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

TL;DR

This paper proves a fractional Hardy's inequality for half spaces in the Heisenberg group, extending previous results and providing new proofs under specific conditions on parameters.

Contribution

It establishes a new fractional Hardy's inequality for half spaces in the Heisenberg group and offers an alternative proof for an existing inequality.

Findings

Proved fractional Hardy's inequality in Heisenberg half spaces.

Derived conditions for parameters $sp>1$ and $eta ext{geq} (2n+sp)/2$.

Provided an alternative proof for a known inequality.

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.