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Weighted local Weyl laws for elliptic operators
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 2, pp. 423-490.

Let A be an elliptic pseudo-differential operator of order m on a closed manifold 𝒳 of dimension n>0, self-ajdoint with respect to some positive smooth density 𝒳 . Then, the spectrum of A is made up of a sequence of eigenvalues (λ k ) k1 whose corresponding (orthogonal) eigenfunctions (e k ) k1 are C . Fix s and define the following integral kernel on 𝒳

K L s (x,y)= 0<λ k L λ k -s e k (x)e k (y) ¯.

We derive asymptotic formulae near the diagonal for the kernels K L s (x,y) when L+ with fixed s. For s=0, K L 0 is the kernel of the spectral projector of A on the energy levels ]0,L], studied by Hörmander in [11]. In the present work we build on Hörmander’s result to study the kernels K L s for s fixed. If s<n m, uniformly in x𝒳, K L s (x,x)L -s+n/m and, at distance L -1/m around the diagonal, the rescaled leading term behaves like the Fourier transform of an explicit function of the symbol of A. If s=n m, under some explicit generic condition on the principal symbol of A, which holds if A is a differential operator, the integral kernel has a logarithmic divergence near the diagonal smoothed at scale L -1/m , so that on the diagonal it is pointwise of order ln(L). Our results also hold when A is an elliptic differential operator on a compact open subset of n and Dirichlet boundary conditions are imposed on the e k .

Published online:
DOI: 10.5802/afst.1699
Alejandro Rivera 1

1 Univ. Grenoble Alpes, UMR5582, Institut Fourier, 38000 Grenoble, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Alejandro Rivera},
     title = {Weighted local {Weyl} laws for elliptic operators},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {423--490},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 31},
     number = {2},
     year = {2022},
     doi = {10.5802/afst.1699},
     zbl = {07549945},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1699/}
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Alejandro Rivera. Weighted local Weyl laws for elliptic operators. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 2, pp. 423-490. doi : 10.5802/afst.1699. https://afst.centre-mersenne.org/articles/10.5802/afst.1699/

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