On H\"ormander's solution of the dbar-equation

TL;DR

This paper explores how Hörmander's classical method for solving the dbar-equation adapts from allowing growth at infinity to requiring decay, emphasizing the necessity of a natural condition on the data, especially in the Fock weight case.

Contribution

It extends Hörmander's solution framework of the dbar-equation to decay conditions at infinity, identifying essential data conditions in the Fock weight context.

Findings

Hörmander's solution adapts to decay conditions at infinity.

A natural and necessary data condition is identified for the Fock weight case.

The approach bridges the gap between growth and decay solutions of the dbar-equation.

Abstract

We explain how H\"ormander's classical solution of the dbar-equation in the plane with a weight which permits growth near infinity carries over to the rather opposite situation when we ask for decay near infinity. Here, however, a natural condition on the datum needs to be imposed. The condition is not only natural but also necessary to have the result at least in the Fock weight case.