Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds

We consider the spectral behavior and noncommutative geometry of commutators [P,f][P,f], where PP is an operator of order 0 with geometric origin and ff a multiplication operator by a function. When ff 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 ff, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.