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Let $\Omega$ be a smooth bounded pseudoconvex domain in $\mathbb{C}^{n}$. Let$1\leq q_{0}\leq (n-1)$. We show that if $q_{0}$--sums of eigenvalues of theLevi form are comparable, then if the Diederich--Forn\ae ss index of $\Omega$is $1$, the $\overline{\partial}$--Neumann operators $N_{q}$ and the Bergmanprojections $P_{q-1}$ are regular in Sobolev norms for $q_{0}\leq q\leq n$. Inparticular, for domains in $\mathbb{C}^{2}$, Diederich--Forn\ae ss index $1$implies global regularity in the $\overline{\partial}$--Neumann problem.

Diederich--Forn\ae ss index and global regularity in the $\overline{\partial}$--Neumann problem: domains with comparable Levi eigenvalues