Geodesic
The $\bar\partial$-Neumann operator and commutators of the Bergman projection and multiplication operators
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1247-1256 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $\bar z$.
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 $\bar z$.
Classification : 32A36, 32W05
Keywords: $\bar{\partial}$-equation; $\bar{\partial}$-Neumann operator; compactness 
@article{CMJ_2008_58_4_a28,
     author = {Haslinger, Friedrich},
     title = {The $\bar\partial${-Neumann} operator and commutators of the {Bergman} projection and multiplication operators},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1247--1256},
     year = {2008},
     volume = {58},
     number = {4},
     mrnumber = {2471181},
     zbl = {1174.32015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2008_58_4_a28/}
}
TY  - JOUR
AU  - Haslinger, Friedrich
TI  - The $\bar\partial$-Neumann operator and commutators of the Bergman projection and multiplication operators
JO  - Czechoslovak Mathematical Journal
PY  - 2008
SP  - 1247
EP  - 1256
VL  - 58
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMJ_2008_58_4_a28/
LA  - en
ID  - CMJ_2008_58_4_a28
ER  - 
%0 Journal Article
%A Haslinger, Friedrich
%T The $\bar\partial$-Neumann operator and commutators of the Bergman projection and multiplication operators
%J Czechoslovak Mathematical Journal
%D 2008
%P 1247-1256
%V 58
%N 4
%U http://geodesic.mathdoc.fr/item/CMJ_2008_58_4_a28/
%G en
%F CMJ_2008_58_4_a28
Haslinger, Friedrich. The $\bar\partial$-Neumann operator and commutators of the Bergman projection and multiplication operators. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1247-1256. http://geodesic.mathdoc.fr/item/CMJ_2008_58_4_a28/

[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

[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 | Zbl

[3] Catlin, D., D'Angelo, J.: Positivity conditions for bihomogeneous polynomials. Math. Res. Lett. 4 (1997), 555-567. | DOI | MR | Zbl

[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 | Zbl

[5] D'Angelo, J.: Real hypersurfaces, orders of contact, and applications. Ann. Math. 115 (1982), 615-637. | DOI | MR | Zbl

[6] Fu, S., Straube, E. J.: Compactness of the $\bar{\partial}$-Neumann problem on convex domains. J. Funct. Anal. 159 (1998), 629-641. | DOI | MR

[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 | Zbl

[8] Folland, G., Kohn, J.: The Neumann problem for the Cauchy-Riemann complex. Annals of Math. Studies 75, Princeton University Press (1972). | MR | Zbl

[9] Haslinger, F.: The canonical solution operator to $\bar{\partial}$ restricted to Bergman spaces. Proc. Amer. Math. Soc. 129 (2001), 3321-3329. | DOI | MR

[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 | MR

[11] Haslinger, F.: Magnetic Schrädinger operators and the $\bar{\partial}$-equation. J. Math. Kyoto Univ. 46 (2006), 249-257. | DOI | MR

[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 | MR

[13] Haslinger, F., Lamel, B.: Spectral properties of the canonical solution operator to $\bar{\partial}$. J. Funct. Anal. 255 (2008), 13-24. | DOI | MR

[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 | MR

[15] Kohn, J.: Subellipticity of the $\bar{\partial}$-Neumann problem on pseudoconvex Domains: sufficient conditions. Acta Math. 142 (1979), 79-122. | DOI | MR

[16] Kohn, J., Nirenberg, L.: Non-coercive boundary value problems. Comm. Pure Appl. Math. 18 (1965), 443-492. | DOI | MR | Zbl

[17] Krantz, St.: Compactness of the $\bar{\partial}$-Neumann operator. Proc. Amer. Math. Soc. 103 (1988), 1136-1138. | DOI | MR | Zbl

[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 | Zbl

[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 | MR | Zbl

[20] Venugopalkrishna, U.: Fredholm operators associated with strongly pseudoconvex domains in $\mathbb C^n$. J. Funct. Anal. 9 (1972), 349-373. | DOI | MR