  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.
