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

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 $\ddot{I}†=n\log |x|$, no higher-dimensional analogue of the two-dimensional Hörmander estimate can be controlled solely by $\hat{I}”\ddot{I}†$; we then establish weighted solvability for the weights $|x{|}^m$ with $m\ne 0$, 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 $\hat{I}”\ddot{I}†$ and $|\nabla \ddot{I}†{|}^2$, a condition that excludes the Gaussian weight and many polynomial weights. The method is based on a weighted identity for the conjugated unknown $U:=u{e}^{-\ddot{I}†/2}$, 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 $\hat{I}”=-D^2$.