Localization operators and exponential weights for modulation spaces

We study localization operators within the framework of ultra-distributions. More precisely, given a symbol a and two windows phi(1), phi(2), we investigate the multilinear mapping from (a, phi(1), phi(2)) is an element of S-(1)'(R-2d) x S-(1) (R-d) x S-(1) (R-d) to the localization operator A(a)(phi 1,phi 2). Results are formulated in terms of modulation spaces with weights which may have exponential growth. We give sufficient and necessary conditions for A(a)(phi 1,phi 2) to be bounded or to belong to a Schatten class. As an application, we study symbols defined by ultra-distributions with compact support, that give trace class localization operators.