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

Laurent-Thiébaut, Christine; Leiterer, Jurgen

doi:10.5802/aif.1338

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.

Résumé
Abstract

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 ${ρ_{i} \leq 0}$ , où les hypersurfaces réelles ${ρ_{i} \leq 0}$ sont transverses et les combinaisons linéaires non nulles à coefficients positifs des formes de Levi des $ρ_{i}$ ont toutes au moins $(q + 1)$ valeurs propres strictement positives, on résoud, en utilisant des formules intégrales, l’équation $\bar{\partial} f = g$ , où $g$ est une $(n, r)$ -forme différentielle continue, $\bar{\partial}$ -fermée dans $D, n - q \leq r \leq n$ , avec les estimées suivantes : si $d$ désigne la distance au bord de $D$ , et si $d^{β} g$ est bornée alors pour tout $ε > 0$ , $f$ est höldérienne d’ordre $1 / 2 - β - ε$ si $0 \leq β < 1 / 2$ et $d^{β + ε - 1 / 2} f$ est bornée si $1 / 2 \leq β < 1$ .

We study the $\bar{\partial}$ -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 ${ρ_{i} \leq 0}$ , where the real hypersurfaces ${ρ_{i} = 0}$ are transversal and any nonzero linear combination with nonnegative coefficients of the Levi form of the $ρ_{i}$ ’s have at least $(q + 1)$ positive eigenvalues, we solve the equation $\bar{\partial} f = g$ for each continuous $(n, r)$ -closed form $g$ in $D$ , $n - q \leq r \leq n$ , with the following estimates: if $d$ denotes the distance to the boundary of $D$ and if $d^{β} g$ is bounded, then for all $ε > 0$ , $f$ is Hölder continuous with exponent $1 / 2 - β - ε$ if $0 \leq β < 1 / 2$ and $d^{β + ε - 1 / 2} f$ is bounded if $1 / 2 \leq β < 1$ .

MR Zbl

DOI : 10.5802/aif.1338

@article{AIF_1993__43_2_383_0,
     author = {Laurent-Thi\'ebaut, Christine and Leiterer, Jurgen},
     title = {Uniform estimates for the {Cauchy-Riemann} equation on $q$-convex wedges},
     journal = {Annales de l'Institut Fourier},
     pages = {383--436},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {43},
     number = {2},
     year = {1993},
     doi = {10.5802/aif.1338},
     mrnumber = {95a:32025},
     zbl = {0782.32014},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1338/}
}

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Laurent-Thiébaut, Christine; Leiterer, Jurgen. Uniform estimates for the Cauchy-Riemann equation on $q$-convex wedges. Annales de l'Institut Fourier, Tome 43 (1993) no. 2, pp. 383-436. doi : 10.5802/aif.1338. https://www.numdam.org/articles/10.5802/aif.1338/

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