# Convex domains, Hankel operators, and maximal estimates

## TL;DR

This paper investigates conditions under which Hankel operators on convex domains are compact, linking boundary geometry, maximal estimates, and the presence of analytic varieties, and extends some results to general pseudoconvex domains.

## Contribution

It establishes a necessary boundary condition for compactness of Hankel operators on convex domains and connects maximal estimates with the absence of boundary analytic varieties, extending to pseudoconvex domains.

## Key findings

- Necessary boundary holomorphicity condition for compact Hankel operators.
- Maximal estimates exclude boundary analytic varieties except top-dimensional ones.
- Equivalence of compactness and subellipticity across all form levels under certain Levi form conditions.

## 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.