Convolution kernels of \((n+1)\)-fold Marcinkiewicz multipliers on the Heisenberg group (Q2777712)

On the Heisenberg group \(H_{n}=C^{n}\times R\), consider the \(n\)-partial sub-Laplacian operators \(L_{1},\cdots,L_{n}\) and the operator \(T=\frac{\partial}{\partial t}\). The corresponding Marcinkiewicz multipliers are functions \(m(L_{1},\dots ,\) \(L_{n}, iT)\) satisfying the following Marcinkiewicz-type condition: NEWLINE\[NEWLINE\mid (\xi_{1}\frac{\partial}{\partial\xi_{1}})^{i_{1}}\cdots(\xi_{n}\frac{\partial}{\partial\xi_{n}})^{i_{n}}(\eta\frac{\partial}{\partial\eta})^{j} m(\xi_{1},\dots ,\xi_{n},\eta)\mid\leq C_{ij}.NEWLINE\]NEWLINE The author of the paper under review proved the \(L^{p}\)-boundedness, \(1<p<+\infty\), of these multipliers, and showed that their convolution kernels have to satisfy some regularity and cancellation conditions [Bull. Aust. Math. Soc. 63, No. 1, 35-58 (2001; Zbl 0979.43004)]. In the paper under review, the author proves that actually these regularity and cancellation conditions are sufficient to completely characterize the convolution kernels of the Marcinkiewicz multipliers \(m(L_{1},\cdots,L_{n},iT)\).