In the recent years so-called Morrey smoothness spaces attracted a lot of interest. They can (also) be understood as generalisations of the classical spaces A s p,q (R n ), A ∈ {B, F }, in R n , where the parameters satisfy s ∈ R (smoothness), 0 < p ≤ ∞ (integrability) and 0 < q ≤ ∞ (summability). In the case of Morrey smoothness spaces additional parameters are involved. In our opinion, among the various approaches at least two scales enjoy special attention, also in view of applications: the scales A s u,p,q (R n ), with A ∈ {N , E}, u ≥ p, and A s,τ p,q (R n ), with τ ≥ 0.We reorganise these two prominent types of Morrey smoothness spaces by adding to (s, p, q) the so-called slope parameter , preferably (but not exclusively) with −n ≤ < 0. It comes out that | | replaces n, and min(| |, 1) replaces 1 in slopes of (broken) lines in the 1 p , sdiagram characterising distinguished properties of the spaces A s p,q (R n ) and their Morrey counterparts. Special attention will be paid to lowslope spaces with −1 < < 0, where corresponding properties are quite often independent of n ∈ N.Our aim is two-fold. On the one hand we reformulate some assertions already available in the literature (many of them are quite recent). On the other hand we establish on this basis new properties, a few of them became visible only in the context of the offered new approach, governed, now, by the four parameters (s, p, q, ).
