Local Sobolev constant estimate for integral Bakry–Émery Ricci curvature

Abstract

We extend several geometrical results in [10] for Riemannian manifolds with integral curvature to complete smooth metric measure spaces with integral Bakry-Émery Ricci curvature.

Comparison Geometry for Integral Bakry–Émery Ricci Tensor Bounds

“…When ∂ r f ≥ −a, we easily see that Ric H f − p,a (R) ≡ 0 if and only if Ric f ≥ (n−1)H. In [22], the author proved many weighted comparison theorems on (M, g, e −f dv) when Ric H f − f,a (R) is bounded and ∂ r f ≥ −a. As applications, classical eigenvalue estimates, Sobolev constant estimates and Myers' type theorems, etc were generalized to the case of some assumptions of Ric H f − f,a (R) and ∂ r f ; see [22,19,10]. However, when f is bounded, there seem to be lack of effective comparison theorems under the integral Bakry-Émery Ricci tensor, though some progress has been made in [22].…”

Lichnerowicz Type Estimate for the $p$-Laplacian Under Weighted Integral Curvature Bounds

“…Lichnerowicz Type Estimate for the p-Laplacian Under Weighted Integral Curvature Bounds Remark 2.1. The case for K ∈ R is established in [6], however for the K > 0, this can be shown using results of Wang-Wei [10] and compactness.…”