Schatten Class Estimates for the Riesz Map of Massless Dirac Operators

Abstract

In this paper, we study the Riesz map g applied to the massless Dirac operator $$\mathcal {D}$$ on $$\mathbb {R}^d$$ , for $$d\ge 2$$ , and study its properties in terms of weak Schatten classes. Our sharp estimates, which are optimal in the scale of weak Schatten classes, show that the decay of singular values of $$g(\mathcal {D}+V)-g(\mathcal {D})$$ differs dramatically for the case when the perturbation V is a purely electric potential and the case when V is a magnetic one. The application of double operator integrals also yields a similar result for the operator $$f(\mathcal {D}+V)-f(\mathcal {D})$$ for an arbitrary monotone function f on $$\mathbb {R}$$ with derivative of Schwartz class.