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).

## 1 Answer

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.

However, for manifold with boundary, the problem becomes more difficult, for dimension 2, see also the Acta paper by Smith and Sogge http://arxiv.org/pdf/math/0605682.pdf.

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 ($\lambda^{\epsilon(n)})$.

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