Two-parameter estimates for joint spectral projections on complex spheres
Let \(S^{2n-1}\) be the unit sphere in \(\mathbb{C}^n\). Denote by \(\mathcal{H}^{\ell, \ell'}\) the space of restrictions to \(S^{2n-1}\) of harmonic polynomials \(p(z, \bar{z}) = p(z_1, \dots ,z_n, \bar{z_1}, \dots \bar{z_n})\) of homogeneous degree \(\ell\) in \(z_1, \dots, z_n\) and of homogeneous degree \(\ell'\) in \((\bar{z_1}, \dots, \bar{z_n})\). The Hilbert space \(L^2(S^{2n-1})\), endowed with the inner product \((f, g) : = \int_{S^{2n-1}} f( \xi) \overline{g(\xi)} d\sigma(\xi)\), where \(d\sigma\) is the measure invariant under the action of the unitary group \(U(n)\), turns out to be the direct sum of the pairwise orthogonal spaces \(\mathcal{H}^{\ell, \ell'}, \ell, \ell' \geq 0\). This decomposition is the joint spectral decomposition with respect to the Laplace-Beltrami operator and the Kohn Laplacian on \(S^{2n-1}\). Let \(\pi_{\ell, \ell'}\) be the orthogonal projector onto \(\mathcal{H}^{\ell, \ell'}\) and let \(1 \leq p \leq 2\). The main result of this paper is that sharp bounds are given for \(\| \pi_{\ell, \ell'} \|_{(p,2)}\) when \(0 \leq \ell' \leq \epsilon_0 \ell\) and \(\epsilon_0 > 0\). This result complements the main result of the author's paper [Can. J. Math, 55, 1134--1154 (2003; Zbl 1039.43013)]. As an application of the main result of this paper the author is able to give a new proof of some estimates obtained by \textit{H. Koch} and \textit{F. Ricci} [Stud. Math. 180, 103--110 (2007; Zbl 1120.43005)] for joint spectral projections on the reduced Heisenberg group \(h^1\).
scientific article; zbMATH DE number 5532446
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