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Norms of Complex Harmonic Projection Operators
Published online by Cambridge University Press: 20 November 2018
Abstract
In this paper we estimate the $\left( {{L}^{p}}-{{L}^{2}} \right)$-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
${{\pi }_{\ell0}}$ 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
${{\mathbb{R}}^{2n}}$.
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- Research Article
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- Copyright © Canadian Mathematical Society 2003
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