Essential norm estimates for Hankel operators on convex domains in $\mathbb{C}^2$

  • Željko Čučković
  • Sönmez Şahutoğlu

Abstract

Let $\Omega \subset \mathbb{C}^2$ be a bounded convex domain with $C^1$-smooth boundary and $\varphi \in C^1(\overline{\Omega})$ such that $\varphi $ is harmonic on the non-trivial disks in the boundary. We estimate the essential norm of the Hankel operator $H_{\varphi }$ in terms of the $\overline{\partial}$ derivatives of $\varphi$ “along” the non-trivial disks in the boundary.

References

Adamjan, V. M., Arov, D. Z., and Kreĭn, M. G., Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem, Mat. Sb. (N.S.) 86(128) (1971), 34–75.

Asserda, S., The essential norm of Hankel operator on the Bergman spaces of strongly pseudoconvex domains, Integral Equations Operator Theory 36 (2000), no. 4, 379–395. https://doi.org/10.1007/BF01232736

Chen, S.-C. and Shaw, M.-C., 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.

Čučković, Ž. and Şahutoğlu, S., Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains, J. Funct. Anal. 256 (2009), no. 11, 3730–3742. https://doi.org/10.1016/j.jfa.2009.02.018

Fu, S. and Straube, E. J., Compactness of the $overline partial $-Neumann problem on convex domains, J. Funct. Anal. 159 (1998), no. 2, 629–641. https://doi.org/10.1006/jfan.1998.3317

Jarnicki, M. and Pflug, P., Invariant distances and metrics in complex analysis, de Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. https://doi.org/10.1515/9783110870312

Lin, P. and Rochberg, R., The essential norm of Hankel operator on the Bergman space, Integral Equations Operator Theory 17 (1993), no. 3, 361–372. https://doi.org/10.1007/BF01200291

Straube, E. J., Lectures on the $mathcal L^2$-Sobolev theory of the $overline partial $-Neumann problem, ESI Lectures in Mathematics and Physics, vol. 7, European Mathematical Society (EMS), Zürich, 2010. https://doi.org/10.4171/076

>

Published
2017-05-27
How to Cite
Čučković, Željko, & Şahutoğlu, S. (2017). Essential norm estimates for Hankel operators on convex domains in $\mathbb{C}^2$. MATHEMATICA SCANDINAVICA, 120(2), 305-316. https://doi.org/10.7146/math.scand.a-25793
Section
Articles