Heat kernel estimates and functional calculi of $-b \Delta$
Heat kernel estimates and functional calculi of $-b \Delta$
We show that the elliptic operator ${\mathcal L} = - b(x) \Delta$ has a bounded $H^\infty$ functional calculus in $L^p(\boldsymbol R^n), 1 < p < \infty$, 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 …