Mercer's Theorem: uniform $L_\infty$ bound on eigenfunctions?

Question

I recently came across a statement of Mercer's theorem in Hermann Koenig's book: Eigenvalue distribution of compact operators. It is interesting that in addition to the usual statement of Mercer's theorem (uniform convergence of kernel in basis of eigenfunctions for continuous kernels on bounded domains) it states that the eigenfunctions are uniformly bounded in the supremum norm (with eigenfunctions normalized in the two norm). His proof does not address this point directly. Does anybody know if this is true and if so how to show a uniform bound on the eigenfunctions $f_n(x)$ over both space $x$ and the index $n$?

Answer 1

This is not true as stated. For example, on compact (connected) Lie groups, or homogeneous spaces for them, such as spheres, the ratio of sup-norm to $L^2$-norm on an eigenspace for Laplacian/Casimir is proportional to the square root of the dimension of the eigenspace. (The standard argument for spherical harmonics, as given, e.g., in Stein-Weiss, generalizes.) Thus, when the multiplicities grow, as they do, polynomially, the ratio grows, to say the least.

In some relatively exotic situations, sharp understanding of such a comparison would prove very serious things, like the Lindelof Hypothesis on zeta and L-functions. :)

Hormander, Seeger-Sogge, and others have made extensive studies of the Laplacian on Riemannian manifolds... On compact Riemannian manifolds, the resolvent is compact, so fits into the question here.

Nevertheless, certainly, one hopes to understand such relationships!

Answer 2

It doesn't seem true to me. Take an orthonormal sequence of continuous functions $\{\phi_n\}_{n\in\mathbb{N}}$ on $[0,1]$ with diverging $L^\infty$ norm. For instance $$\phi_n:=2^{n/2}f(2^n x)$$

with a continuous function $f$ , with $\|f\|_ 2=1 $, and $\operatorname{supp}(f)\subset (1/2,1]$ (so the $\phi_n$ have disjoint supports). They are eigenfunctions of a continuous, symmetric, non-negative kernel, $$K(x,y):=\sum_{n=0}^\infty 3^{-n}\phi_n(x)\phi_n(y)\, . $$

(edit) I don't know if I was clear enough. It's just this:

1. Any orthonormal family of continuous functions on $[a,b]$ can be the family of eigenfunctions of a compact operator with a kernel satisfying the hypotheses of Mercer's theorem (just choose correspondingly the family of positive eigenvalues decaying to zero fast enough.)

2. There are orthonormal families of continuous functions with unbounded uniform norm (for instance, a sequence of continuous functions with unit $L^2$ norm and disjoint support in $[a,b]$).