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

Andreas Debrouwere, Quinten Van Boxstael, Jasson Vindas

TL;DR

This work characterizes when the Cauchy-Riemann operator $\overline{\partial}$ and the Laplacian $\Delta$ are surjective on weighted spaces $\mathcal{K}_{W}(T^{F,G})$ of smooth functions on generalized strips. The authors introduce a weighted Runge approximation technique and leverage the abstract Mittag-Leffler lemma to reduce surjectivity to a growth condition $(\epsilon)_0$ on the weight system $W$ and the non-triviality of the holomorphic subspace $\mathcal{U}_{W}(T^{F,G})$, providing a constructive and elementary proof. The main result yields equivalence among CR surjectivity, Laplace surjectivity, the weight-growth condition, and holomorphic non-triviality, extending previous Kruse results and covering polynomial-type weights as concrete examples. The framework enables explicit analysis for various weight systems and demonstrates how weighted Runge theory controls global solvability of elliptic operators on unbounded domains.

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.

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

TL;DR

This work characterizes when the Cauchy-Riemann operator and the Laplacian are surjective on weighted spaces of smooth functions on generalized strips. The authors introduce a weighted Runge approximation technique and leverage the abstract Mittag-Leffler lemma to reduce surjectivity to a growth condition on the weight system and the non-triviality of the holomorphic subspace , providing a constructive and elementary proof. The main result yields equivalence among CR surjectivity, Laplace surjectivity, the weight-growth condition, and holomorphic non-triviality, extending previous Kruse results and covering polynomial-type weights as concrete examples. The framework enables explicit analysis for various weight systems and demonstrates how weighted Runge theory controls global solvability of elliptic operators on unbounded domains.

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.
Paper Structure (9 sections, 16 theorems, 56 equations)

This paper contains 9 sections, 16 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized strips
  4. Weight function systems
  5. Weighted spaces of smooth and holomorphic functions
  6. The abstract Mittag-Leffler lemma
  7. Surjectivity of the Cauchy-Riemann and Laplace operators on $\mathcal{K}_{W}(T^{F,G})$
  8. A weighted version of the Runge approximation result
  9. Proof of the main result

Key Result

Theorem 1.1

Let $w \colon [0,\infty) \rightarrow [0,\infty)$ be a non-decreasing unbounded function. Suppose that there is $C>1$ such that Let $h\in (0,\infty].$ Define $\mathcal{K}_{(w)}(T_h)$ as the space consisting of all $f \in C^{\infty}(T_h)$ such that Then, the following statements are equivalent:

Theorems & Definitions (36)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 26 more