Multipliers on Pseudoconvex Domains with Real Analytic Boundaries

Joseph J. Kohn

Bollettino dell'Unione Matematica Italiana (2010)

  • Volume: 3, Issue: 2, page 309-324
  • ISSN: 0392-4041

Abstract

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This paper is concerned with (weakly) pseudoconvex real analytic hypersurfaces in 𝐂 n . We are motivated by the study of local boundary regularity of the ∂ ¯ -Neumann problem. Subelliptic estimates in a neighborhood of a point P in the boundary (which imply regularity) are controlled by ideals of germs of real analytic functions I 1 ⁢ ( P ) , ⋯ , I n - 1 ⁢ ( P ) . These ideals have the property that a subelliptic estimate holds for ( p , q ) -forms in a neighborhood of P if and only if 1 ∈ I q ⁢ ( P ) . The geometrical meaning of this is that 1 ∈ I q ⁢ ( P ) if and only if there is a neighborhood of P such that there does not exist a q -dimensional complex analytic manifold contained in the intersection of this neighborhood. Here we present a method to construct these manifolds explicitly. That is, if 1 ∉ I q ⁢ ( P ) then in every neighborhood of P we give an explicit construction of such a manifold. This result is part of a program to give a more precise understanding of regularity in terms of various norms. The techniques should also be useful in the study of other systems of partial differential equations.

How to cite

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Kohn, Joseph J.. "Multipliers on Pseudoconvex Domains with Real Analytic Boundaries." Bollettino dell'Unione Matematica Italiana 3.2 (2010): 309-324. <http://eudml.org/doc/290650>.

@article{Kohn2010,
abstract = {This paper is concerned with (weakly) pseudoconvex real analytic hypersurfaces in $\mathbf\{C\}^\{n\}$. We are motivated by the study of local boundary regularity of the $\bar\partial$-Neumann problem. Subelliptic estimates in a neighborhood of a point $P$ in the boundary (which imply regularity) are controlled by ideals of germs of real analytic functions $I^\{1\}(P),\cdots, I^\{n-1\}(P)$. These ideals have the property that a subelliptic estimate holds for $(p,q)$-forms in a neighborhood of $P$ if and only if $1 \in I^\{q\}(P)$. The geometrical meaning of this is that $1 \in I^\{q\}(P)$ if and only if there is a neighborhood of $P$ such that there does not exist a $q$-dimensional complex analytic manifold contained in the intersection of this neighborhood. Here we present a method to construct these manifolds explicitly. That is, if $1 \notin I^\{q\}(P)$ then in every neighborhood of $P$ we give an explicit construction of such a manifold. This result is part of a program to give a more precise understanding of regularity in terms of various norms. The techniques should also be useful in the study of other systems of partial differential equations.},
author = {Kohn, Joseph J.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {309-324},
publisher = {Unione Matematica Italiana},
title = {Multipliers on Pseudoconvex Domains with Real Analytic Boundaries},
url = {http://eudml.org/doc/290650},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Kohn, Joseph J.
TI - Multipliers on Pseudoconvex Domains with Real Analytic Boundaries
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/6//
PB - Unione Matematica Italiana
VL - 3
IS - 2
SP - 309
EP - 324
AB - This paper is concerned with (weakly) pseudoconvex real analytic hypersurfaces in $\mathbf{C}^{n}$. We are motivated by the study of local boundary regularity of the $\bar\partial$-Neumann problem. Subelliptic estimates in a neighborhood of a point $P$ in the boundary (which imply regularity) are controlled by ideals of germs of real analytic functions $I^{1}(P),\cdots, I^{n-1}(P)$. These ideals have the property that a subelliptic estimate holds for $(p,q)$-forms in a neighborhood of $P$ if and only if $1 \in I^{q}(P)$. The geometrical meaning of this is that $1 \in I^{q}(P)$ if and only if there is a neighborhood of $P$ such that there does not exist a $q$-dimensional complex analytic manifold contained in the intersection of this neighborhood. Here we present a method to construct these manifolds explicitly. That is, if $1 \notin I^{q}(P)$ then in every neighborhood of $P$ we give an explicit construction of such a manifold. This result is part of a program to give a more precise understanding of regularity in terms of various norms. The techniques should also be useful in the study of other systems of partial differential equations.
LA - eng
UR - http://eudml.org/doc/290650
ER -

References

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  9. KOHN, J. J., Boundary behavior of ∂ ¯ on weakly pseudo-convex manifolds of dimension two, J. Differential Geom., 6 (1972), 523-542. Zbl0256.35060MR322365
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  11. KOHN, J. J. - NIRENBERG, L., Noncoercive boundary value problems, Comm. Pure Appl. Math., 18 (1965), 443-492. Zbl0125.33302MR181815DOI10.1002/cpa.3160180305
  12. MORREY, C. B., The analytic embedding of abstract real analytic manifolds, Ann. Math., 40 (1958), 62-70. Zbl0090.38401MR99060DOI10.2307/1970048
  13. NARASIMHAN, R., Introduction to the theory of analytic spaces, Lecture notes in Math., No. 25 (Springer Verlag, 1966). MR217337
  14. SIU, Y.-T., Effective termination of Kohn's algorithm for subeliptic multipliers, arXiv; 0706.411v2 [math CV] 11 Jul 2008. MR2742044DOI10.4310/PAMQ.2010.v6.n4.a11
  15. STRAUBE, E. J., Lectures on the L 2 -Sobolev Theory of the ∂ ¯ -Neumann Problem, preprint (2009). MR2603659DOI10.4171/076

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