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In order to get estimates on the solutions of the equation $\bar \partial u=\omega $ on 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 form open sets in ${\mathbb{C}}^{n}$ to open sets in a Stein manifold.\ \par Using it we get $\displaystyle L^{r}-L^{s}$ results for solutions of equation $\bar \partial u=\omega $ with a gain, $\displaystyle s>r,$ in strictly pseudo convex domains in Stein manifolds.\ \par We also get $\displaystyle L^{r}-L^{s}$ results for domains in ${\mathbb{C}}^{n}$ locally biholomorphic to convex domains of finite type.

The Raising Steps Method: Applications to the $$\bar{\partial }$$ ∂ ¯ Equation in Stein Manifolds