Estimées Lipschitz dans les domaines convexes de type fini de \({\mathbb{C}}^ 2\). (On Lipschitz estimates in convex domains of finite type in \({\mathbb{C}}^ 2)\) (Q1263704)

Let \(\Omega\) be a convex domain of \({\mathbb{C}}^ 2\) with smooth boundary \(\partial \Omega\), and of type \(\leq \ell\). If f is a (0,1)-form in \(L^{\infty}(\Omega)\cap C^{\infty}(\Omega)\) and \({\bar \partial}\)- closed in \(\Omega\), then, from a result of H. Skoda, \({\bar \partial}u=f\) has a solution given on the boundary \(\partial \Omega\) by Skoda's kernels: \[ u(z)=\sum^{2}_{i=1}\int_{\Omega}K_ i(z,\zeta)\wedge f(\zeta) + \int_{\Omega}K_ 3(z,\zeta)\wedge \partial {\bar \rho}(\zeta)\wedge f(\zeta), \] \(\forall z\in \partial \Omega\). Using this expression, the author proves \(u\in L^{\infty}(\partial \Omega)\), \({\bar \partial}_ bu=f\) and the Hölder estimate \[ | u(z)- u(w)| \precsim | z-w|^{1/\ell}(\log | z-w|)^ 2. \] Since domains of finite type properly include that of the uniform total pseudo-convexity of finite order, this result contains a result of M. Range in 1978.