Consider the $n$-dimensional sphere $S^n$. I'm especially interested in the $n=4$ case. The Hilbert space $L^2(S^n)$ can be decomposed into a direct sum of eigenspaces of the Laplacian, which are finite dimensional. I'm looking for non-isometric conformal transformations


$$f: S^n \to S^n$$


s.t. for some $\lambda, \mu > 0$ if $\psi$ is an eigenvector of the Laplacian with eigenvalue $\alpha < \lambda$ then $f(\psi)$ is a sum of eigenvectors with eigenvalues $< \mu$.


Do such $f$ exist? If so, is it possibly to classify them?



Conformal transformations and harmonic analysis on the sphere
The only possibility is the trivial one when $\lambda$ is so small that the only eigenfunctions with eigenvalue less than $\lambda$ are constants (eigenvalue zero). Otherwise the eigenfunctions with eigenvalue less than $\lambda$ span the space of polynomials of degree at most $d$ for some positive integer $d$, and then composition with a non-isometric conformal transformation takes it outside the space of polynomials, and thus outside the span of eigenfunctions of eigenvalue less than $\mu$ for any finite $\mu$.


P.S. In case you've not seen this yet: the group of conformal transformations is described by [this theorem of Liouville](http://en.wikipedia.org/wiki/Liouville%27s_theorem_(conformal_mappings)), identifying it with the group of Möbius transformations.


