On the analysis of anisotropic smoothness

We investigate anisotropic function spaces defined over the multi-level ellipsoid covers of RnRn, where the ellipsoids can quickly change shape from point to point and from level to level. We explicitly define an anisotropic modulus of smoothness (already used implicitly in Dahmen et al. (2010) [4]) and investigate its properties. We show anisotropic variants of classic inequalities such as the Marchaud, Nikolskii and Ul’yanov, relationships with isotropic smoothness and applications to anisotropic Besov space embedding.