Essential Self-Adjointness of the Laplace–Beltrami Operator for a Family of Non-Regular Almost-Riemannian Structures

Oberwolfach Report 47/2019 records the following question, posed in the abstract of L. Rizzi's talk (joint work with V. Franceschi, D. Prandi and M. Seri): is the Laplace–Beltrami operator of the almost-Riemannian structure on R² with orthonormal frame X₁ = ∂ₓ, X₂ = x(x^{2ℓ} + z²)∂_z essentially self-adjoint on the regular region for every ℓ ≥ 1? The case ℓ = 1, and the cases ℓ ≤ n/2 of an n-dimensional version, had been settled by Prandi, Rizzi and Seri in the complete (torus) setting of their Example 7.7; their effective-potential criterion does not apply for larger ℓ. We answer the question affirmatively for every ℓ in the complete setting assumed in the report: the structures X₁ = ∂ₓ, X₂ = c(z) x(x^{2ℓ} + ρ(z))∂_z with c > 0, ρ ≥ 0 and c, cρ bounded, all of which are complete, have an essentially self-adjoint Laplace–Beltrami operator. They include structures that coincide with the model on a strip around the non-regular point; the same holds for the n-dimensional examples of Prandi, Rizzi and Seri for all ℓ and n. We do not prove a localisation theorem for arbitrary complete structures that agree with the model near that point. The proof combines a one-dimensional Hardy inequality obtained with the multiplier δ (the distance from the singular set), whose weight −δΔδ is at least 1 for this family, with a standard Agmon-type argument; the effective potential, in contrast, is not bounded below by 3/(4δ²) when ℓ ≥ 2. The same argument shows that the effective-potential hypothesis in the criteria of Prandi–Rizzi–Seri and Franceschi–Prandi–Rizzi can be replaced by −Δ_ω δ ≥ 1/δ − κ, which is an alternative sufficient condition rather than a strengthening. Taken literally on all of R², the model is incomplete, and its Laplace–Beltrami operator is not essentially self-adjoint for reasons unrelated to the singular set. The more general real-analytic conjecture of the report remains open; we give a real-analytic example without tangency points for which the weak Hardy inequality underlying all these criteria fails. This is an unrefereed note.