Envelopes in function spaces

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/pL 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