Mixed modulation spaces and their application to pseudodifferential operators

This paper uses frame techniques to characterize the Schatten class properties of integral operators. The main result shows that if the coefficients { 〈 k , Φ m , n 〉 } of certain frame expansions of the kernel k of an integral operator are in ℓ 2 , p , then the operator is Schatten p-class. As a corollary, we conclude that if the kernel or Kohn–Nirenberg symbol of a pseudodifferential operator lies in a particular mixed modulation space, then the operator is Schatten p-class. Our corollary improves existing Schatten class results for pseudodifferential operators and the corollary is sharp in the sense that larger mixed modulation spaces yield operators that are not Schatten class.