Abstract
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 p, q. 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 p, q. 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 \(-\Delta ^*\).
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Notes
Consequently, the line segments \([A,A']\) and \([B,B']\) are contained in the lines \(x-y=\frac{2}{d}\) and \(x-y=\frac{2}{d+1}\), respectively.
The number \(p_*\) relates to the range of the oscillatory integral of Carleson–Sjölin type with elliptic phase ( [15, Theorem 1.2]). Being combined with Tao’s bilinear restriction theorem ( [37]) and the bilinear argument in [7, 27], this is one of main ingredients for the results in [25, Theorem 1.4]. The point \(P_\circ \) is the intersection of \({\mathcal {L}}\) and the line connecting \(P_*\) and \((\frac{1}{2}, \frac{d}{2(d+2)})\); see [25, Sects. 2 and 3] for details. If \(d=2\) then \(P_*=P_\circ =D=(1/4,1/4)\).
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Acknowledgements
This work was supported by the KIAS Individual Grant and the National Research Foundation of Korea (NRF) with grant numbers MG073701 and NRF-2020R1F1A1A010735 2012 (Yehyun Kwon), NRF-2021R1A2B5B02001786 (Sanghyuk Lee), and NRF-2019R1F1A1061316 (Ihyeok Seo). The authors would like to thank Alberto Ruiz for informing us of the paper [4] and the anonymous referee for very careful reading and various helpful comments.
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Kwon, Y., Lee, S. & Seo, I. Resolvent Estimates for the Lamé Operator and Failure of Carleman Estimates. J Fourier Anal Appl 27, 53 (2021). https://doi.org/10.1007/s00041-021-09859-6
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DOI: https://doi.org/10.1007/s00041-021-09859-6