Morrey smoothness spaces: A new approach

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 $A^s_{p,q} (\mathbb{R}^n)$, $A\in \{B,F\}$, in $\mathbb{R}^n$, where the parameters satisfy $s\in \mathbb{R}$ (smoothness), $0<p \le \infty$ (integrability) and $0<q \le \infty$ (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 $\mathcal{A}^s_{u,p,q} (\mathbb{R}^n)$, with $\mathcal{A}\in \{\mathcal{N}, \mathcal{E}\}$, $u\geq p$, and $A^{s, \tau}_{p,q} (\mathbb{R}^n)$, with $\tau\geq 0$. We reorganise these two prominent types of Morrey smoothness spaces by adding to $(s,p,q)$ the so--called slope parameter $\varrho$, preferably (but not exclusively) with $-n \le \varrho <0$. It comes out that $|\varrho|$ replaces $n$, and $\min (|\varrho|,1)$ replaces 1 in slopes of (broken) lines in the $( \frac{1}{p}, s)$--diagram characterising distinguished properties of the spaces $A^s_{p,q} (\mathbb{R}^n)$ and their Morrey counterparts. Special attention will be paid to low--slope spaces with $-1 <\varrho <0$, where corresponding properties are quite often independent of $n\in \mathbb{N}$. 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 $(s,p,q,\varrho)$.