Endpoint of the sharp singular concentration-compactness range

Determine whether the supremum in the sharp singular concentration-compactness principle for the space W remains finite at the endpoint parameter ζ=Z_Q(u), where Z_Q(u)=(1-\|u\|^Q)^{-1/(Q-1)} for a nonzero weak limit u in W with \|u\|≤1.

Background

Theorem 1.2 establishes that, for a sequence {u_k} in the space W satisfying |u_k|=1 and u_k converging weakly to a nonzero function u, the weighted exponential supremum is finite whenever 0<ζ<Z_Q(u), and that it is infinite for ζ>Z_Q(u). The result therefore leaves the boundary value ζ=Z_Q(u) unresolved.

The endpoint is significant because it is the exact threshold separating the proven bounded and unbounded regimes. Establishing finiteness or constructing divergence at this value would complete the sharp characterization of the concentration-compactness principle for the singular Moser–Trudinger inequality in the Heisenberg-group setting.