Unboundedness properties of smoothness Morrey spaces of regular distributions on domains

Abstract

We study unboundedness properties of functions belonging to generalised Morrey spaces Mϕ,p(R d ) and generalised Besov-Morrey spaces N s ϕ,p,q (R d ) by means of growth envelopes. For the generalised Morrey spaces we arrive at the same three possible cases as for classical Morrey spaces Mu,p(R d ), i.e., boundedness, the Lp-behaviour or the proper Morrey behaviour for p < u, but now those cases are characterised in terms of the limit of ϕ(t) and t −d/p ϕ(t) as t → 0 + and t → ∞, respectively. For the generalised Besov-Morrey spaces the limit of t −d/p ϕ(t) as t → 0 + also plays a rôle and, once more, we are able to extend to this generalised spaces the known results for classical Besov-Morrey spaces, although some cases are not completely solved. In this context we can completely characterise the situation when N s ϕ,p,q (R d ) consists of essentially bounded functions only, and when it contains regular distributions only.

Compact embeddings in Besov-type and Triebel–Lizorkin-type spaces on bounded domains

“…by the same arguments as above. Conversely, according to [44,Corollary 5.2] (adapted to spaces on bounded domains), Proposition 2.7 and (2.10), we have in this case 11) such that e k (id τ ) e k id B :…”

“…The case τ = 0 is well-known, so we assume τ > 0. Note that the continuity of that embedding was studied in [11,Proposition 2.18] already, with the outcome that A s,τ p,q (Ω) → L ∞ (Ω) if, and only if,…”

Compact embeddings in Besov-type and Triebel-Lizorkin-type Spaces on bounded domains

“…If 0 ≤ τ < 1 p , then Corollaries 3.7 and 3.8 coincide for E-spaces, using (2.18). In view of our continuity result [11,Proposition 2.18] the above outcome can be reformulated for τ > 0 such that A s,τ p,q (Ω) ֒→ L ∞ (Ω) is compact if, and only if, it is bounded. This is different from the case τ = 0.…”

Generalized Besov-type and Triebel-Lizorkin-type spaces

“…In [24] Tang and Xu introduced the corresponding Triebel-Lizorkin-Morrey spaces, thanks to establishing the Morrey version of the Fefferman-Stein vector-valued inequality. We refer for references, historic details and further results to the monographs [36], the nice surveys [22,23], and also the more recent paper [9]. As mentioned above, when p = u, one regains the (classical) Besov and Triebel-Lizorkin spaces, B s p,q (R d ) and F s p,q (R d ), respectively.…”