On estimates for the $\displaystyle \bar \partial $ equation in Stein manifolds

TL;DR

This paper extends $\bar\partial$ estimates to intersections of convex domains in Stein manifolds, using holomorphic retractions and raising steps, enabling solutions in domains with low boundary regularity.

Contribution

It generalizes previous $\bar\partial$ estimates from $\mathbb{C}^n$ to Stein manifolds with less regular boundaries, introducing new techniques like holomorphic retraction and raising steps.

Findings

$L^{r}-L^{s}$ estimates established for convex intersections in Stein manifolds.

Results apply to domains with $\mathcal{C}^3$ boundary regularity.

Method extends previous work to more general geometric settings.

Abstract

We generalize to intersection of strictly $c$ -convex domains in Stein manifold, $ L^{r}-L^{s}$ and Lipschitz estimates for the solutions of the $ \bar \partial $ equation done by Ma and Vassiliadou for domains in $ {\mathbb{C}}^{n}.$ For this we use a Docquier-Grauert holomorphic retraction plus the raising steps method I introduce earlier. This gives results in the case of domains with low regularity, $ {\mathcal{C}}^{3},$ for their boundary.