The raising steps method. Applications to the $\bar{\mathrm{\partial }}$ equation in Stein manifolds

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 $\bar{\mathrm{\partial }}u=\omega$ with a gain, $\displaystyle s>r,$instrictlypseudoconvexdomainsinSteinmanifolds\mathrm{.}\text{par}Wealsoget$\displaystyle L^{r}-L^{s}$resultsfordomainsin${$\mathbb{C}$}^{n}$ locally biholomorphic to convex domains of finite type.