On the surjectivity of the Cauchy-Riemann and Laplace operators on weighted spaces of smooth functions

TL;DR

This paper characterizes when the Cauchy-Riemann and Laplace operators are surjective on weighted spaces of smooth, rapidly decaying functions in complex plane domains, based on growth conditions of weight functions.

Contribution

It provides a complete characterization of surjectivity for these operators on weighted function spaces using growth conditions of the weights.

Findings

Surjectivity depends on specific growth conditions of weight functions.

Complete characterization of operator surjectivity on weighted spaces.

Applicable to domains with strip-like geometries in the complex plane.

Abstract

We study the surjectivity of the Cauchy-Riemann and Laplace operators on certain weighted spaces of smooth functions of rapid decay on strip-like domains in the complex plane that are defined via weight function systems. We fully characterize when these operators are surjective on such function spaces in terms of a growth condition on the defining weight function systems.