Endpoint weak Schatten class estimates and trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups

Along the line of singular value estimates for commutators by Rochberg–Semmes, Lord–McDonald–Sukochev–Zanin and Fan–Lacey–Li, we establish the endpoint weak Schatten class estimate for commutators of Riesz transforms with multiplication operator M_f on Heisenberg groups via homogeneous Sobolev norm of the symbol f. The new tool we exploit is the construction of a singular trace formula on Heisenberg groups, which, together with the use of double operator integrals, allows us to bypass the use of Fourier analysis and provides a solid foundation to investigate the singular values estimates for similar commutators in general stratified Lie groups.