Analysis on quadrics. (English) Zbl 1504.32095
Summary: This paper surveys recent progress on the analysis of the \(\Box_b\) operator on quadric submanifolds of \({\mathbb{C}}^n \times{\mathbb{C}}^m\). We focus our discussion on the (relative) fundamental solution to \(\Box_b\) on quadric submanifolds of arbitrary codimension. We summarize known results regarding \(\Box_b\)-invariance of mappings, necessary and sufficient conditions for solvability and hypoellipticity for \(\Box_b\), and we describe the \(L^p\) regularity of the complex Green operator on quadrics with nonvanishing Levi form. We discuss the ramifications of these results for many examples of quadrics of codimension 1 and 2.
MSC:
| 32V20 | Analysis on CR manifolds |
| 32W10 | \(\overline\partial_b\) and \(\overline\partial_b\)-Neumann operators |
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