Hörmander's solution of the $\bar{\partial }$ -equation with compact support

Abstract

This work is a complement of the study on Hörmander's solution of the $\bar{\partial }$ equation initialised by H. Hedenmalm. Let $\phi$ be a strictly plurisubharmonic function of class C 2 in C n, let $c_{\phi }(z)$ be the smallest eigenvalue of $i\partial \bar{\partial }\phi$ then $\forall z\in {\mathbb{C}}^n$, $c_{\phi }(z)>0$. We denote by $L_{p,q}^2({\mathbb{C}}^n,e^{\phi })$ the $(p,q)$ currents with coefficients in $L_{p,q}^2({\mathbb{C}}^n,e^{\phi })$. We prove that if $\omega \in L_{p,q}^2({\mathbb{C}}^n,e^{\phi })$, $\bar{\partial }$$\omega$ = 0 for q <n then there is a solution u $\in L_{p,q-1}^2({\mathbb{C}}^n,c_{\phi }{e}^{\phi })$ of $\bar{\partial }$u = $\omega$. This is done via a theorem giving a solution with compact support if the data has compact support.