Compact and subelliptic estimates for the \(\overline\partial\)-Neumann operator on \(C^2\) pseudoconvex domains
The following theorems are shown. Theorem 1.1 For \(k\geq 2\), let \(\Omega\) be a bounded \({\mathcal C}^k\) pseudoconvex domain in \(\mathbb{C}^n\) satisfying Catlin's property (P). Then the \(\overline\partial\)-Neumann operator is regular on \(W^{k-1}(\Omega)\). Theorem 1.2 For \(k\geq 2\), let \(\Omega\) be a bounded \({\mathcal C}^k\) pseudoconvex domain in \(\mathbb{C}^n\). Suppose that for some constants \(0< \varepsilon< {1\over 2}\) and \(c> 0\) we have a plurisubharmonic function \(\lambda\in {\mathcal C}^\infty(\overline\Omega)\) for every \(\widetilde\delta> 0\) such that \(0\leq\lambda\leq 1\) on \(\Omega\) and \(i\partial\overline\partial\lambda\geq ic\), \(\widetilde\delta^{-2\varepsilon}\partial\overline\partial|z|^2\) whenever \(\text{dist}(z, b\Omega)\leq\widetilde\delta\). Then we have the subelliptic estimates: \[ \| f\|^2_{k-1+\varepsilon}\precsim \|\overline\partial f\|^2_{k-1}+ \|\overline\partial^* f\|^2_{k-1} \] for all \(f\in\text{Dom}(\overline\partial^*)\cap \text{Dom}(\overline\partial)\) with \(\|\overline\partial f\|^2_{k-1} \|\overline\partial^* f\|^2_{k-1}< \infty\) and: \[ \| f\|_{k-1+\varepsilon}\precsim \|\overline\partial f\|_{k-1} \] for all \(f\in\ker(\overline\partial)^\perp\) with \(\|\overline\partial f\|_{k-1}< \infty\). In particular, the \(\overline\partial\)-Neumann operator is bounded from \(W^{k-1-\varepsilon}(\Omega)\) to \(W^{k-1+\varepsilon}(\Omega)\). The hypotheses of the theorem are redundant in the smooth case. The proof relies on the construction of 2 tangential differential operators \(T_1\) and \(T_2\) with the same principal part. Then one uses the Bergman projection in order to define a new operator \(T= (I- P)T_1+ PT_2\). The advantage is that \(T\) allows the computation of necessary estimates with only second derivatives of the defining function of the domain. This gives estimates that are one derivative better than previously obtained results.
scientific article; zbMATH DE number 5118865
scientific article