course: Different kinds of Riesz transforms on the Heisenberg group and their \(L^p\)-boundedness
speaker: Fulvio Ricci (Scuola Normale Superiore)abstract: On the \((2n+1)\)-dimensional Heisenberg group \(H_n\) we consider two left-invariant metrics and two associated ''laplacians''.
One of them is the Carnot-Carathéodory metric with its sub-Laplacian \(L\), and the other is a Riemannian metric, for which the Laplace-Beltrami operator is the ''full'' Laplacian \(\Delta=L+\partial^2_t\), where \(t\) denotes the vertical coordinate. In each metric we have a family of Riesz transforms, \(XL^{-1/2}\) and \(X\Delta^{-1/2}\) respectively, where \(X\) is a left-invariant vector field. The nature of the Riesz transforms is well understood, as they are Calderón-Zygmund operators satisfying the standard estimates in the appropriate metric.
There are several reasons to study the operators which arise from compositions of Riesz transforms (or more general C-Z operators) of the two kinds and their \(L^p\)-boundendess properties. One of them is that both metrics intervene in the analysis of the (Riemannian) Hodge Laplacian \(\Delta_k\) acting on differential forms of order \(k\) and, in particular of \(L^p\)-boundedness of the Hodge-Riesz transforms \(d\Delta_k^{-1/2}\).
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