H\"ormander's solution of the $\bar{\mathrm{\partial }}$ -equation with compact support

This work is a complement of the study on H"ormander’s solution of the $\bar{\mathrm{\partial }}$ 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 $i\mathrm{\partial }\bar{\mathrm{\partial }}\phi$ then $\forall z\in$\mathbb{C}$^n$,$c_\varphi (z)>0$\mathrm{.}Wedenoteby$L^2_{p,q}($\mathbb{C}$^n, e^\varphi)$the$(p, q)$currentswithcoefficientsin$L^2_{p,q}($\mathbb{C}$^n, e^\varphi)$\mathrm{.}Weprovethatif$\omega\in L^2_{p,q}($\mathbb{C}$^n,e^\varphi)$,$\bar\partial$$\omega$ = 0 for q <n then there is a solution u $\in L ^2_{p,q-1}($\mathbb{C}$^n,c_\varphi e^\varphi)$of$\bar\partial$u=$\omega$. This is done via a theorem giving a solution with compact support if the data has compact support.