Sharp fractional Hardy--Maz'ya remainders on half-spaces: angular reduction and energy stability
Year: 2026
Type: preprint
Abstract: For $d\geq2$, $01$, and $τ>0$, we study the sharp half-space remainder \[ [u]_{s,p,+}^p-\mathcal D_{d,s,p}\int_{\mathbb R^d_+}x_d^{-sp}|u|^p\,dx \geqΛ_{d,s,p,τ}\int_{\mathbb R^d_+} \frac{x_d^{τ-sp}|u|^p}{(x_{d-1}^2+x_d^2)^{τ/2}}\,dx. \] We first obtain exact dimensional and angular reductions by an $L^p$ contraction. For $p=2$, we identify the planar sharp coefficient with the simple first eigenvalue of a singular angular operator with compact resolvent. We prove a ground-state representation and a two-sided estimate for the sharp deficit. These give quantitative stability and a Fourier--Mellin energy criterion for extremizing sequences in all dimensions. For the distance with multiplier $h(ξ)=\min\{ξ^2,1\}$, the best planar stability coefficient is not attained and is strictly smaller than the quadratic dispersion coefficient. The minimizing frequencies are nonzero when $τ=2$ and $s$ is close to one. Finally, we prove normalized local limits for the sharp quadratic constant, each fixed fiber eigenvalue, and the dispersion coefficient.
Source: arXiv (Cornell University)
Authors: Deng, Shengbing, Yang, Dehong
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223414636