Fractional Hardy's inequality for half-spaces in the Heisenberg group

We establish the following fractional Hardy's inequality integral(n)(H+)|f(xi)|(p)/ x(1)(sp)|z|(alpha) d xi <= C integral(H+n)integral(H+n) |f(xi)-f(xi ')|(p) / d(xi(-1)circle xi ')(Q+sp) |z '-z|(alpha)/ d xi ' d xi, for all f is an element of C-c(H-+(n)) for the half-space H-+(n):={xi=(z,t)=(x(1),x(2),...,x(n),y(1),y(2),...,y(n),t)is an element of H-n:x(1)>0} in the Heisenberg group H-n under the conditions sp>1 and alpha >=(Q-2+sp)/2, where n >= 1, Q=2n+2, p >= 1 and 0<s<1. We also provide an alternate proof of a fractional Hardy's inequality in Hn established in an earlier work. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.