Local solvability of the ${\bar{{\it \partial}}}$-equation on ${\mathcal{L}}_1$-spaces

Abstract.

We relate solvability of the ${\bar{{\it \partial}}}$-equation on a Banach space to solvability on its finite dimensional subspaces. From this we obtain both some positive and negative results regarding solvability of the ${\bar{{\it \partial}}}$-equation on various Banach spaces, extending earlier results of Lempert. Moreover, we show how results of Raboin and Mazet on the local solvabilty of ${\bar{{\it \partial}}}$ on nuclear spaces can be seen as consequences of Lempert’s result on $\ell_1$.