Some Weighted Norm Inequalities on Manifolds

Let M be a complete non-compact Riemannian manifold satisfying the volume doubling property and the Gaussian upper bounds. Denote by Δ the Laplace-Beltrami operator and by ? the Riemannian gradient. In this paper, the author proves the weighted reverse inequality \(\left\| {{\Delta ^{\frac{1}{2}}}f} \right\|_{L^p(w)}\leq C\left\| {|\nabla f|} \right\|_{L^p(w)}\), for some range of p determined by M and w. Moreover, a weak type estimate is proved when p = 1. Some weighted vector-valued inequalities are also established.