Compactness of the weighted dbar-Neumann operator and commutators of the Bergman projection with continuous functions

Abstract

Assume that Ω=Ω1∖Ω¯2 is an annulus between two smooth bounded domains Ω1 and Ω2 in ℂn (or a Stein manifold) such that Ω¯2⋐Ω1, Ω1 is weakly q-convex, Ω2 is weakly (n−q−1)-convex and let 1⩽q⩽n−2 with n⩾3. This paper is devoted to studying existence, boundedness and compactness of the weighted ∂¯-Neumann operator Np,qt with weights e−tλ on the space Lp,q2(Ω) (or Wp,qs(Ω)) for sufficiently large t and for λ∈C∞(Ω¯). Moreover, the closedness of the ranges of the operators ∂¯ and ∂¯∗ are proved. Next as an application, we study the regularity of operators associated with Np,qt. Consequently, the global boundary regularity of the ∂¯-equation, ∂¯u=f, is studied on Ω. In the end of the paper, we study compactness of the commutator [Bp,qt,f] of the weighted Bergman projection Bp,qt and the multiplication operator by a function f∈C(Ω¯) on the space Lp,q2(Ω).