Abstract.
The first main result in this article provides uniform estimates for solid Bochner-Martinelli-Koppelman transforms of type (p, n-1), 0 ≤ p ≤ n, of continuous forms on a compact set Ω in the complex space \({\mathbb{C}}^n\), in terms of the Euclidean volume of Ω. In the single variable case this result generalizes a classical inequality for the Cauchy kernel due to Ahlfors and Beurling. The second main result is a quantitative Hartogs-Rosenthal theorem which points out that the uniform distance in the space of continuous (p, n-1)-forms, 0 ≤ p ≤ n, on a compact set Ω in \({\mathbb{C}}^n\) from a smooth form to the subspace of \(\bar{\partial}\)-closed forms on Ω is controlled by the Euclidean volume of Ω. This theorem as well as a third main result are generalizations to higher dimensions of an inequality in single variable complex analysis due to Alexander.
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Received: January, 2006. Accepted: July, 2006.
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Martin, M. Uniform Approximation by Closed Forms in Several Complex Variables. AACA 19, 777 (2009). https://doi.org/10.1007/s00006-009-0189-9
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DOI: https://doi.org/10.1007/s00006-009-0189-9