Explicit sharp gradient stability of the $L^{p}$-Poincaré inequality and fundamental gaps for the Dirichlet $p$-Laplacian operator

In this paper, we prove the gradient stability of the $L^{p}$-Poincaré inequality with explicit constants on bounded convex domains for any $1<p<\infty$. These estimates control the distance to the optimizers in the strongest possible norm with the optimal stability exponent $\max\{2,p\}$, exhibitin