On Hölder and BMO estimates for \(\overline \partial\) on convex domains in \(\mathbb{C}^ 2\) (Q1203504)

Let us consider the equation \(\overline\partial u=f\), where \(f=\sum^ n_{j=1} f_ jd\bar z_ j\) is a \(\overline\partial\)-closed (0,1)-form on a bounded pseudoconvex domain \(D\subset\mathbb{C}^ n\). We denote by \(\Lambda_ \alpha(D)\) the classical Lipschitz space and \(\text{BMO}(D)\) the space of functions of bounded mean oscillation on \(D\). The author proves: Let \(D\subset\mathbb{C}^ 2\) be convex with \(C^ \infty\) boundary. Then there is an integral solution operator \(T: C_{0,1}(\overline D)\to C(D)\) for \(\overline\partial\), such that: (i) \(| Tf|_{\Lambda_ \alpha(D)}\leq C_ \alpha| f|_{\Lambda_ \alpha(D)}\) for all \(f\) with \(\overline\partial f=0\) and all \(\alpha>0\); (ii) \(\| Tf\|_{\text{BMO}(D)}\leq C\| f\|_{L^ \infty}\).