Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group

Abstract

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

Keywords

• 35A23
• 35H20