Quasi-symmetry of \(L^ p\) norms of eigenfunctions.
The aim of this paper is the behavior of the \(L^p\) norms of the positive and negative parts of eigenfunctions of the Laplacian on Riemannian manifolds. Let \(M\) be a smooth compact manifolds, \(\varphi\) a nonconstant real eigenfunction of the Laplacian and \(\varphi_+ = \varphi \chi(\{\varphi \geq 0\})\), \(\varphi_- =\varphi \chi(\{ \varphi \leq 0\})\) the positive or negative part of \(\varphi\). Then the following is proven: For any \(p \geq 1\) there exists \(C>0\), depending only on \(p\) and the manifold \(M\), such that for any nonconstant eigenfunction \(\varphi\) of the Laplacian \[ 1/C \leq \| \varphi_+\| _{L^p}/\| \varphi_-\| _{L^p} \leq C. \]
scientific article; zbMATH DE number 1893851