Lp norms of eigenfunctions and conformal mappings

In France we have a very useful saying “le ridicule ne tue pas” (being ridiculous does not kill), so I’ll attempt to clear my mind on a few topics here, and come what may… 🙂

The first thing that puzzles me is an answer of Terry Tao on his blog to a silly question of mine about eigenfunctions of the Dirichlet Laplacian on a general manifold \Omega. He said that L^p(\Omega) norms describe the distributions of the eigenfunctions and that their understanding is still far from satisfactory.

Let’s pick an eigenfunction \phi_k. I didn’t see at all how a countable sequence of positive numbers, namely the N_{k,p}:=\parallel \phi_k \parallel_p, were going to tell us anything about \phi_k(x) for any x\in\Omega. Could one reconstruct \phi_k from (N_{k,p})_{p\in\mathbb{N}} ? (I heard in algebraic geometry people define objects as the union over all rings of zeros of a given polynomial, maybe there’s a similarity here, we look at \phi_k in all the L^p(\Omega). But I digress…)

So I started digging the arXiv and found a few papers.

A first one is by Zelditch and Toth here which deals with the compact completely integrable case. It looks quite readable, I’ll try in a few months. Their main result (which is sharp in its setting) is:

Theorem (Toth-Zelditch): Let (M,g) be a Riemannian manifold with completely integrable geodesic flow satisfying Eliasson’s non-degeneracy condition. Then, unless (M,g) is a flat torus, for every \epsilon >0 there exists a sequence of eigenfunctions satisfying \parallel \varphi_k\parallel_{\infty}\geq C(\epsilon)\lambda_k^{\frac{1}{4}-\epsilon} and \parallel \varphi_k\parallel_{p}\geq C(\epsilon)\lambda_k^{\frac{p-2}{4p}-\epsilon} (for p=2,3,…).

So this partially answers my question: the norms of the eigenfunctions blow up as a power of the corresponding eigenvalue. But again, what does this tell us on the distribution of the eigenfunctions, other that saying they somehow blow-up ? It could happen in so many ways, no ?

Another paper
is by Schiffman, Tate and Zelditch and deals with some completely integrable systems in complex geometry. The results they obtain there are finer, since do speak of the eigenfunctions themselves by obtaining pointwise estimates. I’m not knowledgeable in complex geometry, but the result that the eigenfunctions behave like gaussians centered around a classical torus was certainly known by physists for a long time, so this is all very sensible and interesting.

The second thing I’d like to understand is related to separation of variables, more precisely to Tao’s question in one of his comments:

    one can phrase this problem as a question of determining how stable the method of separation of variables is in solving PDE. This method trivially lets us determine the eigenfunctions of a rectangle; why shouldn’t some variant of it still give some control on eigenfunctions of a near-rectangle?

This is what leads me to conformal mappings. Riemann’s theorem implies that there exists a conformal mapping f_r of the rectangle to the unit disk, and also a conformal mapping of any near-rectangle \widetilde{f}_r to the unit disk. I’m not a friend of complex analysis, but provided \widetilde{f}_r can be inverted, the map \widetilde{f}_r^{-1}\circ f_r maps the rectangle’s nice separating coordinate system to that of the near-rectangle.

So the question is what does the Laplacian become under this transformation, i.e. are there nasty terms coming in ? Comments welcomed.


2 Responses to “Lp norms of eigenfunctions and conformal mappings”

  1. Terence Tao Says:

    Dear Thomas,

    You are of course welcome to ask questions on my own blog – I only found your question here by accident :).

    The L^p norms of a function u (which, incidentally, can be defined for all real p, not just integer p) are closely connected to the distribution function \lambda(\alpha) := |\{ x: |u(x)| \ge \alpha \}| of u (a concept analogous to the cumulative distribution function in probability theory, by the identity
    \|u\|_{L^p}^p = \int_0^\infty \lambda(\alpha)\ p \alpha^{p-1} d\alpha. Also, tools such as Chebyshev’s inequality can be used to link bounds on L^p norms to bounds on the distribution function. In more fuzzy terms, L^p norms control the “height” or “width” of functions; see my lecture notes on this topic.

    If you apply a conformal transformation in two dimensions, then the Laplacian \Delta gets mapped to e^{-2\omega} \Delta where e^{-2\omega} is the conformal factor. So harmonic functions get mapped to harmonic functions, but eigenfunctions \Delta u= \lambda u get mapped to weighted eigenfunctions \Delta u = e^{2\omega}  \lambda u. It’s not clear that this is an improvement; certainly, the classical billiard ball dynamics become rather weird-looking after conformal transformation, so I would expect the quantum dynamics to do so too.

  2. thomas1111 Says:

    Dear Sir,

    thank you so very much for your reply! I didn’t dare to ask on your blog, first not to bother you and also because it must be read by thousands of experts of all sorts — including you obviously, maybe there are limits to this french saying after all. 😉

    Oh I understand the distribution thing now, yes I’ll read your lectures it looks perfect…

    As for the conformal transformation you’re right that it completely changes the classical dynamics, sigh.

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