A pointwise a-priori estimate for the \(\bar\partial\)-Neumann problem on weakly pseudoconvex domains (Q2346533)

Let \(D\) be a smoothly bounded pseudoconvex domain. Let \(P\in bD\) be a fixed point and \(\omega_1, \omega_2, \dots , \omega_n\) a smooth orthonormal frame for \((1,0)\)-forms on a small neighborhood \(U\) of \(P\) with \(\omega_n =\gamma (\zeta) \partial r\), where \(r\) is a defining function for \(D\). Let \(L_1, \dots , L_n\) be the corresponding dual frame for \((1,0)\)-vector fields. Define \(\mathcal{D}_q^k(D)= C_{(0,q)}^k (\overline D) \cap {\text{dom\,}} \overline \partial^*\) for \(k=1,2,\dots \), and denote by \(\mathcal{D}_{qU}^k\) those forms in \(\mathcal{D}_q^k(D)\) that have compact support in \(\overline D \cap U.\) For a \(C^1\)-form \(f\) of type \((0,q)\) on \(\overline D, \) let \(Q_0(f)= |f|_0 + |\overline \partial f |_0 + | \vartheta f|_0, \) where \(\vartheta\) is the formal adjoint of \(\overline \partial\) and \(|\varphi |_0\) denotes the sum of the supremum norms over \(D\) of the coefficients of \(\varphi.\) For \(0< \delta \leq 1, \;|\varphi|_\delta\) denotes the corresponding Hölder norm of order \(\delta.\) The main result of this paper is a pointwise a-priori estimate for the \(\overline \partial\)-Neumann problem on \(D,\) which reads as follows: there exists an integral operator \(S^{bD}: C_{(0,q)} (bD) \longrightarrow C^\infty _{(0,q)}(D)\) which has the following properties. If \(bD\) is (Levi)pseudoconvex in a neighborhood \(U\) of the point \(P\in bD\) and if \(U\) is sufficiently small, there exist constants \(C_\delta\) depending on \(\delta >0,\) so that one has the following uniform estimates for all \(f\in \mathcal{D}_{qU}^1, \;1\leq q \leq n,\) and \(z\in D\cap U\):{\parindent=6mm \begin{itemize} \item[(i)] \(|f-S^{bD}(f)|_\delta \leq C_\delta Q_0(f)\) for any \(\delta <1\).\item [(ii)] \(|\overline L_j S^{bD}(f)(z)| \leq C_\delta {\text{dist}}(z, bD)^{\delta -1} Q_0(f)\) for \(j=1, \dots ,n\) and any \(\delta <1/2\).\item [(iii)] \(|L_j S^{bD}(f)(z)| \leq C_\delta {\text{dist}}(z, bD)^{\delta -1} Q_0(f)\) for \(j=1, \dots ,n-1\) and any \(\delta <1/3\). \end{itemize}} Furthermore, if \(f_J \overline \omega^J\) is a normal component of \(f\) with respect to the frame \(\overline \omega_1, \dots ,\overline \omega_n,\) one has \(|f_J|_\delta \leq C_\delta Q_0(f)\) for any \(\delta < 1/2\) if \(n\in J\).