Sobolev Estimates for the <inline-formula><math display="inline"><semantics><mover accent="true"><mo>∂</mo><mo>¯</mo></mover></semantics></math></inline-formula> and the <inline-formula><math display="inline"><semantics><mover accent="true"><mo>∂</mo><mo>¯</mo></mover></semantics></math></inline-formula>-Neumann Operator on Pseudoconvex Manifolds

Abstract

Let D be a relatively compact domain in an n-dimensional Kähler manifold with a C2 smooth boundary that satisfies some “Hartogs-pseudoconvexity” condition. Assume that Ξ is a positive holomorphic line bundle over X whose curvature form Θ satisfies Θ≥Cω, where C>0. Then, the ∂¯-Neumann operator N and the Bergman projection P are exactly regular in the Sobolev space Wm(D,Ξ) for some m, as well as the operators ∂¯N, ∂¯⋇N.

Keywords

• ∂ ¯ %22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="equation"> <mml:math id="mm3000"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mo>∂</mml:mo> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:semantics> </mml:math> </named-content>
• ∂ ¯ -Neumann operator%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="equation"> <mml:math id="mm3001"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mo>∂</mml:mo> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:semantics> </mml:math> </named-content>-Neumann operator
• Kähler manifold
• q-convex domain