Weighted local Weyl laws for elliptic operators

Let $A$ be an elliptic pseudo-differential operator of order $m$ on a closed manifold $\mathcal{X}$ of dimension $n>0$, formally positive self-adjoint with respect to some positive smooth density $d\mu_\mathcal{X}$. Then, the spectrum of $A$ is made up of a sequence of eigenvalues $(\lambda_k)_{k\geq 1}$ whose corresponding eigenfunctions $(e_k)_{k\geq 1}$ are $C^\infty$ smooth. Fix $s\in\mathbb{R}$ and define $$K_L^s(x,y)=\sum_{0<\lambda_k\leq L}\lambda_k^{-s} e_k(x)\overline{e_k(y)}\, .$$ We derive asymptotic formulae near the diagonal for the kernels $K_L^s(x,y)$ when $L\rightarrow +\infty$ with fixed $s$. For $s=0$, $K^0_L$ is the kernel of the spectral projector studied by H\"ormander in \cite{ho68}. In the present work we build on H\"ormander's result to study the kernels $K^s_L$. If $s<\frac{n}{m}$, $K_L^s$ is of order $L^{-s+n/m}$ and near 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 kernel has order $\ln(L)$ and the leading term has a logarithmic divergence smoothed at scale $L^{-1/m}$. Our results also hold for elliptic differential Dirichlet eigenvalue problems.