Generalized study of the operator $α\partial^k \bar{\partial}^{k} + β\bar{\partial}^k +γ\partial^k + c$ in weighted Hilbert space $L^2(\mathbb{C}, \mathrm{e}^{-|z|^2})$

TL;DR

The paper analyzes the generalized complex differential operator $H = α∂^k \bar{∂}^k + β \bar{∂}^k + γ ∂^k + c$ on the Gaussian-weighted Hilbert space $L^2(\mathbb{C}, e^{-|z|^2})$. Using Hörmander-type $L^2$ estimates in weighted spaces, it establishes the existence of a bounded right inverse for $H$ and an a priori bound for weak solutions of $Hu=f$. It then specializes to two principal subcases, $α=γ=0$ and $β=γ=0$, obtaining explicit solvability results and quantitative norm bounds, plus several Gaussian-weighted and local-domain corollaries. Overall, the work provides a unified framework for right-inverse construction and weak solvability of this class of operators in Gaussian-weighted spaces, extending prior results on simpler one-derivative or Laplacian-type operators in the same setting.

Abstract

By Hörmander's $L^2$-method, we study the operator $α\partial^k \bar{\partial}^{k} + β\bar{\partial}^k +γ\partial^k + c$ for any order $k$ with $α, β, γ\in \mathbb{R}$ such that $(α, β, γ) \neq(0,0,0)$ in the weighted Hilbert space $L^2(\mathbb{C}, \mathrm{e}^{-|z|^2})$. We prove the existence of its right inverse which is also a bounded operator. Subsequently we will study two cases that arise from this operator, namely: (1) Case where $α= γ=0$: The operator $β\bar{\partial}^{k} + c$ with $\vert β\vert \geq 1$. (2) Case where $β= γ=0$: The operator $α\partial^{k} \bar{\partial}^{k} + c$ with $\vert α\vert \geq 1$.

Theorems & Definitions (17)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 3.1
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3