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Norms of Complex Harmonic Projection Operators

Open Access article
 Printed: Dec 2003
  • Valentina Casarino
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In this paper we estimate the $(L^p-L^2)$-norm of the complex harmonic projectors $\pi_{\ell\ell'}$, $1\le p\le 2$, uniformly with respect to the indexes $\ell,\ell'$. We provide sharp estimates both for the projectors $\pi_{\ell\ell'}$, when $\ell,\ell'$ belong to a proper angular sector in $\mathbb{N} \times \mathbb{N}$, and for the projectors $\pi_{\ell 0}$ and $\pi_{0 \ell}$. The proof is based on an extension of a complex interpolation argument by C.~Sogge. In the appendix, we prove in a direct way the uniform boundedness of a particular zonal kernel in the $L^1$ norm on the unit sphere of $\mathbb{R}^{2n}$.
MSC Classifications: 43A85, 33C55, 42B15 show english descriptions Analysis on homogeneous spaces
Spherical harmonics
43A85 - Analysis on homogeneous spaces
33C55 - Spherical harmonics
42B15 - Multipliers

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