Convex domains, Hankel operators, and maximal estimates

Abstract

Let 1 ≤ q ≤ (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. q s=1 λ j s (z) ≤ λ 1 (z) + · · · + λ n−1 (z) for any q-tuple (j 1 , . . . , j q ) andDate: February 24, 2019. 2010 Mathematics Subject Classification. Primary 32W05; Secondary 47B35. Even if we were to drop the requirement that the basis be orthonormal, the next condition would remain independent of the basis chosen, in view of a theorem of Ostrowski which relates eigenvalues of matrices of the form M and S T MS; see for example [12], Theorem 4.5.9.

Diederich-Fornæss index and global regularity in the $\overline{\partial}$-Neumann problem: domains with comparable Levi eigenvalues

“…for the second inequality we choose Ω η ⊂⊂ Ω such that on Ω \ Ω η , C η ρ 2 ≤ 1. Inserting ( 26) into (12) gives (11). The proof of Proposition 1 is now complete.…”

“…In the proof of Theorem 1, we will 'only' need to estimate Ω |α η (L J u )| 2 , rather than work with the point wise estimate (10). This is essential: once we integrate the right hand side of (10), we can use L 2 methods (and in particular the approximate self bounded gradient condition for (−h η ) mentioned above) to obtain estimate (11) below. This estimate is the crux of the matter.…”

“…We do this via (11) in Proposition 1, applied to the (0, 1)form (X η − X η ) k (N δ,q u) J ω j .Then L u = ((X η − X η ) k (N δ,q u) J L j , and the resulting estimate is (41)…”

“…In[11], footnote 1, the authors point out that thanks to a theorem of Ostrowski ([24], Theorem 4.5.9) the comparability condition remains independent of the basis even when the orthonormality requirement is dropped.…”

Maximal Estimates for the $\bar\partial$-Neumann Problem on Non-pseudoconvex domains

“…Maximal estimates have many applications. In addition to those given by Derridj and Ben Moussa, see also [6], [22], or [23].…”