Propagation of anisotropic Gelfand-Shilov wave front sets

We show a result on propagation of the anisotropic Gelfand–Shilov wave front set for linear operators with Schwartz kernel which is a Gelfand–Shilov ultradistribution of Beurling type. This anisotropic wave front set is parametrized by two positive parameters relating the space and frequency variables. The anisotropic Gelfand–Shilov wave front set of the Schwartz kernel of the operator is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by a partial differential operator defined by a symbol which is a polynomial with real coefficients and order at least two.