Some functional inequalities under lower Bakry-Émery-Ricci curvature bounds with ε-range

For n-dimensional weighted Riemannian manifolds, lower m-Bakry-& Eacute;mery-Ricci curvature bounds with epsilon-range, introduced by Lu-Minguzzi-Ohta (Anal Geom Metr Spaces 10(1):1-30, 2022), integrate constant lower bounds and certain variable lower bounds in terms of weight functions. In this paper, we prove a Cheng type inequality and a local Sobolev inequality under lower m-Bakry-& Eacute;mery-Ricci curvature bounds with epsilon-range. These generalize those inequalities under constant curvature bounds for m is an element of (n, infinity) to m is an element of (-infinity, 1] boolean OR {infinity}.