Fourier coefficients of restrictions of eigenfunctions

Let { e j } be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold ( M, g ). Let H ⊂ M be a submanifold and { ψ k } be an orthonormal basis of Laplace eigenfunctions of H with the induced metric. We obtain joint asymptotics for the Fourier coefficients 〈 γ H e j , ψ k 〉 L 2 ( H ) = ∫ H e j ψ ¯ k d V H of restrictions γh e j of e j to H . In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum { ( μ k , λ j ) } j , k − 0 ∞ of the (square roots of the) Laplacian Δ M on M and the Laplacian Δ H on H in a family of suitably ‘thick’ regions in ℝ 2 . Thick regions include (1) the truncated cone μ k / λ j ∈ [ a, b ] ⊂ (0, 1) and λ j ≼ λ , and (2) the slowly thickening strip ∣ μ k − cλ j ∣ ≼ w ( λ ) and λ j ⩼ λ , where w ( λ ) is monotonic and 1 ≪ w ( λ ) ≾ λ 1/2 . Key tools for obtaining the asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.