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| Some Weighted Norm Inequalities on Manifolds |
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Citation： |
Shiliang ZHAO.Some Weighted Norm Inequalities on Manifolds[J].Chinese Annals of Mathematics B,2018,39(6):1001~1016 |
| Page view： 755
Net amount： 651 |
Authors： |
Shiliang ZHAO; |
Foundation： |
This work was supported by
the China Scholarship Council (No.201406100171). |
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| Abstract： |
Let $M$ be a complete non-compact Riemannian manifold satisfying the volume doubling property
and the Gaussian upper bounds. Denote by $\Delta$ the Laplace-Beltrami operator and by $\nabla$
the Riemannian gradient. In this paper, the author proves the weighted reverse inequality
$\| \Delta^{\frac{1}2} f \|_{L^p(w)} \le C \| |\nabla f| \|_{L^p(w)}$, 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. |
Keywords： |
Weighted norm inequality, Poincar'e inequality, Riesz transform. |
Classification： |
42B20, 58J35 |
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