Hölder and \(L^ p\)-estimates for the \({\bar\partial}\)-equation in some convex domains with real-analytic boundary
It is well known that for convex domains \(D=\{z:r(z)<0\}\) in \({\mathbb{C}}^ n\) and using the ''canonical'' support function \(F(\zeta,z)=\sum^{n}_{i=1}\partial r/\partial \zeta_ i(\zeta)(\zeta_ i-z_ i)\) one can define a reproducing kernel \(H(\zeta\),z), \(\zeta\in bD\), \(z\in D\), for holomorphic functions in \(D\) and also an explicit integral operator \(T\) which satisfies \({\bar \partial}Tf = f\) for \((0,1)\)-forms, \({\bar \partial}f=0.\) In this paper it is shown that: if \(r\) is a convex, real-analytic function of the type \(r(z)=\sum^{n}_{i=1}s_ i(| z_ i|^ 2)-1\), then the kernel \(H(\zeta,z)\) and the operator \(T\) have analogous properties as those in the strictly pseudoconvex case. Namely, \(\int_{bD}|\zeta-z|\;| H(\zeta,z)| d\sigma(\zeta)\leq c\), \(\zeta\in bD\), \(z\in D\), and for a regular, \({\bar\partial}\)-closed (0,1)-form \(\| Tf\|_{L^ p(D)}\leq c\| f\|_{L^ p(D)},\) 1\(\leq p\leq \infty\) and \(\| Tf\|_{Lip(D,1/m)}\leq c\| f\|_{L^{\infty}(D)}.\) The latter property generalizes previous work of \textit{R. M. Range} [cf. Several complex variables, Proc. Int. Conf., Cortona/Italy 1976/77, 247-267 (1978; Zbl 0421.32021)]. The proofs depend on the estimate, for \(\zeta\in bD\), \(z\in D\) and \(| \zeta-z|\) small \(2Re\;F(\zeta,z)\geq -r(z)+c L^r(\zeta)(\zeta -z)^ 2+c| \zeta -z|^ m\) where \(L^r(\zeta)\) is the Levi form of \(r\) at \(\zeta\), saying that the order of contact between \(bD\) and the tangent space at \(\zeta\) is controlled even at points of degeneracy by \(L^r(\zeta)\).
scientific article; zbMATH DE number 3845955