Weighted $L^2$ theory for the Euclidean Dirac operator in higher dimensions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914462663966720 |
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| author | Ren, Guangbin Zhang, Yuchen |
| author_facet | Ren, Guangbin Zhang, Yuchen |
| contents | 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 $φ=n\log|x|$, no higher-dimensional analogue of the two-dimensional Hörmander estimate can be controlled solely by $Δφ$; 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 $Δφ$ and $|\nablaφ|^{2}$, a condition that excludes the Gaussian weight and many polynomial weights. The method is based on a weighted identity for the conjugated unknown $U:=ue^{-φ/2}$, 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 $Δ=-D^{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04504 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weighted $L^2$ theory for the Euclidean Dirac operator in higher dimensions Ren, Guangbin Zhang, Yuchen Complex Variables Analysis of PDEs 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 $φ=n\log|x|$, no higher-dimensional analogue of the two-dimensional Hörmander estimate can be controlled solely by $Δφ$; 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 $Δφ$ and $|\nablaφ|^{2}$, a condition that excludes the Gaussian weight and many polynomial weights. The method is based on a weighted identity for the conjugated unknown $U:=ue^{-φ/2}$, 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 $Δ=-D^{2}$. |
| title | Weighted $L^2$ theory for the Euclidean Dirac operator in higher dimensions |
| topic | Complex Variables Analysis of PDEs |
| url | https://arxiv.org/abs/2604.04504 |