The Diederich–Fornaess index and the global regularity of the $\bar{\partial }$-Neumann problem

Abstract

We describe along the guidelines of Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999), the constant ${\mathcal {E}}_s$ which is needed to control the commutator of a totally real vector field $T_{{\mathcal {E}}}$ with $\bar{\partial }^*$ in order to have $H^s$ a-priori estimates for the Bergman projection $B_k, k\ge q-1$, on a smooth $q$-convex domain $D\subset \subset {\mathbb {C}}^{n}$. This statement, not explicit in Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999), yields regularity of $B_k$ in specific Sobolev degree $s$. Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function $r$ of $D$, the operators $(T^+)^{-\frac{\delta }{2}}$ and $(-r)^{\frac{\delta }{2}}$. We are thus able to extend to general degree $k\ge 0$ of $B_k$, the conclusion of (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999) which only holds for $q=1$ and $k=0$: if for the Diederich–Fornaess index $\delta$ of $D$, we have $(1-\delta )^{\frac{1}{2}}\le {\mathcal {E}}_s$, then $B_k$ is $H^s$-regular.