Uniform estimates for the Cauchy-Riemann equation on q-convex wedges
Laurent-Thiébaut, Christine ; Leiterer, Jurgen
Annales de l'Institut Fourier, Tome 43 (1993), p. 383-436 / Harvested from Numdam
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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⊂ℂ n est défini par des inégalités {ρ i ≤0}, où les hypersurfaces réelles {ρ i ≤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 ∂ ¯f=g, où g est une (n,r)-forme différentielle continue, ∂ ¯-fermée dans D,n-q≤r≤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≤β<1/2 et d β+ε-1/2 f est bornée si 1/2≤β<1.

We study the ∂ ¯-equation with Hölder estimates in q-convex wedges of ℂ n by means of integral formulas. If D⊂ℂ n is defined by some inequalities {ρ i ≤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 ∂ ¯f=g for each continuous (n,r)-closed form g in D, n-q≤r≤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≤β<1/2 and d β+ε-1/2 f is bounded if 1/2≤β<1.

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     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},
     volume = {43},
     year = {1993},
     pages = {383-436},
     doi = {10.5802/aif.1338},
     mrnumber = {95a:32025},
     zbl = {0782.32014},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1993__43_2_383_0}
}

Laurent-Thiébaut, Christine; Leiterer, Jurgen. Uniform estimates for the Cauchy-Riemann equation on $q$-convex wedges. Annales de l'Institut Fourier, Tome 43 (1993) pp. 383-436. doi : 10.5802/aif.1338. http://gdmltest.u-ga.fr/item/AIF_1993__43_2_383_0/

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