On Lp-Resolvent Estimates and the Density of Eigenvalues for Compact Riemannian Manifolds
On Lp-Resolvent Estimates and the Density of Eigenvalues for Compact Riemannian Manifolds
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关于紧致黎曼流形的 L-p 分辨估计和特征值密度
DOI：
10.1007/s00220-014-2077-y
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发表时间：
2015-02-01
影响因子：
2.4
通讯作者：
Yao, Xiaohua
中科院分区：
文献类型：
--
作者：
Bourgain, Jean;Shao, Peng;Yao, Xiaohua
我们解决了Dos Santos Ferreira， Kenig和Salo （Forum Math, 2014）提出的一个有趣的问题，该问题是关于在给定的紧化无边界黎曼流形上，具有度量g的拉普拉斯-贝尔特拉米算子可以有统一的可解估计的区域。这与Kenig， Ruiz和第三位作者（Duke Math J 55:329-347, 1987）关于欧几里得拉普拉斯的早期工作有关，在这种情况下，区域是整个复平面减去以原点为中心的任何圆盘。目前，我们表明，对于球面上的圆形度量S (n), （Dos Santos Ferreira et al. in Forum Math, 2014）中的解决方案估计涉及更小的区域，本质上是最优的。我们通过建立基于从到光谱的距离的明确界限来做到这一点。在另一个方向上，我们还表明（Dos Santos Ferreira et al. In Forum Math, 2014）中的边界可以对非正曲率流形进行对数锐化，对于平面度规的环面，可以通过幂来锐化。后者改进了Shen的早期边界（数学学报，2001）。（Dos Santos Ferreira et al. in Forum Math, 2014）和（Shen in Int Math Res Not 1:1-31, 2001）的工作是基于Hadamard参数。我们的方法是基于相关的Hadamard参数，并遵循（Sogge in Ann Math 126:439-447, 1987）中使用小时间波动方程参数证明L (p)乘法器估计和（Sogge in J Funct Anal 77:123- 138,1988）的谱投影估计的思想。这种方法允许我们调整B， ard （Math Z 155:249-276, 1977）和Hlawka （Monatsh Math 54:1- 36,1950）中的论点，以获得上述改进（Dos Santos Ferreira et al. in Forum Math, 2014）和（Shen in Int Math Res Not 1:1- 31,2001）。利用第一作者（Bourgain in Israel J Math 193(1):441-458, 2013）的最新技术以及他与Guth （Bourgain and Guth in Geom Funct Anal 21:1239-1295, 2011）基于Bennett， Carbery和Tao （Math Z 2:261-302, 2006）的多元线性估计，对环面进行了进一步改进。我们的方法还允许我们给出一个基于谱密度测量的有利分解估计的自然必要条件，此外，基于缩小区间的自然改进谱投影估计的必要和充分条件，这与（Sogge in J Funct Anal 77:123- 138,1988）中对单位长度区间的必要和充分条件相反。我们证明了解离估计对谱内的聚类很敏感，这并不奇怪，因为Sommerfeld在Physikal Zeitschr 11:1057-1066, 1910)关于这些算子的原始猜想。
We address an interesting question raised by Dos Santos Ferreira, Kenig and Salo (Forum Math, 2014) about regions for which there can be uniform resolvent estimates for , , where is the Laplace-Beltrami operator with metric g on a given compact boundaryless Riemannian manifold of dimension . This is related to earlier work of Kenig, Ruiz and the third author (Duke Math J 55:329-347, 1987) for the Euclidean Laplacian, in which case the region is the entire complex plane minus any disc centered at the origin. Presently, we show that for the round metric on the sphere, S (n) , the resolvent estimates in (Dos Santos Ferreira et al. in Forum Math, 2014), involving a much smaller region, are essentially optimal. We do this by establishing sharp bounds based on the distance from to the spectrum of . In the other direction, we also show that the bounds in (Dos Santos Ferreira et al. in Forum Math, 2014) can be sharpened logarithmically for manifolds with nonpositive curvature, and by powers in the case of the torus, , with the flat metric. The latter improves earlier bounds of Shen (Int Math Res Not 1:1-31, 2001). The work of (Dos Santos Ferreira et al. in Forum Math, 2014) and (Shen in Int Math Res Not 1:1-31, 2001) was based on Hadamard parametrices for . Ours is based on the related Hadamard parametrices for , and it follows ideas in (Sogge in Ann Math 126:439-447, 1987) of proving L (p) -multiplier estimates using small-time wave equation parametrices and the spectral projection estimates from (Sogge in J Funct Anal 77:123-138, 1988). This approach allows us to adapt arguments in B,rard (Math Z 155:249-276, 1977) and Hlawka (Monatsh Math 54:1-36, 1950) to obtain the aforementioned improvements over (Dos Santos Ferreira et al. in Forum Math, 2014) and (Shen in Int Math Res Not 1:1-31, 2001). Further improvements for the torus are obtained using recent techniques of the first author (Bourgain in Israel J Math 193(1):441-458, 2013) and his work with Guth (Bourgain and Guth in Geom Funct Anal 21:1239-1295, 2011) based on the multilinear estimates of Bennett, Carbery and Tao (Math Z 2:261-302, 2006). Our approach also allows us to give a natural necessary condition for favorable resolvent estimates that is based on a measurement of the density of the spectrum of , and, moreover, a necessary and sufficient condition based on natural improved spectral projection estimates for shrinking intervals, as opposed to those in (Sogge in J Funct Anal 77:123-138, 1988) for unit-length intervals. We show that the resolvent estimates are sensitive to clustering within the spectrum, which is not surprising given Sommerfeld's original conjecture (Sommerfeld in Physikal Zeitschr 11:1057-1066, 1910) about these operators.