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

Abstract

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