On compactness of the -Neumann problem and Hankel operators

Authors:
Mehmet Çelik and Sönmez Şahutoğlu

Journal:
Proc. Amer. Math. Soc. **140** (2012), 153-159

MSC (2010):
Primary 32W05; Secondary 47B35

Published electronically:
August 29, 2011

MathSciNet review:
2833527

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Abstract: Let , where and are two smooth bounded pseudoconvex domains in such that Assume that the -Neumann operator of is compact and the interior of the Levi-flat points in the boundary of is not empty (in the relative topology). Then we show that the Hankel operator on with symbol is compact for every but the -Neumann operator on is not compact.

**[CD97]**David W. Catlin and John P. D’Angelo,*Positivity conditions for bihomogeneous polynomials*, Math. Res. Lett.**4**(1997), no. 4, 555–567. MR**1470426**, 10.4310/MRL.1997.v4.n4.a11**[CS01]**So-Chin Chen and Mei-Chi Shaw,*Partial differential equations in several complex variables*, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR**1800297****[ČŞ09]**Željko Čučković and Sönmez Şahutoğlu,*Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains*, J. Funct. Anal.**256**(2009), no. 11, 3730–3742. MR**2514058**, 10.1016/j.jfa.2009.02.018**[FS01]**Siqi Fu and Emil J. Straube,*Compactness in the \overline∂-Neumann problem*, Complex analysis and geometry (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 9, de Gruyter, Berlin, 2001, pp. 141–160. MR**1912737****[Has08]**Friedrich Haslinger,*The \overline∂-Neumann operator and commutators of the Bergman projection and multiplication operators*, Czechoslovak Math. J.**58(133)**(2008), no. 4, 1247–1256. MR**2471181**, 10.1007/s10587-008-0084-x**[HI97]**Gennadi M. Henkin and Andrei Iordan,*Compactness of the Neumann operator for hyperconvex domains with non-smooth 𝐵-regular boundary*, Math. Ann.**307**(1997), no. 1, 151–168. MR**1427681**, 10.1007/s002080050028**[Hör65]**Lars Hörmander,*𝐿² estimates and existence theorems for the ∂ operator*, Acta Math.**113**(1965), 89–152. MR**0179443****[Koh63]**J. J. Kohn,*Harmonic integrals on strongly pseudo-convex manifolds. I*, Ann. of Math. (2)**78**(1963), 112–148. MR**0153030****[Sha10]**Mei-Chi Shaw,*The closed range property for on domains with pseudoconcave boundary*, Complex analysis, Trends Math., Birkhäuser, Basel, 2010, pp. 307-320.**[ŞS06]**Sönmez Şahutoğlu and Emil J. Straube,*Analytic discs, plurisubharmonic hulls, and non-compactness of the \overline∂-Neumann operator*, Math. Ann.**334**(2006), no. 4, 809–820. MR**2209258**, 10.1007/s00208-005-0737-0**[Str97]**Emil J. Straube,*Plurisubharmonic functions and subellipticity of the \overline∂-Neumann problem on non-smooth domains*, Math. Res. Lett.**4**(1997), no. 4, 459–467. MR**1470417**, 10.4310/MRL.1997.v4.n4.a2**[Str10]**Emil J. Straube,*Lectures on the ℒ²-Sobolev theory of the \overline{∂}-Neumann problem*, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2010. MR**2603659****[Ven72]**U. Venugopalkrishna,*Fredholm operators associated with strongly pseudoconvex domains in 𝐶ⁿ*, J. Functional Analysis**9**(1972), 349–373. MR**0315502**

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Additional Information

**Mehmet Çelik**

Affiliation:
Department of Mathematics and Information Sciences, University of North Texas at Dallas, 7300 Houston School Road, Dallas, Texas 75241

Email:
Mehmet.Celik@unt.edu

**Sönmez Şahutoğlu**

Affiliation:
Department of Mathematics & Statistics, University of Toledo, 2801 West Bancroft Street, Toledo, Ohio 43606

Email:
sonmez.sahutoglu@utoledo.edu

DOI:
https://doi.org/10.1090/S0002-9939-2011-11350-9

Keywords:
$\overline{\partial}$-Neumann problem,
Hankel operators,
non-pseudoconvex domains

Received by editor(s):
August 24, 2010

Published electronically:
August 29, 2011

Additional Notes:
The second author is supported in part by the University of Toledo’s Summer Research Awards and Fellowships Program

Communicated by:
Mei-Chi Shaw

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.