Morrey smoothness spaces: A new approach

Abstract

In the recent years, the so-called Morrey smoothness spaces attracted a lot of interest. They can (also) be understood as generalisations of the classical spaces A(skp,q/s}(ℝⁿ) with A ∈{B,F} in ℝⁿ, 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 $\cal{A}_{u,p,q}^{s}(\mathbb{R}^{n})$ with $\cal{A}\in\{\mathcal{N},\cal{E}\}$ and u ⩾ p, and A ^s,τ_p,q (ℝⁿ) with A ∈ {B, F} and τ ⩾ 0.

We reorganise these two prominent types of Morrey smoothness spaces by adding to (s,p, q) the so-called slope parameter ϱ, preferably (but not exclusively) with −n ⩽ ϱ < 0. It comes out that ∣ϱ∣ replaces n, and min(∣ϱ∣, 1) replaces 1 in slopes of (broken) lines in the ($1\over p$, s)-diagram characterising distinguished properties of the spaces A ^s_p,q (ℝⁿ) and their Morrey counterparts. Special attention will be paid to low-slope spaces with −1 < ϱ < 0, where the 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 which are quite recent). On the other hand, we establish on this basis new properties, a few of which become visible only in the context of the offered new approach, governed, now, by the four parameters (s, p, q, ϱ).