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Uniform estimates for the Cauchy-Riemann equation on \(q\)-convex wedges. (English) Zbl 0782.32014

We study the \(\overline\partial\)-equation with Hölder estimates in \(q\)- convex wedges of \(\mathbb{C}^ n\) by means of integral formulas. If \(D \subset \mathbb{C}^ n\) is defined by some inequalities \(\{\rho_ i \leq 0\}\), 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 \(D\), \(n-q \leq r \leq n\), with the following estimates: if \(d\) denotes the distance to the boundary of \(D\) and if \(d^ \beta g\) is bounded, then for all \(\varepsilon>0\), \(f\) is Hölder continuous with exponent \(1/2-\beta-\varepsilon\) if \(0 \leq \beta<1/2\) and \(d^{\beta+\varepsilon-1/2}f\) is bounded if \(1/2 \leq \beta<1\).

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32F10 \(q\)-convexity, \(q\)-concavity
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
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