Sharp fractional Hardy's inequality for half-spaces in the Heisenberg group
Haripada Roy
TL;DR
This paper advances fractional Hardy inequalities on the Heisenberg group by establishing a sharp-constant Hardy inequality for the half-space \\mathbb{H}^n_+. It develops a nonlocal framework with a kernel involving the homogeneous distance d(ξ^{-1}\\circ ξ')^{Q+sp} and a weight |z-z'|^α, and proves the inequality for all admissible parameters, computing the sharp constant precisely when sp+α>1. The core approach combines distance-function analysis, weak harmonicity of (δ_{\\mathbb{H}^n_+})^s, a carefully designed test-function method, and a one-dimensional reduction to transfer optimal constants from Euclidean to the Heisenberg setting. The main contribution is the exact sharp constant for the half-space, expressed as \\mathcal{C}_{n,s,p,α}(\\mathbb{H}^n_+)=C_{n,s,p,α} Λ_{s,p,α}, with explicit forms for Λ_{s,p,α} and the dimension-dependent prefactor, enriching the PDE toolkit on sub-Riemannian geometries.
Abstract
In this work we establish the following fractional Hardy's inequality $$C\int_{\mathbb{H}^n_+}\frac{|f(ξ)|^p}{x_1^{sp+α}}dξ\leq \int_{\mathbb{H}^n}\int_{\mathbb{H}^n}\frac{|f(ξ)-f(ξ')|^p}{d(ξ^{-1}\circ ξ')^{Q+sp}|z'-z|^α}dξ'dξ,\ \ \forall f\in C_c^{\infty}(\mathbb{H}^n_+)$$ for the half-space $\mathbb{H}^n_+=\{ξ=(x,y,t)=(x_1,\ldots,x_n,y_1,\ldots,y_n)\in\mathbb{H}^n:x_1>0\}$ 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 $sp+α$, we are only able to compute the corresponding sharp constant when $sp+α>1$.