Let be a bounded q-pseudoconvex domain in , and let . If is smooth, we find sufficient conditions for the -Neumann operator to be compact. If is non-smooth and if , we show that compactness of the -Neumann operator, , on square integrable (0, p+1)-forms is equivalent to compactness of the commutators , , on square integrable -closed (0, p)-forms, where is the Bergman projection on (0, p)-forms. Moreover, we prove that compactness of the commutator of with bounded functions percolates up in the -complex on -closed forms and square integrable holomorphic forms. Furthermore, we find a characterization of compactness of the canonical solution operator, , of the -equation restricted on (0, p+1)-forms with holomorphic coefficients in terms of compactness of commutators , , on (0, p)-forms with holomorphic coefficients, where is the Bergman-Toeplitz operator acting on (0, p)-forms with symbol . This extends to domains which are not necessarily pseudoconvex.
Submitted August 4, 2017. Published May 10, 2018.
Math Subject Classifications: 32F10, 32W05.
Key Words: and -Neumann operator; Bergman-Toeplitz operator; q-convex domains.
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| Sayed Saber |
Faculty of Science
Beni-Suef University, Egypt
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