Local Sobolev Constant Estimate for Integral Ricci Curvature Bounds
  We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the $L^2$ Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions. 
Xianzhe Dai, Guofang Wei, Zhenlei Zhang
math.DG