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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 studiedbecause of its importance in the analysis of the classic equations of fluiddynamics. When $\Omega$ is a bounded domain which is star-shaped with respectto a ball $B$, a right inverse given by an integral operator was introduced byBogovskii, who also proved the continuity using the Calder\'on-Zygmund theoryof singular integrals.  In this paper we give an alternative elementary proof using the Fouriertransform. As a consequence, we obtain estimates of the constant in thecontinuity in terms of the ratio between the diameters of $\Omega$ and $B$.Moreover, using the relation between the existence of right inverses of thedivergence with the Korn and improved Poincar\'e inequalities, we obtainestimates for the constants in these two inequalities.

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