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If $\Omega\subset\R^n$ is a bounded domain, the existence of solutions ${\bfu}\in H^1_0(\Omega)^n$ of ${div} {\bf u} = f$ for $f\in L^2(\Omega)$ withvanishing mean value, is a basic result in the analysis of the Stokesequations.  In particular it allows to show the existence of a solution $({\bf u},p)\inH^1_0(\Omega)^n\times L^2(\Omega)$, where ${\bf u}$ is the velocity and $p$ thepressure.  It is known that the above mentioned result holds when $\Omega$ is aLipschitz domain and that it is not valid for arbitrary H\"older-$\alpha$domains.  In this paper we prove that if $\Omega$ is a planar simply connectedH\"older-$\alpha$ domain, there exist right inverses of the divergence whichare continuous in appropriate weighted spaces, where the weights are powers ofthe distance to the boundary. Moreover, we show that the powers of the distancein the results obtained are optimal.  In our results, the zero boundary condition is replaced by a weaker one. Forthe particular case of domains with an external cusp of power type, we provethat our weaker boundary condition is equivalent to the standard one. In thiscase we show the well posedness of the Stokes equations in appropriate weightedSobolev spaces obtaining as a consequence the existence of a solution $({\bfu},p)\in H^1_0(\Omega)^n\times L^r(\Omega)$ for some $r<2$ depending on thepower of the cusp.

A right inverse of the divergence for planar H\"older-$\alpha$ domains