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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?

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# Conformal transformations and harmonic analysis on the sphere

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 -> 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?