Gelfand-Shilov Spaces, Pseudo-differential Operators and Localization Operators.

We present new results concerning pseudo-differential operators in the function spaces Sμμ(ℝn) of Gelfand and Shilov. In particular we discuss Sνμ(ℝn)-regularity of solutions to SG-elliptic pseudo-differential equations, allowing lower order semilinear perturbations. The results apply to SG-elliptic partial differential equations with polynomial coefficients. We also study the action of Weyl operators and localization operators on Sνμ(ℝn). [ABSTRACT FROM AUTHOR]