Two-sided estimates on the density of the feynman-kac semigroups of stable-like processes

Suppose that α ∈ (0, 2) and that X is an α-stable-like process on R^d. Let μ be a signed measure on Rd belonging to the class K_d,α and A^μ_t be the continuous additive functional of X associated with μ. In this paper we show that the Feynman-Kac semigroup (T^μ_t : t ≥ 0) defined by T^μ_tf(x) = E_x(e^−Aμ_t f(X_t)) has a density q^μ and that there exist positive constants c₁, c₂, c₃, c₄ such that c₁e^−c₂tt^−d/α (1 ⋀ t^1/α/|x − y|)^d+α ≤ q^μ(t, x, y) ≤ c₃e^c₄tt^−d/α (1 ⋀ t^1/α/|x − y|)^d+α for all (t, x, y) ∈ (0, ∞)×R^d×R^d. We also provide similar estimates for the densities of two other kinds of Feynman-Kac semigroups of X.