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 ${\left({\lambda }_{k}\right)}_{k\ge 1}$ whose corresponding (orthogonal) eigenfunctions ${\left({e}_{k}\right)}_{k\ge 1}$ are ${C}^{\infty }$. Fix $s\in ℝ$ and define the following integral kernel on $𝒳$

 ${K}_{L}^{s}\left(x,y\right)=\sum _{0<{\lambda }_{k}\le L}{\lambda }_{k}^{-s}{e}_{k}\left(x\right)\overline{{e}_{k}\left(y\right)}\phantom{\rule{0.166667em}{0ex}}.$

We derive asymptotic formulae near the diagonal for the kernels ${K}_{L}^{s}\left(x,y\right)$ when $L\to +\infty$ with fixed $s$. For $s=0$, ${K}_{L}^{0}$ is the kernel of the spectral projector of $A$ on the energy levels $\right]0,L\right]$, studied by Hörmander in [11]. In the present work we build on Hörmander’s result to study the kernels ${K}_{L}^{s}$ for $s\in ℝ$ fixed. If $s<\frac{n}{m}$, uniformly in $x\in 𝒳$, ${K}_{L}^{s}\left(x,x\right)\asymp {L}^{-s+n/m}$ 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 $s=\frac{n}{m}$, 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 $ln\left(L\right)$. Our results also hold when $A$ is an elliptic differential operator on a compact open subset of ${ℝ}^{n}$ and Dirichlet boundary conditions are imposed on the ${e}_{k}$.

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

1 Univ. Grenoble Alpes, UMR5582, Institut Fourier, 38000 Grenoble, France
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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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