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We establish well-posedness results for multidimensional non degenerate$\alpha$-stable driven SDEs with time inhomogeneous singular drifts in$\mathbb{L}^r-{\mathbb B}_{p,q}^{-1+\gamma}$ with $\gamma<1$ and $\alpha$ in$(1,2]$, where $\mathbb{L}^r$ and ${\mathbb B}_{p,q}^{-1+\gamma} $ stand forLebesgue and Besov spaces respectively. Precisely, we first prove thewell-posedness of the corresponding martingale problem and then give a precisemeaning to the dynamics of the SDE. This allows us in turn to define an ad hocnotion of weak solution, for which well-posedness holds as well. Our resultsrely on the smoothing properties of the underlying PDE, which is investigatedby combining a perturbative approach with duality results between Besov spaces.

On Multidimensional stable-driven Stochastic Differential Equations with Besov drift