Two-sided estimates on Dirichlet heat kernels for time-dependent parabolic operators with singular drifts in C1,α-domains

In this paper, we establish sharp two-sided estimates for the Dirichlet heat kernels of a large class of time-dependent parabolic operators with singular drifts in C1,α-domain in Rd, where d⩾1 and α∈(0,1]. Our operator is L+μ⋅∇x, where L is a time-dependent uniformly elliptic divergent operator with Dini continuous coefficients and μ is a signed measure on (0,∞)×Rd belonging to parabolic Kato class. Along the way, a gradient estimate is also established. Our method employs a combination of partial differential equations and perturbation techniques.