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spinorial Sobolev inequality on the unit sphere states \\begin{equation*}\n  \\Big(\\int| D\\psi|^{\\frac{2n}{n+1}}\\Big)^{\\frac{n+1}{n}}-\\frac{n}{2}\\omega_{n}^{1/n}\\int\\langle D\\psi,\\psi\\rangle\n  \\geq 0, \\end{equation*} with equality if and only if $\\psi \\in {\\mathcal M}$, the set of all $-\\frac 12$-Killing spinors and their conformal transformations.\n  Our main result in this paper is to refine this inequality by establishing a stability inequality\n  \\begin{equation*} \\Big(\\int| D\\psi|^{\\frac{2n}{n+1}}\\Big)^{\\frac{n+1}{n}}-\\frac{n}{2}\\omega_{n}^{1/n}\\int\\langle D\\psi,\\psi\\rangle\n  \\geq {\\bf c}_","authors_text":"Guofang Wang, Mingwei Zhang","cross_cats":["math-ph","math.AP","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-08-12T16:09:08Z","title":"Stability of spinorial Sobolev inequalities on 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