Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation

Let $L={\mathrm{\Delta }}^{\alpha \mathrm{/}2}+b\cdot \mathrm{\nabla }$ 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.