Derivatives of the Spectral Function and Sobolev Norms of Eigenfunctions on a Closed Riemannian Manifold

Let e(x, y, λ) be the spectral function and χλ the unit spectral projection operator, with respect to the Laplace–Beltrami operator on a closed Riemannian manifold M. We generalize the one-term asymptotic expansion of e(x, x, λ) by Hörmander (Acta Math.88 (1968), 341–370) to that of ∂xα∂yβe(x,y,λ)|x=y for any multiindices α, β in a sufficiently small geodesic normal coordinate chart of M. Moreover, we extend the sharp (L2,Lp) (2 ≤p≤∞) estimates of χλ by Sogge (J. Funct. Anal.77 (1988), 123–134; London Math. Soc. Lecture Note Ser. 137, Cambridge University Press, Cambridge, 1989; Vol. 1, pp. 416–422) to the sharp (L2, Sobolev Lp) estimates of χλ.