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Weighted $L^2$ theory for the Euclidean Dirac operator in higher dimensions

Guangbin Ren, Yuchen Zhang

Abstract

We study weighted $L^{2}$ solvability for the Euclidean Dirac operator in dimensions $n\ge 3$. We prove that, on the exterior domain $\mathbb{R}^{n}\setminus\overline{B(0,1)}$ with logarithmic weight $\varphi=n\log|x|$, no higher-dimensional analogue of the two-dimensional Hörmander estimate can be controlled solely by $Δ\varphi$; we then establish weighted solvability for the weights $|x|^{m}$ with $m\neq 0$, for the quadratic weight $x_{1}^{2}$, and for sufficiently small anisotropic perturbations of the Gaussian weight, with sharp constant $1/4$ in the Gaussian case. The obstruction arises because, in dimensions $n\ge 3$, the classical weighted identity is coercive only under a structural relation between $Δ\varphi$ and $|\nabla\varphi|^{2}$, a condition that excludes the Gaussian weight and many polynomial weights. The method is based on a weighted identity for the conjugated unknown $U:=ue^{-\varphi/2}$, together with suitable scalar and Clifford-valued multipliers; this identity yields the required coercive estimates and also gives weighted $L^{2}$ solvability for the Poisson equation through the factorization $Δ=-D^{2}$.

Weighted $L^2$ theory for the Euclidean Dirac operator in higher dimensions

Abstract

We study weighted solvability for the Euclidean Dirac operator in dimensions . We prove that, on the exterior domain with logarithmic weight , no higher-dimensional analogue of the two-dimensional Hörmander estimate can be controlled solely by ; we then establish weighted solvability for the weights with , for the quadratic weight , and for sufficiently small anisotropic perturbations of the Gaussian weight, with sharp constant in the Gaussian case. The obstruction arises because, in dimensions , the classical weighted identity is coercive only under a structural relation between and , a condition that excludes the Gaussian weight and many polynomial weights. The method is based on a weighted identity for the conjugated unknown , together with suitable scalar and Clifford-valued multipliers; this identity yields the required coercive estimates and also gives weighted solvability for the Poisson equation through the factorization .

Paper Structure

This paper contains 16 sections, 23 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Clifford algebras and Dirac operators
  3. The Dirac operator and its $L^2$ theory
  4. Definitions and basic properties
  5. The maximal operator defined by $D$ and its adjoint
  6. Functional analysis preliminaries for the $L^2$ existence theorem
  7. Classical weighted $L^{2}$ theory for the Dirac operator
  8. The fundamental identity
  9. A priori estimate and the existence theorem
  10. An obstruction in higher dimensions
  11. The counterexample
  12. Proof of Theorem \ref{['thm:sharp-obstruction']}
  13. Interpretation and implications
  14. A weighted identity and coercive estimates
  15. Weighted $L^2$ existence theorems for the Dirac equation
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^{2}$ be a domain and let $\varphi \in C^{2}(\Omega,\mathbb{R})$ be subharmonic. Suppose that for some $f \in L^{2}_{\varphi}(\Omega,\mathbb{R}_{2})$. Then there exists $u \in L^{2}_{\varphi}(\Omega,\mathbb{R}_{2})$ such that and $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2: Obstruction in higher dimensions
  • Theorem 1.3: Weighted $L^2$ solvability
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Definition 3.1: Dirac operator
  • Definition 3.2: Weighted $L^2$ space
  • Lemma 3.3
  • proof
  • ...and 41 more