zbMATH — the first resource for mathematics

The null space of the $$\overline \partial$$-Neumann operator. (English) Zbl 1083.32033
In a first part, the author proves results concerning closed range of $$\overline{\partial}$$ and invertibility of $$\square$$ (and concerning its null space on $$(n-1)$$-forms) on spherical shells, by systematic use of spherical harmonics. Some of these results are known, but the author’s methods yield more precise constants.
In section 3, the author proves a theorem that precises the notion that when the Levi form has the critical signature at some point of the boundary ($$(n-q-1,q)$$), then the $$\overline{\partial}$$-Neumann problem is flawed on the domain at the level of $$(0,q)$$-forms: the dimension of the null space of $$\square_{q}$$ is infinite, or $$\overline{\partial}$$ does not have closed range on $$L^{2}_{(0,q)}$$ (possibly both).
One of the results in section 2 is that the null space of $$\square_{n-1}$$ has $$n$$ independent multipliers. In section 4, it is shown that this property characterizes shells bounded by two confocal ellipsoids. A description of the null space of $$\square_{n-1}$$ is obtained for these domains that allows to discuss the boundary behavior of the kernel of the orthogonal projection on this null space when the range of $$\square_{n-1}$$ is closed, at a boundary point where the Levi form has signature $$(0,n-1)$$.

MSC:
 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
Full Text:
References:
 [1] H.P. Boas & M.-C. Shaw, Sobolev estimates for the lewy operator on weakly pseudo-convex boundaries, Math. Ann274 (1986) p. 221-231 · Zbl 0588.32023 [2] L. Boutet de Monvel & J. Sjöstrand, Sur la singularité des noyaux de Bergman et de szegö, Astérisque34-35 (1976) p. 123-164 · Zbl 0344.32010 [3] S.-S. Chern & J.-K. Moser, Real hypersurfaces in complex manifolds, Acta Math133 (1974) p. 219-271 · Zbl 0302.32015 [4] S.-C. Chen & M.-C. Shaw, Partial differential equations in several complex variables, AMS/IP Studies in advanced mathematics Vol. 19, Amer. Math. Soc, 2001 · Zbl 0963.32001 [5] L. Hörmander $$, L^2$$ estimates and existence theorems for the $$\bar{∂ }$$ operator, Acta Math.113 (1965) p. 89-152 · Zbl 0158.11002 [6] L. Hörmander, The multinomial distribution and some Bergman kernels, Geometric analysis of PDE and several complex variables. Contemporary Mathematics Proceedings (to appear) · Zbl 1102.41028 [7] L. Hörmander, The analysis of linear partial differential operators I, Springer Verlag, 1983 · Zbl 0521.35001 [8] J.-J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. I., Ann. of Math.78 (1963) p. 112-148 · Zbl 0161.09302 [9] J.-J. Kohn & L. Nirenberg, Non-coercive boundary problems, Comm. Pure Appl. Math18 (1965) p. 443-492 · Zbl 0125.33302 [10] J.-J. Kohn & D.C. Spencer, Complex Neumann problems, Ann. of Math66 (1957) p. 89-140 · Zbl 0099.30605 [11] M.-C. Shaw, Global solvability and regularity for $$\bar{∂ }$$ on an annulus between two weakly pseudo-convex domains, Trans. Amer. Math. Soc291 (1985) p. 255-267 · Zbl 0594.35010 [12] M.-C. Shaw $$, L^2$$ estimates and existence theorems for the tangential Cauchy-Riemann complex, Invent. Math82 (1985) p. 133-150 · Zbl 0581.35057 [13] L. Hörmander, The analysis of linear partial differential operators III, Springer-Verlag, 1985 · Zbl 0601.35001 [14] J.J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. II, Ann. Math. (2)79 (1964) p. 450-472 · Zbl 0178.11305
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.