On regularity of ∂-solutions on 𝑎𝑞 domains with 𝐶² boundary in complex manifolds

This is the authors' abstract. We don't add key points for this paper.

We study regularity of solutions u u to ∂ ¯ u = f ∂ u=f on a relatively compact C 2 C^2 domain D D in a complex manifold of dimension n n , where f f is a ( 0 , q ) (0,q) form. Assume that there are either ( q + 1 ) (q+1) negative or ( n − q ) (n-q) positive Levi eigenvalues at each point of boundary ∂ D ∂ D . Under the necessary condition that a locally L 2 L^2 solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain 1 / 2 1/2 derivative when q = 1 q=1 and f f is in the Hölder–Zygmund space Λ r ( D ) Λ ^r( D) with r > 1 r>1 . For q > 1 q>1 , the same regularity for the solutions is achieved when ∂ D ∂ D is either sufficiently smooth or of ( n − q ) (n-q) positive Levi eigenvalues everywhere on ∂ D ∂ D .