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

When studying non-symmetric nonlocal operators on Rd:Lf(x)=∫Rd(f(x+z)−f(x)−∇f(x)⋅z1{|z|⩽1})κ(x,z)|z|d+αdz, where 0<α<2, d⩾1, and κ(x,z) is a function on Rd×Rd 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 Lt 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),‖L1/2f‖p≍‖Γ(f)1/2‖p, where Γ(f):=12L(f2)−fLf is the carré du champ operator associated with L, and L1/2 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.