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

摘要

We prove geometric (L^p) versions of Hardy’s inequality for the sub-elliptic Laplacian on convex domains (Omega ) in the Heisenberg group (mathbb {H}^n), where convex is meant in the Euclidean sense. When (p=2) and (Omega ) is the half-space given by (langle xi , u angle > d) this generalizes an inequality previously obtained by Luan and Yang. For such p and (Omega ) the inequality is sharp and takes the form $$egin{aligned} int _Omega |abla _{mathbb {H}^n}u|^2 , dxi ge rac{1}{4}int _{Omega } sum _{i=1}^nrac{langle X_i(xi ), u angle ^2+langle Y_i(xi ), u angle ^2}{{{mathrm{ext {dist}}}}(xi , partial Omega )^2}|u|^2, dxi , end{aligned}$$where ({{mathrm{ext {dist}}}}(, cdot ,, partial Omega )) denotes the Euclidean distance from (partial Omega ).