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

Abstract

We establish the following fractional Hardy's inequality $${\int }_{{\mathbb{H}}_{+}^n}\frac{|f(\xi ){|}^p}{x_1^{sp}|z{|}^{\alpha }}d\xi \le C{\int }_{{\mathbb{H}}_{+}^n}{\int }_{{\mathbb{H}}_{+}^n}\frac{|f(\xi )-f({\xi }^{\prime }){|}^p}{d({\xi }^{-1}\circ {\xi }^{\prime }{)}^{Q+sp}|z^{\prime }-z{|}^{\alpha }}d{\xi }^{\prime }d\xi ,\forall f\in C_c({\mathbb{H}}_{+}^n)$$ for the half space ${\mathbb{H}}_{+}^n:=\{\xi =(z,t)=(x_1,x_2,\dots ,x_n,y_1,y_2,\dots ,y_n,t)\in {\mathbb{H}}^n:x_1>0\}$ in the Heisenberg group ${\mathbb{H}}^n$ under the conditions $sp>1$ and $\alpha \ge (2n+sp)/2$. We also provide an alternate proof of a fractional Hardy's inequality in ${\mathbb{H}}^n$ established in an earlier work.

The paper has 13 pages