Anisotropic function spaces and related semi–linear hypoelliptic equations

Looking for the best possible smoothness (in terms of the upper index of the Besov spaces) for the solution of some semi–linear equations we consider a model case of a hypoelliptic operator, which acts between anisotropic Besov spaces. To obtain the best regularity we need some properties for the corresponding spaces, which we prove here. In particular we prove Fatou, Fubini and truncation properties. We give also some characterisations of the Besov and Triebel–Lizorkin spaces.