ABSTRACT
We already mentioned in our introductory remarks that characterisations like (1.4) gave reason to study the behaviour of the non-increasing rearrangement f∗ of a function f , in particular, when these spaces contain essentially unbounded functions. This leads to the concept of growth envelopes. Our results for spaces of type Bsp,q or F sp,q are postponed to Part II; we start with some simple features and examples in order to give a better feeling for what is really “measured” by growth envelopes. For that reason we test our new envelope tool on rather classical spaces like Lorentz (-Zygmund) spaces first – before arriving at more surprising results in Part II. Moreover, there is also an interesting point at the end of this chapter: the recognition of growth envelope functions in terms of fundamental functions in rearrangement-invariant spaces. Finally, preparing some later discussion of global versus local behaviour in Section 10.3 we already add some “higher-order” and “weighted” examples in Section 3.4.
