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

Eric Amar (1)

1 IMB - Institut de Mathématiques de Bordeaux

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.

Mots clés: d_bar; Stein manifold

Domaines: Variables complexes [math.CV]

Eric Amar. The raising steps method. Applications to the $\bar \partial $ equation in Stein manifolds.. The Journal of Geometric Analysis, 2016, 26 (2), pp.898-913. ⟨hal-00922689v2⟩