The ${\mathrm{\square }}_b$ Heat Equation and Multipliers via the Wave Equation

Recently, Nagel and Stein studied the ${\mathrm{\square }}_b$-heat equation, where ${\mathrm{\square }}_b$ is the Kohn Laplacian on the boundary of a weakly-pseudoconvex domain of finite type in ${\text{C}}^2$. They showed that the Schwartz kernel of $e^{-t{\mathrm{\square }}_b}$ satisfies good “off-diagonal” estimates, while that of $e^{-t{\mathrm{\square }}_b}-\pi$ satisfies good “on-diagonal” estimates, where $\pi$ is the Szeg"o projection. We offer a simple proof of these results, which easily generalizes to other, similar situations. Our methods involve adapting the well-known relationship between the heat equation and the finite propagation speed of the wave equation to this situation. In addition, we apply these methods to study multipliers of the form $m\text{l}({\mathrm{\square }}_b\mathring{)}$. In particular, we show that $m\text{l}({\mathrm{\square }}_b\mathring{)}$ is an NIS operator, where $m$ satisfies an appropriate Mihlin-H"ormander condition.