Rough spectral asymptotics for commutators of general singular integrals

Abstract: Recently, Schatten class membership of commutators arising from Connes' quantised calculus has been characterised in a general framework, which covers multiple concrete situations of interest. In contrast to this, related results dealing with the asymptotic behaviour of the singular values of these commutators have been restricted to a relatively short list of specific examples only. In this work, we show that a version of the spectral asymptotic formula, involving comparable lower and upper limits in place of an exact limit, remains valid in great generality of singular integrals over metric measure spaces. A key proof ingredient of independent interest is a new asymptotic version of the Marcinkiewicz interpolation theorem. We also present a new approach to commutator upper bounds at the critical index; while not quite as general as more elaborate methods, it is general enough to reproduce the known upper bounds in Heisenberg and Carnot groups in a much simpler way.