We consider the spectral behavior and noncommutative geometry of commutators$[P,f]$, where$P$is an operator of order 0 with geometric origin and$f$a multiplication operator by a function. When$f$is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes’ residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Hölder continuous functions$f$, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.
An analog of the Weyl law for the Kohn Laplacian on spheres
