The tangential Cauchy-Riemann complex on spheres

Abstract

This paper investigates the ∂ ¯ b {\overline \partial _b} complex of Kohn and Rossi on the unit sphere in complex n n -space (considered as the boundary of the unit ball). The methods are Fourier-analytic, exploiting the fact that the unitary group U ( n ) U(n) acts homogeneously on the complex. We decompose the spaces of sections into irreducible components under the action of U ( n ) U(n) and compute the action of ∂ ¯ b {\overline \partial _b} on each irreducible piece. We then display the connection between the ∂ ¯ b {\overline \partial _b} complex and the Dolbeault complexes of certain line bundles on complex projective space. Precise global regularity theorems for ∂ ¯ b {\overline \partial _b} are proved, including a Sobolev-type estimate for norms related to ∂ ¯ b {\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.