Heat kernel estimates and functional calculi of $-b Δ$ - McIntosh, Alan; Nahmod, Andrea (Mathematica Scandinavica, 2000)
We show that the elliptic operator ${\mathcal L} = - b(x) Δ$ has a bounded $H^∞$ functional calculus in $L^p(\boldsymbol R^n), 1 < p < ∞$, where $b$ is a bounded measurable complex-valued function with positive real part. In the process, we prove quadratic estimates for ${\mathcal L}$, 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 $\partial_t u + {\mathcal L} u = 0$.