Extremals for Heisenberg-group Moser–Trudinger inequalities

Construct or characterize extremal functions attaining the sharp constants in Moser–Trudinger-type inequalities on the Heisenberg group, including the singular and nonsingular inequalities discussed in the paper.

Background

The paper explains that Euclidean proofs and extremal-function analyses commonly rely on Schwarz symmetrization and the Pólya–Szegő inequality. In the Heisenberg group, the relevant Pólya–Szegő inequality fails in its classical form, preventing the direct transfer of those methods.

Although the paper proves sharp singular Moser–Trudinger inequalities and related concentration-compactness results, it does not determine whether the sharp suprema are attained or identify their extremals. The existence and structure of extremals therefore remain unresolved.