Published Paper

Inserted: 30 jul 2023
Last Updated: 17 oct 2026

Journal: Math. Ann.
Volume: 396
Number: Art. no. 61
Pages: 59pp
Year: 2026
Doi: 10.1007/s00208-026-03550-1

Abstract:

We study the loss of compactness for optimal functions associated with subcritical approximations of the sharp Folland–Stein–Sobolev embedding in the Heisenberg group. A key ingredient is a fine asymptotic control of the optimal functions by the Jerison–Lee extremals, which attain equality in the critical Sobolev inequality (JAMS, 1988). When boundary blow-up is ruled out, we show that centered-symmetric maximizers in a Korányi ball concentrate at its center, and we determine the exact blow-up rate and the pointwise asymptotic profile away from the pole, thereby obtaining a natural counterpart of the classical Atkinson–Brezis–Peletier asymptotics for nearly critical Sobolev problems. We then introduce a class of smooth domains geometrically regular near their characteristic set and extend the analysis to this setting, proving one-point concentration, describing the profile by the Dirichlet Green function, and expressing the exact blow-up rate through a Green-function boundary functional arising from the anisotropic Pohozaev identity.

Keywords: Heisenberg group, Sobolev embeddings, CR Yamabe

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