Consider the n-dimensional $n$-dimensional sphere S^n. $S^n$. I'm especially interested in the n=4 $n=4$ case. The Hilbert space L^2(S^n) $L^2(S^n)$ 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 ->
$$f: S^n \to S^n$$
s.t. for some lambda, $\lambda, \mu > 0 0$ if psi $\psi$ is an eigenvector of the Laplacian with eigenvalue alpha $\alpha < lambda \lambda$ then f(psi) $f(\psi)$ is a sum of eigenvectors with eigenvalues $< mu.\mu$.
Do such f $f$ exist? If so, is it possibly to classify them?