Amar, Eric Estimées Lipschitz dans les domaines convexes de type fini de \({\mathbb{C}}^ 2\). (On Lipschitz estimates in convex domains of finite type in \({\mathbb{C}}^ 2)\). (French) Zbl 0688.32015 Publ. Mat., Barc. 33, No. 1, 69-83 (1989). 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. Reviewer: Na Jisheng MSC: 32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) Keywords:convex domain of finite type; Skoda kernel; (partial d)-bar equation; Hölder estimate PDFBibTeX XMLCite \textit{E. Amar}, Publ. Mat., Barc. 33, No. 1, 69--83 (1989; Zbl 0688.32015) Full Text: DOI EuDML