On a bridge connecting Lebesgue and Morrey spaces in view of their growth properties

Abstract

We study unboundedness properties of functions belonging to generalised Morrey spaces ${\mathcal{M}}_{\phi ,p}({\mathbb{R}}^d)$ and generalised Besov-Morrey spaces ${\mathcal{N}}_{\phi ,p,q}^s({\mathbb{R}}^d)$ by means of growth envelopes. For the generalised Morrey spaces we arrive at the same three possible cases as for classical Morrey spaces ${\mathcal{M}}_{u,p}({\mathbb{R}}^d)$, i.e., boundedness, the $L_p$-behaviour or the proper Morrey behaviour for $p<u$, but now those cases are characterised in terms of the limit of $\phi (t)$ and $t^{-d/p}\phi (t)$ as $t\to 0^{+}$ and $t\to \infty$, respectively. For the generalised Besov-Morrey spaces the limit of $t^{-d/p}\phi (t)$ as $t\to 0^{+}$ 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 ${\mathcal{N}}_{\phi ,p,q}^s({\mathbb{R}}^d)$ consists of essentially bounded functions only, and when it contains regular distributions only.

28 pages