The Diederich-Fornaess index I: For domains of non-trivial index

We study bounded pseudoconvex domains in complex Euclidean space. We define an index associated to the boundary and show this new index is equivalent to the Diederich-Fornss index defined in 1977. This connects the Diederich-Fornss index to boundary conditions and refines the Levi pseudoconvexity. We also prove the beta-worm domain is of index pi/(2 beta). It is the first time that a precise non-trivial Diederich-Fornwss index in Euclidean spaces is obtained. This finding also indicates that the Diederich-Fornwss index is a continuum in (0, 1], not a discrete set. The ideas of proof involve a new complex geometric analytic technique on the boundary and detailed estimates on differential equations. (C) 2018 Elsevier Inc. All rights reserved.