The tangential Cauchy-Riemann complex on spheres

Folland, G. B.

doi:10.1090/S0002-9947-1972-0309156-X

The tangential Cauchy-Riemann complex on spheres
HTML articles powered by AMS MathViewer

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

Hermann Boerner, Representations of groups. With special consideration for the needs of modern physics, Second English edition, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1970. Translated from the German by P. G. Murphy in cooperation with J. Mayer-Kalkschmidt and P. Carr. MR 0272911
Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745
F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. MR 0202713
K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. III. Stability theorems for complex structures, Ann. of Math. (2) 71 (1960), 43–76. MR 115189, DOI 10.2307/1969879
J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. I, Ann. of Math. (2) 78 (1963), 112–148. MR 153030, DOI 10.2307/1970506
J. J. Kohn, Boundaries of complex manifolds, Proc. Conf. Complex Analysis (Minneapolis, 1964) Springer, Berlin, 1965, pp. 81–94. MR 0175149
J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492. MR 181815, DOI 10.1002/cpa.3160180305
J. J. Kohn and Hugo Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of Math. (2) 81 (1965), 451–472. MR 177135, DOI 10.2307/1970624
Hans Lewy, On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. of Math. (2) 64 (1956), 514–522. MR 81952, DOI 10.2307/1969599
Hans Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. (2) 66 (1957), 155–158. MR 88629, DOI 10.2307/1970121
Charles N. Moore, The summability of the developments in Bessel functions, with applications, Trans. Amer. Math. Soc. 10 (1909), no. 4, 391–435. MR 1500847, DOI 10.1090/S0002-9947-1909-1500847-3
Claus Müller, Spherical harmonics, Lecture Notes in Mathematics, vol. 17, Springer-Verlag, Berlin-New York, 1966. MR 0199449, DOI 10.1007/BFb0094775
Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory. , Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
William J. Sweeney, The $D$-Neumann problem, Acta Math. 120 (1968), 223–277. MR 226662, DOI 10.1007/BF02394611
François Trèves, On local solvability of linear partial differential equations, Bull. Amer. Math. Soc. 76 (1970), 552–571. MR 257550, DOI 10.1090/S0002-9904-1970-12443-0
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
André Weil, L’intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 869, Hermann & Cie, Paris, 1940 (French). [This book has been republished by the author at Princeton, N. J., 1941.]. MR 0005741
Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158