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Let $\Omega$ be an arbitrary bounded domain of $\R^n$. We study the rightinvertibility of the divergence on $\Omega$ in weighted Lebesgue and Sobolevspaces on $\Omega$, and rely this invertibility to a geometric characterizationof $\Omega$ and to weighted Poincar\'e inequalities on $\Omega$. We recover, inparticular, well-known results on the right invertibility of the divergence inSobolev spaces when $\Omega$ is Lipschitz or, more generally, when $\Omega$ isa John domain, and focus on the case of $s$-John domains.

Divergence operator and Poincare inequalities on arbitrary bounded domains