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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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