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

Abstract

In this work we establish the following fractional Hardy's inequality $$C\int_{\mathbb{H}^n_+}\frac{|f(\xi)|^p}{x_1^{sp+\alpha}}d\xi\leq \int_{\mathbb{H}^n}\int_{\mathbb{H}^n}\frac{|f(\xi)-f(\xi')|^p}{d({\xi}^{-1}\circ \xi')^{Q+sp}|z'-z|^\alpha}d\xi'd\xi,\ \ \forall f\in C_c^{\infty}(\mathbb{H}^n_+)$$ for the half-space $\mathbb{H}^n_+=\{\xi=(x,y,t)=(x_1,\ldots,x_n,y_1,\ldots,y_n)\in\mathbb{H}^n:x_1>0\}$ in the Heisenberg group $\mathbb{H}^n$ without any restriction on parameters, and compute the corresponding sharp constant. In a previous joint work, we established a variant of Hardy's inequality for the same half-space, but with certain parameter restrictions. However, all integrals in that work were considered over half-spaces, and here the seminorm is taken over the entire $\mathbb{H}^n$. Although this inequality holds for all values of the quantity $sp+\alpha$, we are only able to compute the corresponding sharp constant when $sp+\alpha>1$.