Special Issue — Sanya School on Complex Analysis and Complex Geometry; Guest Editors: John Erik Fornæss and Franc ForstneričNo Access

Sobolev spaces for the weighted $\bar{\partial}$ -Neumann operator

Friedrich Haslinger

Friedrich Haslinger

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria

E-mail Address: friedrich.haslinger@univie.ac.at

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https://doi.org/10.1142/S0129167X17400079Cited by:0 (Source: Crossref)

Abstract

We discuss compactness of the $\bar{\partial}$ -Neumann operator in the setting of weighted $L^{2}$ -spaces on $ℂ^{n} .$ In addition we describe an approach to obtain the compactness estimates for the $\bar{\partial}$ -Neumann operator. For this purpose we have to define appropriate weighted Sobolev spaces and prove an appropriate Rellich–Kondrachov lemma.

Keywords:

AMSC: 32W05, 35N15, 46E35

References

1. R. A. Adams and J. J. F. Fournier, Sobolev spaces, Pure and Applied Math. 140 (2006). Google Scholar
2. P. Bolley, M. Dauge and B. Helffer, Conditions suffisantes pour l’injection compacte d’espace de Sobolev à poids, Séminaire Équation aux Dérivées Partielles (France), Université de Nantes 1 (1989) 1–14. Google Scholar
3. E. B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, Vol. 42 (Cambridge University Press, Cambridge, 1995). Google Scholar
4. F. Haslinger, The $\bar{\partial}$ -Neumann problem and Schrödinger operators, de Gruyter Expositions in Mathematics 59 (2014). Google Scholar
5. F. Haslinger and B. Helffer, Compactness of the solution operator to $\bar{\partial}$ in weighted $L^{2}$ -spaces, J. Funct. Anal. 243 (2007) 679–697. Web of Science, Google Scholar
6. J. Johnsen, On the spectral properties of Witten Laplacians, their range projections and Brascamp-Lieb’s inequality, Integral Equations Operator Theory 36 (2000) 288–324. Web of Science, Google Scholar
7. J.-M. Kneib and F. Mignot, Equation de Schmoluchowski généralisée, Ann. Math. Pura Appl. (IV) 167 (1994) 257–298. Web of Science, Google Scholar
8. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure and Appl. Math. 18 (1965) 443–492. Web of Science, Google Scholar
9. E. Straube, The $L^{2}$ -Sobolev theory of the $\bar{\partial}$ -Neumann problem, ESI Lectures in Mathematics and Physics EMS (2010). Google Scholar