Eigenfunction bounds for the Laplacian on the \(n\)-torus. (English) Zbl 0779.58039

Bourgain, J.

Int. Math. Res. Not. 1993, No. 3, 61-66 (1993).

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.

Reviewer: M.Puta (Timişoara)

MSC:

58J50

Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

eigenfunctions; Laplacian; \(n\)-torus; \(L^ p\)-norm