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

@misc{deng2026sharp,
  title = {{Sharp fractional Hardy--Maz'ya remainders on half-spaces: angular reduction and energy stability}},
  author = {Shengbing Deng and Dehong Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31595},
  eprint = {2610.31595},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31595},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Sharp fractional Hardy--Maz'ya remainders on half-spaces: angular reduction and energy stability
AU  - Shengbing Deng
AU  - Dehong Yang
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31595
UR  - https://arxiv.org/abs/2610.31595
AB  - For d≥2, 0<s<1, p>1, and τ>0, we study the sharp half-space remainder [u]_s,p,+^p−𝒟_d,s,p∫_ℝ^d_+x_d^−sp|u|^p dx ≥Λ_d,s,p,τ∫_ℝ^d_+ (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.
ER  -