Bogovskii's right inverse of the divergence
  The existence of right inverses of the divergence as an operator form $H^1_0(\Omega)^n$ to $L_0^2(\Omega)$ is a problem that has been widely studied because of its importance in the analysis of the classic equations of fluid dynamics. When $\Omega$ is a bounded domain which is star-shaped with respect to a ball $B$, a right inverse given by an integral operator was introduced by Bogovskii, who also proved the continuity using the Calder\'on-Zygmund theory of singular integrals.   In this paper we give an alternative elementary proof using the Fourier transform. As a consequence, we obtain estimates of the constant in the continuity in terms of the ratio between the diameters of $\Omega$ and $B$. Moreover, using the relation between the existence of right inverses of the divergence with the Korn and improved Poincar\'e inequalities, we obtain estimates for the constants in these two inequalities. 
Ricardo G. Duran
math.AP