Sharp remainder of the $L^p$-Poincaré inequality for Baouendi-Grushin vector fields

In this paper, we establish a sharp remainder formula for the Poincaré inequality for Baouendi-Grushin vector fields in the setting of $L^p$ for complex-valued functions. In special cases, we recover previously known results. Consequently, we also derive the $L^p$-Poincaré inequality with an explicit optimal constant under a certain assumption. Additionally, we provide estimates of the remainder term for $p\ge 2$ and $1<p<2\le n<\mathrm{\infty }$. As an application, we obtain a blow-up in finite time and global existence of the positive solutions to the initial-boundary value problem of the doubly nonlinear porous medium equation involving a degenerate nonlinear operator ${\mathrm{\Delta }}_{\gamma ,p}$.