Critical Hardy inequality on the half-space via the harmonic transplantation

Abstract

We prove a critical Hardy inequality on the half-space $${\mathbb {R}}^N_+$$ by using the harmonic transplantation for functions in $${\dot{W}}_0^{1,N}({\mathbb {R}}^N_+)$$ . Also we give an improvement of the subcritical Hardy inequality on $${\dot{W}}_0^{1,p}({\mathbb {R}}^N_+)$$ for $$p \in [2, N)$$ , which converges to the critical Hardy inequality when $$p \nearrow N$$ . Sobolev type inequalities are also discussed.