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

2025/04/08 by Roy, Haripada
#26D15 #43A80 #46E35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.05949

Abstract

In this work we establish the following fractional Hardy's inequality C∫_ℍⁿ+\frac|f(ξ)|^px₁^sp+αdξ≤ ∫_ℍⁿ∫_ℍⁿ\frac|f(ξ)-f(ξ')|^pd(ξ^-1∘ ξ')^Q+sp|z'-z|^αdξ'dξ, ∀ f∈ C_c^∞(ℍⁿ₊) for the half-space ℍⁿ₊=\ξ=(x,y,t)=(x₁,…,x_n,y₁,…,y_n)∈ℍⁿ:x₁>0\ in the Heisenberg group ℍⁿ 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 ℍⁿ. 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.

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• Fractional Hardy's inequality for half spaces in the Heisenberg group
• Geometric Hardy and Hardy-Sobolev inequalities on Heisenberg groups
• Sharp constants in several inequalities on the Heisenberg group
• Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group
• Hardy-Littlewood-Sobolev inequality revisit on Heisenberg group