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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The tangential Cauchy-Riemann complex on spheres
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by G. B. Folland PDF
Trans. Amer. Math. Soc. 171 (1972), 83-133 Request permission

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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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 171 (1972), 83-133
  • MSC: Primary 43A75; Secondary 35N15, 58G05
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0309156-X
  • MathSciNet review: 0309156