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.