An andreotti-grauert theorem with $l^r$ estimates

TL;DR

This paper extends the Andreotti-Grauert theorem by establishing $L^r$ estimates for solutions to the $ar{ ext{d}}$-equation with compact support on Stein manifolds, using weighted $L^r$ space techniques.

Contribution

It introduces $L^r$ estimates with weights for solutions to the $ar{ ext{d}}$-problem, generalizing classical results to $L^r$ spaces with explicit integrability conditions.

Findings

Solutions exist in $L^s$ with explicit relation to $L^r$ data.

Weighted $L^r$ estimates are established for $ar{ ext{d}}$ solutions.

The method applies to currents with compact support in Stein manifolds.

Abstract

By a theorem of Andreotti and Grauert if $\omega $ is a $(p,q)$ current, $q < n,$ in a Stein manifold $\displaystyle \Omega ,\ \bar \partial $ closed and with compact support, then there is a solution $u$ to $\bar \partial u=\omega $ still with compact support in $\displaystyle \Omega .$ The main result of this work is to show that if moreover $\displaystyle \omega \in L^{r}(m),$ where $m$ is a suitable Lebesgue measure on the Stein manifold, then we have a solution $u$ with compact support {\sl and} in $L^{s}(m),\ \frac{1}{s}=\frac{1}{r}-\frac{1}{2(n+1)}.$ We prove it by estimates in $L^{r}$ spaces with weights.