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

Shengbing Deng, Dehong Yang

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_{\mat…