Global boundary regularity for the $ \bar \partial $‐equation on q‐pseudoconvex domains

Abstract

We introduce a notion of q-pseudoconvex domain of new type for a bounded domain of C n and prove that for given a ∂-closed (p, r)-form, r ≥ q, that is smooth up to the boundary, there exists a (p, r − 1)-form smooth up to the boundary which is a solution of ∂-equation on a bounded q-pseudoconvex domain.

Compactness of the complex Green operator

“…Here, ∂ * ϕ is the adjoint of ∂ with respect to the weighted L 2 inner product with weight e −ϕ . The computations that lead to (3.23) in [27] are valid for general weight functions, and there is the following analogue of (3.23) (this is made explicit in [1], Proposition 2.1):…”

Subelliptic estimates and regularity of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\bf \bar{\partial}$\end{document} at the boundary of a Q‐pseudoconvex domain of finite type

“…Our first choice is a = q o , b = n and k = q for a pair q and q o satisfying q ≥ q o + 1. Following [18] and [1] we introduce Definition 2. 2 We say that ∂Ω is q-pseudoconvex (for the orientation induced from Ω) at a point z o ∈ ∂Ω, if there exists a local patch B and a C 2 smooth bundle…”

Estimates for the complex Green operator: Symmetry, percolation, and interpolation

“…III.3 for a discussion of the Hermitian structure on C n that pays attention to norms of the dz j , etc.). 1 After this paper was posted and submitted for publication, we became aware of [13], where a closely related result is shown in the case where M is an actual hypersurface. We thank Ken Koenig for pointing out this reference to us.…”

“…Because P 0 G (p,q) is elliptic, the full estimates then interpolate: if the complex Green operator is compact on (p, q 1 )-forms and on (p, q 2 )-forms (q 1 ≤ q 2 ), then it is compact on (p, q)-forms for q 1 ≤ q ≤ q 2 . 1 The simultaneous validity of compactness or subellipticity for the complex Green operator mentioned above has been shown with the help of a somewhat ad hoc local 'Hodge-⋆-like' operator that, unlike the actual Hodge- * , maps (p, q)-forms to (p, m − 1 − q)-forms and intertwines ∂ M and ∂ * M modulo terms of order zero ( [17,15,6]). These terms are of no consequence for compactness and subellipticity, they can simply be absorbed.…”