Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay Multilinear estimates for the Laplace spectral projectors on compact manifolds. (English. Abridged French version) Zbl 1040.58011 C. R., Math., Acad. Sci. Paris 338, No. 5, 359-364 (2004). Let \(\Delta\) be the scalar Laplacian on a closed \(m\) dimensional Riemannian manifold. Denote by \(\chi_\lambda:=\chi(\sqrt{-\Delta}-\lambda)\) the spectral projector around \(\lambda\).The authors show there is a constant \(C\) so that for \(\lambda,\mu\geq1\) and \(f,g\in L^2(M)\), one has the estimate \[ \| \chi_\lambda f\chi_\mu g\| _{L^2}\leq C\Lambda(m,\min(\lambda,\mu))\| f\| _{L^2} \| g\| _{L^2} \] where \(\Lambda(2,\nu)=\nu^{1/4}\), \(\Lambda(3,\nu)=\nu^{1/2}\log^{1/2}(\nu)\), and \(\Lambda(m,\nu)=\nu^{(m-2)/2}\) for \(m\geq4\). The authors also establish a corresponding trilinear estimate. Applying this to spherical harmonics \(H_p\) and \(H_q\) of degrees \(p\geq1\) and \(q\geq1\) on \(S^m\) then yields \[ \| H_pH_q\| _{L^2}\leq C\Lambda(m,\min(p,q))\| H_p\| _{L^2}\| H_q\| _{L^2}. \] Reviewer: Peter B. Gilkey (Eugene) Cited in 1 ReviewCited in 8 Documents MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 33C55 Spherical harmonics Keywords:spectral projection; bilinear estimate; trilinear estimate; spherical harmonics PDFBibTeX XMLCite \textit{N. Burq} et al., C. R., Math., Acad. Sci. Paris 338, No. 5, 359--364 (2004; Zbl 1040.58011) Full Text: DOI References: [1] Burq, N.; Gérard, P.; Tzvetkov, N., Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Preprint, 2003 · Zbl 1092.35099 [2] Gallot, S.; Hulin, D.; Lafontaine, J., Riemannian Geometry. Riemannian Geometry, Universitext (1990), Springer-Verlag: Springer-Verlag Berlin · Zbl 0636.53001 [3] Hörmander, L., Oscillatory integrals and multipliers on \(FL^p \), Ark. Math., 11, 1-11 (1973) · Zbl 0254.42010 [4] Sogge, C., Oscillatory integrals and spherical harmonics, Duke Math. J., 53, 43-65 (1986) · Zbl 0636.42018 [5] Sogge, C., Concerning the \(L^p\) norm of spectral clusters for second order elliptic operators on compact manifolds, J. Funct. Anal., 77, 123-138 (1988) · Zbl 0641.46011 [6] Sogge, C., Fourier Integrals in Classical Analysis. Fourier Integrals in Classical Analysis, Cambridge Tracts in Math. (1993) · Zbl 0783.35001 [7] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Monographs Harmon. Anal., III (1993), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0821.42001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.