Compactness of commutators of Toeplitz operators on q-pseudoconvex domains
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Let Ω be a bounded q-pseudoconvex domain in ℂn, n ≥ 2 and let 1 ≤ q ≤ n - 1. If Ω is smooth, we find sufficient conditions for the ∂ˉ-Neumann operator to be compact. If Ω is non-smooth and if q ≤ p ≤ n - 1, we show that compactness of the ∂ˉ-Neumann operator, Np+1, on square integrable (0, p + 1)-forms is equivalent to compactness of the commutators [Bp, z̅j], 1 ≤ j ≤ n, on square integrable ∂ˉ-closed (0, p)-forms, where Bp is the Bergman projection on (0, p)-forms. Moreover, we prove that compactness of the commutator of Bp with bounded functions percolates up in the ∂ˉ-complex on ∂ˉ-closed forms and square integrable holomorphic forms. Furthermore, we find a characterization of compactness of the canonical solution operator, Sp+1, of the ∂ˉ-equation restricted on (0, p + 1)-forms with homomorphic coefficients in terms of compactness of commutators [Tpzj*, Tpzj], 1 ≤ j ≤ n, on (0, p)-forms with holomorphic coefficients, where Tpzj is the Bergman-Toeplitz operator acting on (0, p)-forms with symbol zj. This extends to domains which are not necessarily pseudoconvex.
CitationSaber, S. (2018). Compactness of commutators of Toeplitz operators on q-pseudoconvex domains. Electronic Journal of Differential Equations, 2018(111), pp. 1-17.
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