Multilinear estimates for the Laplace spectral projectors on compact manifolds.
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 \[ \| H_pH_q\| _{L^2}\leq C\Lambda(m,\min(p,q))\| H_p\| _{L^2}\| H_q\| _{L^2}. \]
scientific article; zbMATH DE number 2057114
scientific article