Morrey smoothness spaces: A new approach

• Published in: Science China. Mathematics 2023-06, Vol.66 (6), p.1301-1358
• Main Authors: Haroske, Dorothee D., Triebel, Hans
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Mathematics; Mathematics and Statistics; Parameters; Smoothness
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-021-1960-0

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 (sk p,q/s }(ℝ n ) with 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 u , p , q s ( R n ) with A ∈ { N , E } and u ⩾ p , and A p,q s,τ (ℝ n ) 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 p , s )-diagram characterising distinguished properties of the spaces A p,q s (ℝ n ) 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 ).