Sharp remainder of the Poincaré inequality for Baouendi–Grushin vector fields

Abstract

In this note, we establish a sharp remainder formula for the Poincaré inequality for the Baouendi–Grushin vector fields. We give a simple proof for it without using the variational principle. As an application, we obtain a blow-up result for solutions to the Dirichlet initial-boundary value problem for the Baouendi–Grushin heat operator.