Sharp subcritical Moser–Trudinger inequalities on Heisenberg groups and subelliptic PDEs

The aim of this paper is to prove a sharp subcritical Moser–Trudinger inequality on the whole Heisenberg group. Let H=Cn×RH=Cn×R be the n−n−dimensional Heisenberg group, Q=2n+2Q=2n+2 be the homogeneous dimension of HH, Q′=QQ−1, and ρ(ξ)=(|z|4+t2)14 be the homogeneous norm of ξ=(z,t)∈Hξ=(z,t)∈H. Then we establish the following inequality on HH (Theorem 1.1): there exists a positive constant αQ=Q(2πnΓ(12)Γ(Q−12)Γ(Q2)−1Γ(n)−1)Q′−1 such that for any pair β,α satisfying 0≤β