Sharp fractional Hardy--Maz'ya remainders on half-spaces: angular reduction and energy stability

Abstract

For $d\ge 2$, $0<s<1$, $p>1$, and $\tau >0$, 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 $h(\xi )=\min \{{\xi }^2,1\}$, the best planar stability coefficient is not attained and is strictly smaller than the quadratic dispersion coefficient. The minimizing frequencies are nonzero when $\tau =2$ 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.