Abstract
In order to have estimates on the solutions of the equation \(\bar{\partial }u=\omega \) on a Stein manifold, we introduce a new method, the “raising steps method”, to get global results from local ones. In particular, it allows us to transfer results from open sets in \({\mathbb {C}}^{n}\) to open sets in a Stein manifold. Using it, we get \(\displaystyle L^{r}-L^{s}\) results for solutions of the equation \(\bar{\partial }u=\omega \) with a gain, \(\displaystyle s>r\), in strictly pseudo convex domains in Stein manifolds. We also get \(\displaystyle L^{r}-L^{s}\) results for domains in \({\mathbb {C}}^{n}\) locally biholomorphic to convex domains of finite type.
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Amar, E. The Raising Steps Method: Applications to the \(\bar{\partial }\) Equation in Stein Manifolds. J Geom Anal 26, 898–913 (2016). https://doi.org/10.1007/s12220-015-9576-8
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DOI: https://doi.org/10.1007/s12220-015-9576-8