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Resolvent Estimates for the Lamé Operator and Failure of Carleman Estimates
Journal of Fourier Analysis and Applications ( IF 1.2 ) Pub Date : 2021-05-26 , DOI: 10.1007/s00041-021-09859-6
Yehyun Kwon , Sanghyuk Lee , Ihyeok Seo

In this paper, we consider the Lamé operator \(-\Delta ^*\) and study resolvent estimate, uniform Sobolev estimate, and Carleman estimate for \(-\Delta ^*\). First, we obtain sharp \(L^p\)\(L^q\) resolvent estimates for \(-\Delta ^*\) for admissible pq. This extends the particular case \(q=\frac{p}{p-1}\) due to Barceló et al. [4] and Cossetti [8]. Secondly, we show failure of uniform Sobolev estimate and Carleman estimate for \(-\Delta ^*\). For this purpose we directly analyze the Fourier multiplier of the resolvent. This allows us to prove not only the upper bound but also the lower bound on the resolvent, so we get the sharp \(L^p\)\(L^q\) bounds for the resolvent of \(-\Delta ^*\). Strikingly, the relevant uniform Sobolev and Carleman estimates turn out to be false for the Lamé operator \(-\Delta ^*\) even though the uniform resolvent estimates for \(-\Delta ^*\) are valid for certain range of pq. This contrasts with the classical result regarding the Laplacian \(\Delta \) due to Kenig, Ruiz, and Sogge [23] in which the uniform resolvent estimate plays a crucial role in proving the uniform Sobolev and Carleman estimates for \(\Delta \). We also describe locations of the \(L^q\)-eigenvalues of \(-\Delta ^*+V\) with complex potential V by making use of the sharp \(L^p\)\(L^q\) resolvent estimates for \(-\Delta ^*\).



中文翻译:

Lamé运算符的溶剂估计和Carleman估计的失败

在本文中,我们考虑瘸子运营商\( - \三角洲^ * \) ,并研究解决方法估计,统一索伯列夫估计,并Carleman型估计为\( - \三角洲^ * \) 。首先,我们针对可允许的p,  q获得\(-\ Delta ^ * \)的锐利\(L ^ p \)\(L ^ q \)可分辨的估计。由于Barceló等人的缘故,这扩展了特殊情况\(q = \ frac {p} {p-1} \)。[4]和科塞蒂[8]。其次,我们证明\(-\ Delta ^ * \)的统一Sobolev估计和Carleman估计失败。为此,我们直接分析分解物的傅立叶乘数。这使我们不仅可以证明分解体的上限,还可以证明其下限,因此我们可以得到\(-\ Delta ^的分解体的\(L ^ p \)\(L ^ q \)尖锐边界* \)。令人惊讶的是,即使对于\(-\ Delta ^ * \)的统一分解估计在p的特定范围内有效,相关的统一Sobolev和Carleman估计对于Lamé运算符\(-\ Delta ^ * \)也是错误的。,  q。这与关于Laplacian \(\ Delta \)的经典结果形成对比由于Kenig,Ruiz和Sogge [23]的观点,其中统一的分解估计在证明\(\ Delta \)的统一的Sobolev和Carleman估计中起着至关重要的作用。我们还描述了的位置\(L ^ Q \)的-eigenvalues \( - \德尔塔^ * + V \)与复势V通过利用锋利的\(L ^ P \) - \(L ^ Q \)\(-\ Delta ^ * \)的可分辨估计。

更新日期:2021-05-26
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