The fundamental solution to \Box_{𝑏} on quadric manifolds – Part 1. General formulas

Boggess, Albert; Raich, Andrew

doi:10.1090/bproc/77

The fundamental solution to $\Box _b$ on quadric manifolds – Part 1. General formulas
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Proc. Amer. Math. Soc. Ser. B 9 (2022), 186-203

Abstract:

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 Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] 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.

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