Abstract
Recently, Nagel and Stein studied the \({\square_{b}}\) -heat equation, where \({\square_{b}}\) is the Kohn Laplacian on the boundary of a weakly pseudoconvex domain of finite type in \({\mathbb{C}^2}\) . They showed that the Schwartz kernel of \({e^{-t\square_{b}}}\) satisfies good “off-diagonal” estimates, while that of \({e^{-t\square_{b}}-\pi}\) 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 \({m(\square_{b})}\) . In particular, we show that \({m(\square_{b})}\) is an NIS operator, where m satisfies an appropriate Mihlin–Hörmander condition.
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Street, B. The \({\square_{b}}\) heat equation and multipliers via the wave equation. Math. Z. 263, 861–886 (2009). https://doi.org/10.1007/s00209-008-0443-1
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DOI: https://doi.org/10.1007/s00209-008-0443-1