\(C^{k}\)-estimates for the \(\bar\partial\)-equation on convex domains of finite type
Let \(\Omega\) be a bounded convex domain in \({\mathbb C}^n\) with smooth boundary of finite type \(m\). Let \(C^0_{0,q}(\bar\Omega)\) be the Banach space of \((0,q)\)-forms with continuous coefficients on \(\bar\Omega\). Here \(1\leq q\leq n-1\). The author constructs a bounded linear operator \[ T_q: C^0_{0,q}(\bar\Omega)\,\rightarrow\, C^{\frac{1}{m}}_{0,q-1}(\bar\Omega) \] such that \(\bar\partial \,T_q(f)=f\) for all \(f\in C^0_{0,q}(\bar\Omega)\) with \(\bar\partial f=0\). The author also obtains estimates in Sobolev spaces. More precisely, there exists a linear operator \[ T_q^\ast:\, C_{0,q}(\bar\Omega)\,\rightarrow\, C_{0,q-1}(\bar\Omega) \] such that for all \(k\in {\mathbb N}\) and all \(\bar\partial\)-closed \(f\in C^k_{0,q}(\bar\Omega)\), one has \(\bar\partial \,T_q^\ast(f)=f\) and \[ \| T_q^\ast(f)\| _{\bar\Omega, k+\frac{1}{m}}\leq c_k\| f\| _{\bar\Omega, k}. \]
scientific article; zbMATH DE number 5013682