Some functional inequalities under lower Bakry-Émery-Ricci curvature bounds with \({\varepsilon }\)-range (Q6593243)

In this nice paper, the author proves some functional inequalities under lower Bakry-Émery-Ricci curvature bounds with \(\epsilon\)-range.\N\NMore precisely, under certain assumptions on an \(n\)-dimensional complete weighted Riemannian manifold \(M\), the author proves an upper bound for the spectrum with \(\epsilon\)-range. To this end, a volume comparison result is used. Moreover, a local Sobolev inequality is established. The main tool used to prove it is the local Poincaré inequality and the volume doubling property. Finally, an upper bound of the \(L^p\) spectrum for deformed measure is obtained.