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Eigenfunction bounds for the Laplacian on the \(n\)-torus. (English) Zbl 0779.58039
The author proves the following result: Let \(n\geq 4\) and \(p \geq 2(n+1)/(n-3)\). Then \[ M_{n,p} = \lim_{\Delta\psi + \lambda\psi = 0}{\| \psi \|_ p\over \| \psi \|_ 2}\ll \lambda^{(n- 2)/4-n/2p+\varepsilon} \] where \(\Delta\) stands for the Laplacian on the \(n\)-torus \(\Pi^ n = \mathbb{R}^ n/\mathbb{Z}^ n\) and \(\|\;\|_ p\) the usual \(L^ p\)-norm.

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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