Weighted local Weyl laws for elliptic operators

Rivera, Alejandro

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 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 $𝒳$

$$K_L^s(x,y)=\sum _{0<{\lambda }_k\le L}{\lambda }_k^{-s}{e}_k\left(x\right)\overline{e_k\left(y\right)}.$$

We derive asymptotic formulae near the diagonal for the kernels $K_L^s(x,y)$ 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 $]0,L]$, studied by Hörmander in [11]. In the present work we build on Hörmander’s result to study the kernels $K_L^s$ for $s\in ℝ$ fixed. If $s<\frac{n}{m}$, uniformly in $x\in 𝒳$, $K_L^s(x,x)\asymp L^{-s+n/m}$ 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 $e_k$.

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