Annals of Global Analysis and Geometry

, Volume 26, Issue 3, pp 231–252 | Cite as

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 ∂ x α y βe(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.

Laplace–Beltrami operator spectral function unit spectral projection operator Sobolev norms of eigenfunctions 


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© Kluwer Academic Publishers 2004

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  • Xu Bin

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