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Let $$L=\Delta ^{\alpha /2}+ b\cdot \nabla $$ with $$\alpha \in (1,2)$$ . We prove the Martin representation and the Relative Fatou Theorem for non-negative singular L-harmonic functions on $$\mathcal{C }^{1,1}$$ bounded open sets.
The Green function of the fractional Laplacian of the differential order bigger than one and the Green function of its gradient perturbations are comparable for bounded smooth multidimensional open sets if the drift function is in an appropriate Kato class.
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