Notes on scalar curvature lower bounds of steady gradient Ricci solitons

We provide new type of decay estimate for scalar curvatures of steady gradient Ricci solitons. We also give certain upper bound for the diameter of a Riemannian manifold whose $\mathrm{\infty }$-Bakry–Emery Ricci tensor is bounded by some positive constant from below. For the proofs, we use $\mu$-bubbles introduced by Gromov.