Comparison Geometry for Integral Bakry–Émery Ricci Tensor Bounds

We prove mean curvature and volume comparison estimates on smooth metric measure spaces when their integral Bakry–Émery Ricci tensor bounds, extending Wei–Wylie’s comparison results to the integral case. We also apply comparison results to get diameter estimates, eigenvalue estimates, and volume growth estimates on smooth metric measure spaces with their normalized integral smallness for Bakry–Émery Ricci tensor. These give generalizations of some work of Petersen–Wei, Aubry, Petersen–Sprouse, Yau and more.