A partial differential equation characterization of anisotropic Hardy spaces

We obtain a differential characterization for the anisotropic Hardy space HAp$H_A^p$ by identifying it with a parabolic Hardy space associated with a general continuous group. This allows HAp$H_A^p$ to be defined using a parabolic differential equation of Calderón and Torchinsky. We also provide a classification of dilations corresponding to equivalent anisotropic Hardy spaces with respect to linear transformations.