Sharp gradient stability for a class of Hardy-Sobolev-Maz'ya inequalities

Abstract

In this paper, we proved the sharp gradient stability for a class of Hardy-Sobolev-Maz'ya inequalities with partial (stronger) singular weight and non-radial extremal functions. Our result seems to be the first stability result for non-radial extremal functions. The presence of partial (stronger) singular weight brings substantial new challenges, requiring us to significantly refine the techniques from Deng-Tian 2025, Figalli-Neumayer 2019 and Figalli-Zhang 2022, and introduce some new ideas to handle both the cylindrical symmetry of non-radial extremal functions and the partial (stronger) singular weight structure. Key technical innovations include new compact embedding with strong singularity, non-degeneracy and spectral property of the linearized operator $\mathcal{L}_{v}$ generated by non-radial extremal function $v$ and new refined spectral inequalities, which are crucial for our analysis. Since the extremal function $v$ is non-radial, ODE approach fails, we use binary PDE to prove the spectral property of $\mathcal{L}_{v}$. Surprisingly, the sharp exponent $γ=\max\{2,p\}$ in our sharp gradient stability inequality (1.12) is independent of the partial weight dimension $k$, while the extremal manifold depends on $k$.