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The fundamental solution to $\Box_b$ on quadric manifolds with nonzero eigenvalues and null variables

Albert Boggess and Andrew Raich

Vol. 327 (2023), No. 2, 233–266
Abstract

We prove sharp pointwise bounds on the complex Green operator and its derivatives on a class of embedded quadric manifolds of high codimension. In particular, we start with the class of quadrics that we previously analyzed (Trans. Amer. Math. Soc. Ser. B 10 (2023), 507–541) — ones whose directional Levi forms are nondegenerate, and add in null variables. The null variables do not substantially affect the estimates or analysis at the form levels for which b is solvable and hypoelliptic. In the nonhypoelliptic degrees, however, the estimates and analysis are substantially different. In the earlier paper, when hypoellipticity of b failed, so did solvability. Here, however, we show that if there is at least one null variable, b is always solvable, and the estimates are qualitatively different than in the other cases. Namely, the complex Green operator has blow-ups off of the diagonal. We also characterize when a quadric M whose Levi form vanishes on a complex subspace admits a b-invariant change of coordinates so that M presents with a null variable.

Keywords
quadric submanifolds, higher codimension, nonzero eigenvalues, complex Green operator, null variables, zero eigenvalues
Mathematical Subject Classification
Primary: 32V20, 32W10, 35R03
Secondary: 42B37, 43A80
Milestones
Received: 26 August 2023
Revised: 24 December 2023
Accepted: 4 February 2024
Published: 12 March 2024
Authors
Albert Boggess
School of Mathematical and Statistical Sciences
Arizona State University
Tempe, AZ
United States
Andrew Raich
Department of Mathematical Sciences
University of Arkansas
Fayetteville, AR
United States

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