Fourier–Laplace transformation of functionals on a weighted space of infinitely smooth functions on $\mathbb R^n$

The dual of the space of infinitely smooth functions on $\mathbb R^n$ with partial derivatives satisfying certain weighted estimates is described in terms of the Fourier–Laplace transformation. An integral representation is obtained for the solutions of a homogeneous linear partial differential equation with constant coefficients that belong to this space.