# Solutions of the $\bar \partial $-equation on Stein and on K\"ahler manifold with compact support

## TL;DR

This paper investigates solutions to the $ar ext{d}$-equation on Stein and K"ahler manifolds, providing $L^{r}$ and Sobolev estimates for solutions with compact support, and linking these to Poisson and Hodge Laplacian estimates.

## Contribution

It establishes new $L^{r}$ and Sobolev estimates for $ar ext{d}$-equation solutions with compact support on Stein and K"ahler manifolds, extending previous results and improving Andreotti-Grauert type theorems.

## Key findings

- Existence of solutions with compact support in Stein manifolds for $L^{r}$ $ar ext{d}$-closed forms.
- Estimates on solutions to the Poisson equation with compact support in K"ahler manifolds.
- Connection between $ar ext{d}$-equation solutions and Hodge Laplacian in K"ahler geometry.

## Abstract

We study the $\bar \partial $-equation first in Stein manifold then in complete K\"ahler manifolds. The aim is to get $L^{r}$ and Sobolev estimates on solutions with compact support. In the Stein case we get that for any $(p,q)$-form $\omega $ in $L^{r}$ with compact support and $\bar \partial $-closed there is a $(p,q-1)$-form $u$ in $W^{1,r}$ with compact support and such that $\bar \partial u=\omega .$ In the case of K\"ahler manifold, we prove and use estimates on solutions on Poisson equation with compact support and the link with $\bar \partial $ equation is done by a classical theorem stating that the Hodge laplacian is twice the $\bar \partial $ (or Kohn) Laplacian in a K\"ahler manifold. This uses and improves, in special cases, our result on Andreotti-Grauert type theorem.