Uniform Sobolev Resolvent Estimates for the Laplace-Beltrami Operator on Compact Manifolds
Uniform Sobolev Resolvent Estimates for the Laplace-Beltrami Operator on Compact Manifolds
复制标题
紧凑流形上 Laplace-Beltrami 算子的统一 Sobolev 分解估计
DOI：
10.1093/imrn/rnt051
复制
发表时间：
2014
影响因子：
1
通讯作者：
Yao Xiaohua
中科院分区：
文献类型：
--
作者：
Shao Peng;Yao Xiaohua
In this paper, we continue the study on the resolvent estimates of the Laplace–Beltrami operator Δgon a compact manifold M with dimension n≥3. On the Sobolev line 1/p−1/q=2/n, we can prove that the resolvent (Δg+ζ)−1is uniformly bounded from Lpto Lqwhen (p,q) are within the range: p≤2(n+1)/(n+3) and q≥2(n+1)/(n−1) and ζ is outside a parabola opening to the right and a small disk centered at the origin. This naturally generalizes the previous results in [2, 3] which addressed only the special case when p=2n/(n+2),q=2n/(n−2). Using the shrinking spectral estimates between Lpand Lq, we also show that when (p,q) are within the interior of the range mentioned above, one can obtain a logarithmic improvement over the parabolic region for resolvent estimates on manifolds equipped with Riemannian metric of nonpositive sectional curvature, and a power improvement depending on the exponent (p,q) for flat torus. The latter therefore partially improves Shen's work in [5] on the Lp→L2uniform resolvent estimates on the torus. Similar to the case as proved in [2] when (p,q)=(2n/(n+2),2n/(n−2)), the parabolic region is also optimal over the round sphere Snwhen (p,q) are now in the range. However, we may ask whether the range is sharp in the sense that it is the only possible range on the Sobolev line for which a compact manifold can have uniform resolvent estimate for ζ being outside a parabola.
