Dimension-free Maz’ya–Shaposhnikova limiting formulas in Grushin spaces

Motivated by the paper [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XXIII (2022) 837–875], we provide a simplified and robust proof of the dimension-free Maz’ya–Shaposhnikova limiting formula for seminorms associated with the Baouendi–Grushin operator. This operator, which is generally non-hypoelliptic, arises frequently in the study of Carnot–Carathéodory spaces. Our approach not only streamlines the original argument but also extends a recent result from [arXiv:2401.03409].