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

TL;DR

This paper investigates weighted $L^2$ solvability for the Euclidean Dirac operator in higher dimensions, revealing limitations of classical estimates and establishing new solvability results for specific weights.

Contribution

It identifies obstructions to weighted $L^2$ estimates in higher dimensions and proves solvability for certain polynomial and Gaussian weights with sharp constants.

Findings

No higher-dimensional analogue of Hörmander estimate controlled solely by $ riangle$.

Weighted solvability established for weights $|x|^{m}$, $x_1^2$, and small Gaussian perturbations.

Classical weighted identities are coercive only under specific structural conditions, excluding Gaussian and polynomial weights.

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 H\"ormander estimate can be controlled solely by $\Delta\varphi$; 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 $\Delta\varphi$ 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 $\Delta=-D^{2}$.