The raising steps method . Applications to the ∂̄ equation in Stein manifolds .

In order to get estimates on the solutions of the equation ∂̄u = ω 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 C to open sets in a Stein manifold. Using it we get L − L results for solutions of equation ∂̄u = ω with a gain, s > r, in strictly pseudo convex domains in Stein manifolds. We also get L −L results for domains in C locally biholomorphic to convex domains of finite type.