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

TL;DR

This paper extends H"ormander's method for solving the ar equation, providing solutions with compact support under certain conditions involving plurisubharmonic functions and $L^2$ spaces.

Contribution

It offers a new theorem that guarantees solutions with compact support for the ar equation in weighted $L^2$ spaces, complementing prior work by Hedenmalm.

Findings

Existence of solutions with compact support for ar equations.

Solutions lie in weighted $L^2$ spaces with eigenvalue-dependent weights.

The method applies to strictly plurisubharmonic functions with positive eigenvalues.

Abstract

This work is a complement of the study on H\"ormander's solution of the $\bar\partial$ equation initialised by H. Hedenmalm. Let $\varphi$ be a strictly plurisubharmonic function of class C 2 in C n, let $c_\varphi(z)$ be the smallest eigenvalue of $i\partial\bar\partial\varphi$ then $\forall z\in\mathbb{C}^n$, $c_\varphi (z)>0$. We denote by $L^2_{p,q}(\mathbb{C}^n, e^\varphi)$ the $(p, q)$ currents with coefficients in $L^2_{p,q}(\mathbb{C}^n, e^\varphi)$. We prove that if $\omega\in L^2_{p,q}(\mathbb{C}^n,e^\varphi)$, $\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.