Unboundedness properties of smoothness Morrey spaces of regular distributions on domains (Q681912)

Let \(X\) be a quasi-Banach space on the domain \(\Omega\) in \(\mathbb R^n\) consisting entirely of regular distributions. Then \[ E^X_G (t) = \sup \big\{ f^* (t): \| f \mid X \| \leq 1 \big\}, \quad t>0, \] is called the growth envelope function. Here, \(f^* (t)\) is the usual rearrangement of \(f \in X\). If \(X\) is not embedded in \(L_\infty (\Omega)\), then the growth envelope function describes the possible unboundedness of the elements of \(X\). The corresponding theory for the well-known spaces \(B^s_{p,q} (\Omega)\) and \(F^s_{p,q} (\Omega)\), which are based on \(L_p\)-spaces, has been developed in detail, mainly by the first-named author. The paper deals with the corresponding theory where the basic Lebesgue spaces \(L_p (\Omega)\) are replaced by the Morrey spaces \(M_{u,p} (\Omega)\), resulting in related smoothness Morrey spaces. It turns out that a substantial theory can only be expected if the domain \(\Omega\) is bounded. Applications to approximation numbers for related compact embeddings are given.