Weighted local Weyl laws for elliptic operators

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Let A be an elliptic pseudo-differential operator of order m on a closed manifold X of dimension n>0, formally positive self-adjoint with respect to some positive smooth density dμₓ. Then, the spectrum of A is made up of a sequence of eigenvalues (λₖ)ₖ≥ ₁ whose corresponding eigenfunctions (eₖ)ₖ≥ ₁ are C∞ smooth. Fix s and define Kₗˢ(x,y)=∑₀<λₖ≤ ₗλₖ⁻ˢ eₖ(x)eₖ(y). We derive asymptotic formulae near the diagonal for the kernels Kₗˢ(x,y) when L→ +∞ with fixed s. For s=0, K⁰ₗ is the kernel of the spectral projector studied by Hörmander in ho68. In the present work we build on Hörmander's result to study the kernels Kˢₗ. If s<n/m, Kₗˢ 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=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 (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.