  We obtain sharp quantitative Laplacian upper and lower estimates under no
assumption on curvatures. As a result, we derive quantitative Laplacian, area
and volume comparison theorems for tubes in Riemannian and K\"ahler manifolds
under weak integral curvature assumptions. We also give some applications, such
as a general Bonnet-Myers theorem and Cheng's eigenvalue estimate under weak
integral curvature assumptions.
