# Laplacian eigenfunction $L^p$ norms

Suppose I have a compact surface (possibly with boundary), and consider the eigenfunctions of the Laplacian, normalized so that their $L^2$ norms are $1.$ Is there some general result or conjecture on how the $L^p$ norm of $f_\lambda$ (where $\lambda$ is the eigenvalue) behaves as function of $\lambda?$ Presumably this depends on the geometry of the surface (and, of course, nothing about this question is special to two dimensions, except that this case might be easier).

• Search for Sogge's results on $L^p$ bounds. I believe this is an active area of study, especially when one restricts to manifolds with interesting geometric properties. Sep 5, 2014 at 1:22
• I haven't read this paper myself, but I believe the work of Sogge (J. Funct. Analysis 1988) may have results of the kind you want? Sep 5, 2014 at 1:24
• For "interesting" cases, much better bounds are known (and expected) than the general theory by Sogge. See for example the result of Iwaniec-Sarnak and all the recent work on Berry's random wave model.
– Asaf
Sep 5, 2014 at 7:42
• @Asaf Any particular paper you recommend (on the Berry random wave thing)? Sep 5, 2014 at 13:29
• This arxiv.org/pdf/0903.3420v1.pdf very nice survey article of Zelditch contains quite a bit of discussion about this very topic, including an overview of some of the work of Sogge. Sep 5, 2014 at 14:08

As pointed out already by the comments, Sogge has indeed made a lot of contributions in this area. Consider a 2-dimensional compact Riemannian manifold without boundary, then the $L^2$ normalized eigenfunction of Laplacian $e_{\lambda}$ which sastisfy $$-\Delta e_{\lambda}=\lambda^2e_{\lambda}$$ has the following estimate $$\|e_{\lambda}\|_{L^p}\leq C\lambda^{\sigma(p)},~~\lambda\ge 1,$$ here $\sigma(p)=\frac{1}{2}(\frac{1}{2}-\frac{1}{p})$, if $2\leq p\leq 6$, and $\sigma(p)=2(\frac{1}{2}-\frac{1}{p})-1$, if $6\leq p\leq \infty$. For dimension $n>2$, there are similar results.
It's also interesting to improve the bound above. The above estimates is sharp in general. The $L^{\infty}$ norm is saturated by the Zonal function (which is concentrated near the pole) on the round sphere. For lower $p$(below the critical point, which is 6 here), it's saturated by the highest weight hamonics (which is concentrated near the equator ). However, for manifold with nonpositive curvature, on can get some improvement($\log \lambda$), and for flat torus $\mathbb{T}^n$, one can get a further improvement ($\lambda^{\epsilon(n)})$.