Some functional inequalities under lower Bakry–Émery–Ricci curvature bounds with \({\varepsilon }\)-range

For n-dimensional weighted Riemannian manifolds, lower m-Bakry–Émery–Ricci curvature bounds with \({\varepsilon }\) $\epsilon$ -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–Émery–Ricci curvature bounds with \({\varepsilon }\) $\epsilon$ -range. These generalize those inequalities under constant curvature bounds for \(m \in (n,\infty )\) $m\in (n,\infty )$ to \(m\in (-\infty ,1]\cup \{\infty \}\) $m\in (-\infty ,1]\cup \{\infty \}$ .