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

Abstract

In this work we establish the following fractional Hardy's inequality $$C{\int }_{{\mathbb{H}}_{+}^n}\frac{|f(\xi ){|}^p}{x_1^{sp+\alpha }}d\xi \le {\int }_{{\mathbb{H}}^n}{\int }_{{\mathbb{H}}^n}\frac{|f(\xi )-f({\xi }^{\prime }){|}^p}{d({\xi }^{-1}\circ {\xi }^{\prime }{)}^{Q+sp}|z^{\prime }-z{|}^{\alpha }}d{\xi }^{\prime }d\xi ,\forall f\in C_c^{\infty }({\mathbb{H}}_{+}^n)$$ for the half-space ${\mathbb{H}}_{+}^n=\{\xi =(x,y,t)=(x_1,\dots ,x_n,y_1,\dots ,y_n)\in {\mathbb{H}}^n:x_1>0\}$ 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 $sp+\alpha >1$.

21 pages