Heat Equation and Multipliers via the Wave Equation

Recently, Nagel and Stein studied the b-heat equation, where b is the Kohn Laplacian on the boundary of a weakly-pseudoconvex domain of finite type in C. They showed that the Schwartz kernel of e b satisfies good “off-diagonal” estimates, while that of e b −π satisfies good “on-diagonal” estimates, where π denotes the Szegö 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 ( b). In particular, we show that m ( b) is an NIS operator, where m satisfies an appropriate Mihlin-Hörmander condition.