Heat kernels for non-symmetric diffusion operators with jumps

For d⩾2, we prove the existence and uniqueness of heat kernels to the following time-dependent second order diffusion operator with jumps: L_t:=[Formula presented]∑i,j=1da_ij(t,x)∂_ij ²+∑i=1db_i(t,x)∂_i+L_t ^κ, where a=(a_ij) is a uniformly bounded, elliptic, and Hölder continuous matrix-valued function, b belongs to some suitable Kato's class, and L_t ^κ is a non-local α-stable-type operator with bounded kernel κ. Moreover, we establish sharp two-sided estimates, gradient estimate and fractional derivative estimate for the heat kernel under some mild conditions.