Let $f :S^{n-1}\longrightarrow R_+$ , $P_{j,k}. j,k \in N$ be a spherical harmonics (http://en.wikipedia.org/wiki/Spherical_harmonics)

and $\displaystyle{f\left(\frac{x}{|x|}\right)|x|^{1-\frac n2}=\sum_{j,k}c_{j,k}P_{j,k}\left(\frac{x}{|x|}\right)|x|^{1-\frac n2}}$, where $c_{j,k}$ are some coefficients. Consider $$ I=I_{\frac n2}\left(f\left(\frac{x}{|x|}\right)|x|^{1-\frac n2}\right)(\xi), $$ where $I_{\alpha}$ is a Riesz potential (http://en.wikipedia.org/wiki/Riesz_potential).

Is it true that $$ I=\sum_{j,k}\frac{c_{j,k}}{(j^2+k^2)^{n/4}}P_{j,k}\left(\frac{x}{|x|}\right)|x|^{1-\frac n2}? $$

Any help or nice reference would be very appreciated.

Thank you.

  • $\begingroup$ I think the sum is unnecessary here: you are asking whether $I_{n/2} (|x|^{1-n/2}P_{j,k}) = a_{j,k} |x|^{1-n/2}P_{j,k}$ where $a_{j,k}=(j^2+k^2)^{-n/4}$. In other words, whether spherical harmonics, extended with homogeneity degree $1-n/2$, are eigenvectors for $I_{n/2}$. $\endgroup$ – 40 votes Jul 11 '13 at 20:44

The homogeneity is not correct: $I$ would have to be homogeneous of degree +1. Also it's not clear what exactly you mean by $P_{j,k}$ (on the Wikipedia page you reference, it refers to an associated Legendre polynomial). In general, if $Y_\ell$ is a homogeneous harmonic polynomial of degree $\ell$, one would expect $$I_{\frac{n}{2}} \left( |\cdot|^{1-\frac{n}{2}} Y_\ell \left(\frac{\cdot}{|\cdot|} \right) \right)(x) = c_{n,\ell} Y_\ell \left(\frac{x}{|x|} \right) |x|$$ when both sides are interpreted in the Fourier sense. The easiest way to find the exact constant would be to look up formulas for Fourier transforms of homogeneous distributions in some place like Grafakos. I would say that it's a reasonable guess that the constant will scale like $\ell^{-\frac{n}{2}}$ as $\ell \rightarrow \infty$.


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