Morrey spaces on domains: different approaches and growth envelopes (Q1651350)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\), satisfying some regularity (avoiding interior cusps). The authors compare three types of Morrey spaces \(M_{u,p} (\Omega)\), \(0<p\leq u<\infty\): restriction of \(M_{u,p} (\mathbb{R}^n)\) to \(\Omega\) and two intrinsic modifications. It is the first aim of this paper to compare these different approaches. Furthermore, the authors study the so-called growth envelope, consisting of the growth envelope function \[ \sup \{ f^* (t): \;\| f \, | X \| \leq 1 \}, \qquad 0<t<\epsilon, \] where \(f^* (t)\) is the usual rearrangement of the function \(f\) belonging to the space \(X\) (one of the three types of Morrey spaces considered), and an additional fine index.