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The □b Heat Equation and Multipliers via the Wave Equation

By Brian Street

Abstract

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 2. They showed that the Schwartz kernel of e −t □ b satisfies good “off-diagonal ” estimates, while that of e −t □ 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.

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