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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-17290-002 (Oberwolfach Reports 47/2019, Problem 1).