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 ${\mathcal M}_{\varphi,p}({\mathbb R}^d)$ and generalised Besov-Morrey spaces ${\mathcal N}^{s}_{\varphi,p,q}({\mathbb R}^d)$ 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 $\varphi(t)$ and $t^{-d/p} \varphi(t)$ as $t \to 0^+$ and $t\to\infty$, respectively. For the generalised Besov-Morrey spaces the limit of $t^{-d/p} \varphi(t)$ as $t \to 0^+$ also plays a r\^ole 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 ${\mathcal N}^{s}_{\varphi,p,q}({\mathbb R}^d)$ consists of essentially bounded functions only, and when it contains regular distributions only.