*Wissenschaftliches Programm* † *Liste der Vortragenden*
14.30–14.50 Uhr, POT 251
Dorothee D. Haroske, Friedrich-Schiller-Universitšt 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 Bp, qs, in ‘limiting’ situations. It is, for
instance, well-known that Bp, qn/pL if, and only if, 0 < p < , 0 < q < 1 – but what can be said
about the growth of functions f Bp, qn/p otherwise, i.e. when Bp, qn/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, EGX(t) := sup ||f|X||<1 f*(t), 0 < t < 1. It
turns out that in rearrangement-invariant spaces there is a connection between EGX and the
fundamental function X; we derive further properties and give some examples. The pair
G(X) = is called growth envelope of X, where uX, 0 < uX < , is the smallest number
some c > 0 and all f X, and G is the Borel measure associated with - log EGX. One verifies for the
Lorentz spaces G = (t-1/p, q), but we also obtain characterisations for spaces of type Bp, qs,
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 ECX and the continuity envelope C are introduced completely parallel to EGX 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.