*Wissenschaftliches Programm* † *Liste der Vortragenden*

Sektion 4
Montag, 18.09.2000, 14.30–14.50 Uhr, POT 251

Envelopes in function spaces

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/p-->L oo if, and only if, 0 < p <  oo , 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 fX; we derive further properties and give some examples. The pair EG(X) = (  X      )
 EG (t),uX is called growth envelope of X, where uX, 0 < uX <  oo , is the smallest number satisfying

(                   )
   integral e[ *   ]v         1/v
      f-(t)-  m (dt)     <  c ||f |X ||
      EXG (t)    G
  0
for some c > 0 and all f  (- X, and mG is the Borel measure associated with - log EGX. One verifies for the Lorentz spaces EG(Lpq) = (t-1/p, q), but we also obtain characterisations for spaces of type Bp, qs, where n(1
p - 1)+ < s < n
p.
Instead of investigating the growth of functions one can also focus on their smoothness, i.e. when X-->C it makes sense to replace f*(t) by w(f,tt)-, where w(f, t) is the modulus of continuity. Now the continuity envelope function ECX and the continuity envelope EC are introduced completely parallel to EGX and EG, 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.