On temperate distributions decaying at infinity

We describe classes of temperate distributions with prescribed decay properties at infinity. The definition of the elements of such classes is given in terms of the Schwartz’s bounded distributions, and we discuss their characterization in terms of convolution and of decomposition as a finite sum of derivatives of suitable functions. We also prove mapping properties under the action of a class of Fourier integral operators, with inhomogeneous phase function and polynomially bounded symbol globally defined on R^d.