Heat kernels for time-dependent non-symmetric stable-like operators

When studying non-symmetric nonlocal operators on R^d: Lf(x)=∫R^d(f(x+z)−f(x)−∇f(x)⋅z1_{|z|⩽1})[Formula presented]dz, where 0<α<2, d⩾1, and κ(x,z) is a function on R^d×R^d that is bounded between two positive constants, it is customary to assume that κ(x,z) is symmetric in z. In this paper, we study heat kernel of L and derive its two-sided sharp bounds without the symmetric assumption κ(x,z)=κ(x,−z). In fact, we allow the kernel κ to be time-dependent and x→κ(t,x,z) to be only locally β-Hölder continuous with Hölder constant possibly growing at a polynomial rate in |z|. We also derive gradient estimate when β∈(0∨(1−α),1) as well as fractional derivative estimate of order θ∈(0,(α+β)∧2) for the heat kernel. Moreover, when α∈(1,2), drift perturbation of the time-dependent non-local operator L_t with drift in Kato's class is also studied in this paper. As an application, when κ(x,z)=κ(z) does not depend on x, we show the boundedness of nonlocal Riesz's transformation: for any p>2d/(d+α), ‖L^1/2f‖_p≍‖Γ(f)^1/2‖_p, where Γ(f):=[Formula presented]L(f²)−fLf is the carré du champ operator associated with L, and L^1/2 is the square root operator of L defined by using Bochner's subordination. Here ≍ means that both sides are comparable up to a constant multiple.