Stable process with singular drift

Suppose that d≤2 and αε(1,2). Let μ=( μ₁, ⋯, μ^d) be such that each μⁱ is a signed measure on R^d belonging to the Kato class K_d,α-1. In this paper, we consider the stochastic differential equation dx _t=dS_t+dA_t, where S^t is a symmetric α-stable process on R^d and for each j=1,⋯,d, the jth component A_t^j of A_t is a continuous additive functional of finite variation with respect to X whose Revuz measure is μ^j. The unique solution for the above stochastic differential equation is called an α-stable process with drift μ. We prove the existence and uniqueness, in the weak sense, of such an α-stable process with drift μ and establish sharp two-sided heat kernel estimates for such a process.