Convex domains, Hankel operators, and maximal estimates

Çeli̇k, Mehmet; Şahutoğlu, Sönmez; Straube, Emil

doi:10.1090/proc/14729

Convex domains, Hankel operators, and maximal estimates
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by Mehmet Çeli̇k, Sönmez Şahutoğlu and Emil J. Straube PDF

Proc. Amer. Math. Soc. 148 (2020), 751-764 Request permission

Abstract:

Let $1\leq q\leq (n-1)$. We first show that a necessary condition for a Hankel operator on $(0,q-1)$-forms on a convex domain to be compact is that its symbol is holomorphic along $q$-dimensional analytic varieties in the boundary. Because maximal estimates (equivalently, a comparable eigenvalues condition on the Levi form of the boundary) turn out to be favorable for compactness of Hankel operators, this result then implies that on a convex domain, maximal estimates exclude analytic varieties from the boundary, except ones of top dimension $(n-1)$ (and their subvarieties). Some of our techniques apply to general pseudoconvex domains to show that if the Levi form has comparable eigenvalues, or equivalently, if the domain admits maximal estimates, then compactness and subellipticity hold for forms at some level $q$ if and only if they hold at all levels.

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