Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group - Simon Larson (Bulletin of Mathematical Sciences, 2016)
Abstract We prove geometric $$L^p$$ L p versions of Hardy’s inequality for the sub-elliptic Laplacian on convex domains $$Ω $$ Ω in the Heisenberg group $$\mathbb {H}^n$$ H n , where convex is meant in the Euclidean sense. When $$p=2$$ p = 2 and $$Ω $$ Ω is the half-space given by $$\langle ξ , ν \rangle > d$$ ⟨ ξ , ν ⟩ > d this generalizes an inequality previously obtained by Luan and Yang. For such p and $$Ω $$ Ω the inequality is sharp and takes the form $$\begin{aligned} \int _Ω |\nabla _{\mathbb {H}^n}u|^2 \, dξ \ge \frac{1}{4}\int _{Ω } \sum _{i=1}^n\frac{\langle X_i(ξ ), ν \rangle ^2+\langle Y_i(ξ ), ν \rangle ^2}{{{\mathrm{\text {dist}}}}(ξ , \partial Ω )^2}|u|^2\, dξ , \end{aligned}$$ ∫ Ω | ∇ H n u | 2 d ξ ≥ 1 4 ∫ Ω ∑ i = 1 n ⟨ X i ( ξ ) , ν ⟩ 2 + ⟨ Y i ( ξ ) , ν ⟩ 2 dist ( ξ , ∂ Ω ) 2 | u | 2 d ξ , where $${{\mathrm{\text {dist}}}}(\, · \,, \partial Ω )$$ dist ( · , ∂ Ω ) denotes the Euclidean distance from $$\partial Ω $$ ∂ Ω .