Optimal Lipschitz estimates for the \(\bar\partial\) equation on a class of convex domains
For mult-index \(N:=(n_1,\dots,n_m)\) let \(| N| =n_1 + \dots +n_m\) and let \(\Omega \subset \mathbb C^{| N| }\) be defined as \[ \Omega_N = \biggl\{(Z_{(1)}, \dots ,Z_{(m)}) \in \mathbb C^{n_1}\times \dots \times \mathbb C^{n_m} : \sum_{j=1}^m (| Z_{(j)}| ^2+| Z_{(j)}*Z_{(j)}| ) <1\biggr\}, \] where on \(\mathbb C^n\), \(Z*W=\sum_1^nz_kw_k\) and \(| Z| ^2=\sum_1^n| z_k| ^2\). So \(\Omega_N\) is convex circular, non strictly pseudoconvex, and a not even piecewise smooth boundary bounded domain in \(\mathbb C^{| N| }\). In this paper the authors construct an explicit linear integral solution operator for \(\overline \partial\) to obtain the following estimates for the \(\overline \partial\) equation on \(\Omega_N\). Now let \(\| u\| _{\Lambda_\alpha (\Omega_N)}\) be the classical Lipschitz norm and let \(\| u\| _{\Lambda_{\overline \alpha } (\Omega_N)} = \| u\| _{\infty} +\sup_{| h| <\frac 12} \frac {| u(x+h)-u(x)| }{| h| ^\alpha| \ln| h| | }.\) The main results are : For any \(\overline \partial\)-closed \((0,1)\) form \(f\) with coefficients in \(L^p(\Omega_N)\), there exists a function \(u\) defined on \(\Omega_N\), that satisfies \(\overline \partial u=f\) (in distribution sense) and \[ \begin{aligned} \| u\| _{\Lambda_\alpha (\Omega_N)} &\leq C_p\| f\| _{L^p(\Omega_N)}, \quad\text{if } p<\infty.\\ \| u\| _{\Lambda_{\overline {1/2}} (\Omega_N)} &\leq C_\infty \| f\| _{L^\infty (\Omega_N)}, \end{aligned} \] where \(p\) and \(\alpha\) are related in the following way: \[ \begin{aligned} \alpha =\frac 12 - \frac {| N| +m-l+1} p, &\quad\text{ if } N\neq(2,\dots ,2) \text{ and } p>2(| N| +m-l+1),\\ \alpha =\frac 12 - \frac {3m} p, &\quad\text{ if } N=(2,\dots ,2) \text{ and } p>6m. \end{aligned} \] The authors also give concrete examples to show that the estimates above are optimal.
scientific article; zbMATH DE number 2074290