Estimates of the Green function for the fractional Laplacian perturbed by gradient (Q411536)

Let \(\Delta^{\alpha/2}\) with \(\alpha\in(1,2)\) denote the fractional Laplacian on a non-empty, bounded, open \(C^{1,1}\) set \(D\subset\mathbb{R}^d\), \(d\in\{2,3,\dots\}\), and consider the operator \(L= \Delta^{\alpha/2}+ b(x)\cdot\nabla\), where \(b\) is a function in the Kato class \(K^{\alpha-1}_d\). Define \(G_D\) and \(G_D\) as the Green functions of \(\Delta^{\alpha/2}\) and \(L\), respectively. The authors prove that there exists a constant \(C= C(\alpha,b,D)\) such that \(C^{-1}G_D(x,y)\leq G_D\leq CG_D(x,y)\) for all \(x,y\in D\), thus extending a result of \textit{M. Cranston} and \textit{Z. Zhao} [Commun. Math. Phys. 112, 613--625 (1987; Zbl 0647.60071)]. The proof is based on the perturbation formula \(G_D= G_D+ G_Db\cdot\nabla G_D\) and uses estimates for \(G_D\) and the gradient of \(G_D\), the boundary Harnack inequality for \(\Delta^{\alpha/2}\), and the Kato condition for the drift function \(b\). For sharp, explicit estimates for \(G_D\) see \textit{T. Kulczycki} [Probab. Math. Stat. 17, No. 2, 339--364 (1997; Zbl 0903.60063)], \textit{Z.-Q. Chen} and \textit{R. Song} [Math. Ann. 312, No. 3, 465--501 (1998; Zbl 0918.60068)].