Uniform estimates for the Cauchy-Riemann equation on $q$-convex wedges

Laurent-Thiébaut, Christine; Leiterer, Jurgen

Annales de l'Institut Fourier, Tome 43 (1993) no. 2, pp. 383-436.

Dans ce travail on résoud l’équation de Cauchy-Riemann avec des estimées höldériennes dans une intersection de domaines $q$-convexes. Plus précisément si $D\subset {ℂ}^n$ est défini par des inégalités $\{{\rho }_i\le 0\}$, où les hypersurfaces réelles $\{{\rho }_i\le 0\}$ sont transverses et les combinaisons linéaires non nulles à coefficients positifs des formes de Levi des ${\rho }_i$ ont toutes au moins $(q+1)$ valeurs propres strictement positives, on résoud, en utilisant des formules intégrales, l’équation $\overline{\partial }f=g$, où $g$ est une $(n,r)$-forme différentielle continue, $\overline{\partial }$-fermée dans $D,n-q\le r\le n$, avec les estimées suivantes : si $d$ désigne la distance au bord de $D$, et si $d^{\beta }g$ est bornée alors pour tout $\epsilon >0$, $f$ est höldérienne d’ordre $1/2-\beta -\epsilon$ si $0\le \beta <1/2$ et $d^{\beta +\epsilon -1/2}f$ est bornée si $1/2\le \beta <1$.

We study the $\overline{\partial }$-equation with Hölder estimates in $q$-convex wedges of ${ℂ}^n$ by means of integral formulas. If $D\subset {ℂ}^n$ is defined by some inequalities $\{{\rho }_i\le 0\}$, where the real hypersurfaces $\{{\rho }_i=0\}$ are transversal and any nonzero linear combination with nonnegative coefficients of the Levi form of the ${\rho }_i$’s have at least $(q+1)$ positive eigenvalues, we solve the equation $\overline{\partial }f=g$ for each continuous $(n,r)$-closed form $g$ in $D$, $n-q\le r\le n$, with the following estimates: if $d$ denotes the distance to the boundary of $D$ and if $d^{\beta }g$ is bounded, then for all $\epsilon >0$, $f$ is Hölder continuous with exponent $1/2-\beta -\epsilon$ if $0\le \beta <1/2$ and $d^{\beta +\epsilon -1/2}f$ is bounded if $1/2\le \beta <1$.

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