Fourier coefficients of restrictions of eigenfunctions

Let {ej} be an orthonormal basis of Laplace eigenfunctions a compact Riemannian manifold (M, g). H ⊂ M submanifold and {ψk} with the induced metric. We obtain joint asymptotics for Fourier coefficients $${\left\langle {{\gamma _H}{e_j},{\psi _k}} \right\rangle _{{L^2}(H)}} = {\int_H {{e_j}\overline \psi } _k}d{V_H}$$ restrictions γhej ej to H. In particular, we sums norm-squares over spectrum $$\left\{ {({\mu _k},{\lambda _j})} \right\}_{j,k - 0}^\infty $$ (square roots the) Laplacian ΔM on ΔH in family suitably ‘thick’ regions ℝ2. Thick include (1) truncated cone μk/λj ∈ [a, b] (0, 1) λj ≼ λ, (2) slowly thickening strip ∣μk − cλj∣ w(λ) ⩼ where is monotonic 1 ≪ ≾ λ1/2. Key tools obtaining composition calculus integral operators new multidimensional Tauberian theorem.