| Résume | Let $\Omega \subset \mathbb{C}^{n} $ be a smooth bounded pseudoconvex domain. The $\overline{\partial }$-Neumann problem is Hodge theory with boundary on Ω. Its solution operator $N_{q}$ is globally regular, i.e. preserves $C^{\infty}(\Omega)$ , on strongly pseudoconvex domains (Kohn), but not on the worm domains (Christ). Since the 1990s it has been conjectured that global regularity follows when the Diederich–Fornæss index of $\Omega$ equals 1. This index is the supremum of $η \in]0,1]$ such that $-(-ρ )^{η}$ is plurisubharmonic near $b\Omega$ for some defining function ρ. I will present joint work with Emil Straube. Suppose that, for some $1\leq q_{0}\leq n-1$, the $q_{0}$-sums of the Levi eigenvalues are comparable, and the index is 1. Then $N_{q}$ and the Bergman projections $P_{q-1}$, for $q_{0}\leq q\leq n,$, are continuous in every Sobolev space $W^s(Ω)$. In particular, the conjecture holds in $C^2$, and there having index one is preserved under biholomorphisms. The key point is that index one gives a self-bounded gradient condition, which is weak pointwise but suffices in an integrated $L^2$ form.
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