Heat kernel estimates and functional calculi of -b Δ - McIntosh, Alan; Nahmod, Andrea (MATHEMATICA SCANDINAVICA, 2000)
We show that the elliptic operator L = - b(x) Δ has a bounded H^∞ functional calculus in L^p(R^n), 1 < p < ∞, where b is a bounded measurable complex-valued function with positive real part. In the process, we prove quadratic estimates for L, and obtain bounds with fast decay and Hölder continuity estimates for k_t(x,y) b(y) and its gradient, where k_t(x,y) is the heat kernel of -b(x) Δ. This implies L^p regularity of solutions to the parabolic equation _t u + L u = 0.