Weyl's law and estimates of eigenvalues for fractional sub-Laplacian operators on Carnot groups

This paper investigates the Dirichlet eigenvalue problem for fractional sub-Laplacian ( − △ G ) s on a bounded open domain Ω within a homogeneous Carnot group G = ( R n , ∘ ) . Let λ k ( s ) denote the k -th Dirichlet eigenvalue. Employing the abstract theory of Dirichlet forms and heat semigroups, we construct the associated Dirichlet heat kernel and establish explicit estimates for its trace. As a primary application, these estimates are used to prove a Weyl-type asymptotic formula for λ k ( s ) . Moreover, under a quantitative boundary regularity condition on Ω, we derive an estimate for the remainder term in the trace of Dirichlet heat kernel. We then present explicit lower and upper bounds for λ k ( s ) , which exhibit the optimal growth order with respect to both the index k and the homogeneous dimension Q . Finally, a monotonicity result for the Dirichlet eigenvalues is established, showing that the map s ↦ λ k ( s ) 1 s is increasing on the interval ( 0 , 1 ] .