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Compactness of Hankel operators with continuous symbols on convex domains (2005.14323v2)
Published 28 May 2020 in math.CV and math.FA
Abstract: Let $\Omega$ be a bounded convex domain in $\mathbb{C}{n}$, $n\geq 2$, $1\leq q\leq (n-1)$, and $\phi\in C(\bar{\Omega})$. If the Hankel operator $H{q-1}_{\phi}$ on $(0,q-1)$--forms with symbol $\phi$ is compact, then $\phi$ is holomorphic along $q$--dimensional analytic (actually, affine) varieties in the boundary. We also prove a partial converse: if the boundary contains only `finitely many' varieties, $1\leq q\leq n$, and $\phi\in C(\bar{\Omega})$ is analytic along the ones of dimension $q$ (or higher), then $H{q-1}_{\phi}$ is compact.