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?