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

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.

💡 Research Summary

The paper investigates the surjectivity of two fundamental constant‑coefficient partial differential operators—the Cauchy‑Riemann operator ∂ = ½(∂/∂x + i∂/∂y) and the Laplace operator Δ = ∂²/∂x² + ∂²/∂y²—on a class of weighted spaces of smooth functions that exhibit rapid decay at infinity. The underlying domains are “generalized strips” in the complex plane, defined by two uniformly continuous bounding functions F and G:
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