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Keywords:
$\bar{\partial}$-equation; $\bar{\partial}$-Neumann operator; compactness
Summary:
We prove that compactness of the canonical solution operator to $\bar \partial $ restricted to $(0,1)$-forms with holomorphic coefficients is equivalent to compactness of the commutator $[\mathcal P,\bar M]$ defined on the whole $L^2_{(0,1)}(\Omega ),$ where $\bar M$ is the multiplication by $\bar z$ and $\mathcal P $ is the orthogonal projection of $L^2_{(0,1)}(\Omega )$ to the subspace of $(0,1)$ forms with holomorphic coefficients. Further we derive a formula for the $\bar \partial $-Neumann operator restricted to $(0,1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by $z $ and $\bar z$.
References:
[1] Boas, H. P., Straube, E. J.: Global regularity of the $\bar{\partial} $-Neumann problem: a survey of the $L^2$-Sobolev theory. Several Complex Variables (M. Schneider and Y.-T. Siu, eds.) MSRI Publications, vol. 37, Cambridge University Press (1999), 79-111. MR 1748601
[2] Catlin, D.: Global regularity of the $\bar{\partial} $-Neumann problem. Proc. Sympos. Pure Math. 41 39-49; A.M.S. Providence, Rhode Island, 1984. MR 0740870 | Zbl 0578.32031
[3] Catlin, D., D'Angelo, J.: Positivity conditions for bihomogeneous polynomials. Math. Res. Lett. 4 (1997), 555-567. DOI 10.4310/MRL.1997.v4.n4.a11 | MR 1470426 | Zbl 0886.32015
[4] Chen, So-Chin, Shaw, Mei-Chi: Partial differential equations in several complex variables. Studies in Advanced Mathematics, Vol. 19, Amer. Math. Soc. (2001). MR 1800297 | Zbl 0963.32001
[5] D'Angelo, J.: Real hypersurfaces, orders of contact, and applications. Ann. Math. 115 (1982), 615-637. DOI 10.2307/2007015 | MR 0657241 | Zbl 0488.32008
[6] Fu, S., Straube, E. J.: Compactness of the $\bar{\partial}$-Neumann problem on convex domains. J. Funct. Anal. 159 (1998), 629-641. DOI 10.1006/jfan.1998.3317 | MR 1659575
[7] Fu, S., Straube, E. J.: Compactness in the $\bar{\partial}$-Neumann problem. Complex Analysis and Geometry (J. McNeal, ed.), Ohio State Math. Res. Inst. Publ. 9 (2001), 141-160. MR 1912737 | Zbl 1011.32025
[8] Folland, G., Kohn, J.: The Neumann problem for the Cauchy-Riemann complex. Annals of Math. Studies 75, Princeton University Press (1972). MR 0461588 | Zbl 0247.35093
[9] Haslinger, F.: The canonical solution operator to $\bar{\partial}$ restricted to Bergman spaces. Proc. Amer. Math. Soc. 129 (2001), 3321-3329. DOI 10.1090/S0002-9939-01-05953-6 | MR 1845009
[10] Haslinger, F.: The canonical solution operator to $\bar{\partial}$ restricted to spaces of entire functions. Ann. Fac. Sci. Toulouse Math. 11 (2002), 57-70. DOI 10.5802/afst.1018 | MR 1986383
[11] Haslinger, F.: Magnetic Schrädinger operators and the $\bar{\partial}$-equation. J. Math. Kyoto Univ. 46 (2006), 249-257. MR 2284342
[12] Haslinger, F., Helffer, B.: Compactness of the solution operator to $\bar{\partial}$ in weighted $L^2$-spaces. J. Funct. Anal. 243 (2007), 679-697. DOI 10.1016/j.jfa.2006.09.004 | MR 2289700
[13] Haslinger, F., Lamel, B.: Spectral properties of the canonical solution operator to $\bar{\partial}$. J. Funct. Anal. 255 (2008), 13-24. DOI 10.1016/j.jfa.2008.03.013 | MR 2417807
[14] Henkin, G., Iordan, A.: Compactness of the $\bar{\partial}$-Neumann operator for hyperconvex domains with non-smooth B-regular boundary. Math. Ann. 307 (1997), 151-168. DOI 10.1007/s002080050028 | MR 1427681
[15] Kohn, J.: Subellipticity of the $\bar{\partial}$-Neumann problem on pseudoconvex Domains: sufficient conditions. Acta Math. 142 (1979), 79-122. DOI 10.1007/BF02395058 | MR 0512213
[16] Kohn, J., Nirenberg, L.: Non-coercive boundary value problems. Comm. Pure Appl. Math. 18 (1965), 443-492. DOI 10.1002/cpa.3160180305 | MR 0181815 | Zbl 0125.33302
[17] Krantz, St.: Compactness of the $\bar{\partial}$-Neumann operator. Proc. Amer. Math. Soc. 103 (1988), 1136-1138. DOI 10.1090/S0002-9939-1988-0954995-2 | MR 0954995 | Zbl 0736.35071
[18] Ligocka, E.: The regularity of the weighted Bergman projections. Seminar on deformations, Proceedings, Lodz-Warsaw, 1982/84, Lecture Notes in Math. {\it 1165}, Springer-Verlag, Berlin (1985), 197-203. MR 0825756 | Zbl 0594.35049
[19] Salinas, N., Sheu, A., Upmeier, H.: Toeplitz operators on pseudoconvex domains and foliation $C^{\ast}$-algebras. Ann. of Math. 130 (1989), 531-565. DOI 10.2307/1971454 | MR 1025166 | Zbl 0708.47021
[20] Venugopalkrishna, U.: Fredholm operators associated with strongly pseudoconvex domains in $\mathbb C^n$. J. Funct. Anal. 9 (1972), 349-373. DOI 10.1016/0022-1236(72)90007-9 | MR 0315502
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