Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications

AbstractLet H=Cn×R be the n-dimensional Heisenberg group, Q=2n+2 be the homogeneous dimension of H, Q′=QQ−1, and ρ(ξ)=(|z|4+t2)14 be the homogeneous norm of ξ=(z,t)∈H. Then we prove the following sharp Moser–Trudinger inequality on H (Theorem 1.6): 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<Q, 0<α≤αQ(1−βQ) there holds sup‖u‖1,τ≤1∫H1ρ(ξ)β{exp(α|u|Q/(Q−1))−∑k=0Q−2αkk!|u|kQ/(Q−1)}≤C(Q,β,τ)<∞. The constant αQ(1−βQ) is best possible in the sense that the supremum is infinite if α>αQ(1−βQ). Here τ is any positive number, and ‖u‖1,τ=[∫H|∇Hu|Q+τ∫H|u|Q]1/Q.Our result extends the sharp Moser–Trudinger inequality by Cohn and Lu (2001) [19] on domains of finite measure on H and sharpens the recent result of Cohn et al. (2012) [18] where such an inequality was studied for the subcritical case α