*Wissenschaftliches Programm* *Liste der Vortragenden*

**Sektion 4**

Montag, 18.09.2000,
14.30–14.50 Uhr, POT 251

#### Dorothee D. Haroske, Friedrich-Schiller-Universität Jena

We present our recently developed concept of envelopes in function spaces – a relatively simple tool for
the study of rather complicated spaces, say, of Besov type B_{p, q}^{s}, in ‘limiting’ situations. It is, for
instance, well-known that B_{p, q}^{n/p}L_{} if, and only if, 0 < p < , 0 < q __<__ 1 – but what can be said
about the growth of functions f B_{p, q}^{n/p} otherwise, i.e. when B_{p, q}^{n/p} contains essentially
unbounded functions ? Edmunds and Triebel proved that one can characterize such spaces by sharp
inequalities involving the non-increasing rearrangement f^{*} of a function f. This led us to the introduction
of the growth envelope function of a function space X, E_{G}^{X}(t) := sup _{||f|X||<1} f^{*}(t), 0 < t < 1. It
turns out that in rearrangement-invariant spaces there is a connection between E_{G}^{X} and the
fundamental function _{X}; we derive further properties and give some examples. The pair
_{G}(X) = is called growth envelope of X, where u_{X}, 0 < u_{X} __<__ , is the smallest number
satisfying

for
some c > 0 and all f X, and _{G} is the Borel measure associated with - log E_{G}^{X}. One verifies for the
Lorentz spaces _{G} = (t^{-1/p}, q), but we also obtain characterisations for spaces of type B_{p, q}^{s},
where n( - 1)_{+} < s __<__ .

Instead of investigating the growth of functions one can also focus on their smoothness, i.e. when XC
it makes sense to replace f^{*}(t) by , where (f, t) is the modulus of continuity. Now the continuity
envelope function E_{C}^{X} and the continuity envelope _{C} are introduced completely parallel to E_{G}^{X} and
_{G}, respectively, and similar questions are studied.

Naturally these sharp assertions imply a lot of interesting (new) inequalities; in our opinion, however, the
essential advantage of this new approach rather results from its simplicity when establishing (so far final)
answers to relatively difficult questions.