Two-sided estimates on Dirichlet heat kernels for time-dependent parabolic operators with singular drifts in C(1,alpha)-domains

Abstract

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 C-1,C-alpha-domain in R-d, where d >= 1 and alpha is an element of (0,1]. Our operator is L + mu . del(x), where L is a time-dependent uniformly elliptic divergent operator with Dini continuous coefficients and mu is a signed measure on (0, infinity) x R-d 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. (C) 2011 Elsevier Inc. All rights reserved.