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

Abstract

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

Revised, to appear in J. Funct. Anal

References in corpus (1)

• Analytic discs, plurisubharmonic hulls, and non-compactness of the d-bar-Neumann operator

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• Compactness of Hankel operators with continuous symbols
• On compactness of the dbar-Neumann problem and Hankel operators
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• On compactness of Hankel and the $\bar{\partial }$-Neumann operators on Hartogs domains in ${\mathbb{C}}^2$
• Convex domains, Hankel operators, and maximal estimates
• Compactness of products of Hankel operators on the polydisk and some product domains in ${\mathbb{C}}^2$
• Schatten class Hankel and $\overline{\partial }$-Neumann operators on pseudoconvex domains in ${\mathbb{C}}^n$
• Compactness of Hankel operators with symbols continuous on the closure of pseudoconvex domains
• Essential norm estimates for Hankel operators on convex domains in ${\mathbb{C}}^2$
• On spectra of Hankel operators on the polydisc
• Compactness of composition operators on the Bergman spaces of convex domains and analytic discs