Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-16T04:12:22.006Z Has data issue: false hasContentIssue false
  • English
  • Français

L∞-continuity of Henkin operators solving in certain weakly pseudoconvex domains of ℂ2

Published online by Cambridge University Press:  14 November 2011

Joan Verdera
Joan Verdera
Affiliation:
Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, Barcelona 7, Spain
Get access Rights & Permissions [Opens in a new window]

Extract

Let ψ ∈ C2[0,1] be a positive function on (0, 1]. Under certain assumptions on ψ, the set

is a pseudoconvex domain with C2-boundary, for which it is possible to construct a Henkin-type operator Hψ = Kψ + Bψ solving in Dψ. The operator Bψ, is L∞-continuous because it has a Riesz potential type kernel, while the L∞-continuity of Kψ depends on the flatness of ψ at 0. Our main result states that Kψ is continuous from L∞(∂Dψ) into L∞(Dψ) if and only if

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 99 , Issue 1-2 , 1984 , pp. 25 - 33
Copyright
Copyright © Royal Society of Edinburgh 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Henkin, G.. Integral representations of functions in strictly pseudoconvex domains and applications to the 3 problem. Math. USSR Sb. 11 (1970), 273–281.CrossRefGoogle Scholar
2Henkin, G. and Cirka, E.. Boundary properties of holomorphic functions of several complex variables. J. Soviet Math. 5 (1976), 612–687.CrossRefGoogle Scholar
3Range, R.. Holder estimates for on convex domains in ℂ2 with real analytic boundary. Proc. Sympos. Pure Appl. Math. 30, 31–33 (Providence, R.I.: Amer. Math. Soc, 1977).Google Scholar
4Range, R.. On Hölder estimates for u=f on weakly pseudoconvex domains. Proc. Int. Conf., Cortona, Italy 1976–77, 247–267 (Pisa: Editrice Tecnico Scientifica EX, 1978).Google Scholar