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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The tangential Cauchy-Riemann complex on spheres


Author: G. B. Folland
Journal: Trans. Amer. Math. Soc. 171 (1972), 83-133
MSC: Primary 43A75; Secondary 35N15, 58G05
MathSciNet review: 0309156
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Abstract: This paper investigates the $ {\overline \partial _b}$ complex of Kohn and Rossi on the unit sphere in complex $ n$-space (considered as the boundary of the unit ball). The methods are Fourier-analytic, exploiting the fact that the unitary group $ U(n)$ acts homogeneously on the complex. We decompose the spaces of sections into irreducible components under the action of $ U(n)$ and compute the action of $ {\overline \partial _b}$ on each irreducible piece. We then display the connection between the $ {\overline \partial _b}$ complex and the Dolbeault complexes of certain line bundles on complex projective space. Precise global regularity theorems for $ {\overline \partial _b}$ are proved, including a Sobolev-type estimate for norms related to $ {\overline \partial _b}$. Finally, we solve the $ \overline \partial $-Neumann problem on the unit ball and obtain a proof by explicit calculations of the noncoercive nature of this problem.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0309156-X
Keywords: Tangential Cauchy-Riemann operators, subelliptic operators, transversally elliptic operators, representations of $ U(n)$, spherical harmonics, Sobolev estimates, $ \overline \partial $-Neumann problem, Bessel functions
Article copyright: © Copyright 1972 American Mathematical Society