## 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.## References

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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