ABSTRACT
We shall need a finer characterisation than that provided by the growth envelope functions only. By Proposition 3.15 it is obvious, for instance, that EXG alone cannot distinguish between different spaces like Lp,q1 (logL)a and Lp,q2 (logL)a . So it is desirable to complement EXG by some expression, naturally belonging to EXG , but yielding – as a test – the number q (or a related quantity) in case of Lp,q(logL)a spaces. Again a more substantial justification for complementing EXG by this additional expression results from more complicated spaces (like Bsp,q and F
s p,q) than Lp,q(logL)a; but in these classical
cases the outcome can be checked immediately. The missing link is obtained by the introduction of some “characteristic” index uXG , which gives a finer measure of the (local) integrability of functions belonging to X. Moreover, the definition below is also motivated by (sharp) inequalities of type (1.4) with κ ≡ 1.
