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Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains

Zeljko Cuckovic, Sonmez Sahutoglu
Arxiv ID: 0809.1901Last updated: 3/8/2021
Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let f be a function smooth up to the boundary on a smooth bounded pseudoconvex domain D in C^n. We show that, if D is convex or the Levi form of the boundary of D is of rank at least n-2, then compactness of the Hankel operator H_f implies that f is holomorphic "along" analytic discs in the boundary. Furthermore, when D is convex in C^2 we show that the condition on f is necessary and sufficient for compactness of H_f

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