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The fundamental solution to \(\Box_b\) on quadric manifolds. I: General formulas. (English) Zbl 1487.32201

Summary: This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of \(\mathbb{C}^n\times\mathbb{C}^m\). In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator \(N\) and the projection onto the nullspace of \(\Box_b\). The main application of our formulas is the critical case of codimension two quadrics in \(\mathbb{C}^4\) where we discuss the known solvability and hypoellipticity criteria of M. M. Peloso and F. Ricci [Funct. Anal. 203, No. 2, 321–355 (2003; Zbl 1043.32021)]. We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups.

MSC:

32W10 \(\overline\partial_b\) and \(\overline\partial_b\)-Neumann operators
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)

Citations:

Zbl 1043.32021
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References:

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