Envelopes in function spaces - a first approach.

Preprint series: 01-09, Analysis

The paper is published: Jenaer Schriften zur Mathematik und Informatik, Math/Inf/16/01, 2001

MSC:
46E35 Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}
26A16 Lipschitz (Hölder) classes
46E30 Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Abstract: 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 $B^s_{p,q}$, in `{\em limiting}' situations. It is, for instance, well-known that $B^{1+n/p}_{p,q} \hookrightarrow L_\infty$ if, and only if, $0<p<\infty$, $0<q\leq 1$ -- but what can be said about the growth of functions $f\in B^{1+n/p}_{p,q}$ otherwise, i.e. when $B^{1+n/p}_{p,q}$ contains essentially unbounded functions ?
We introduce the {\em growth envelope function} of a function space $X$, ${\mathcal E}^X_{\mathsf G}(t) := \sup\left\{ f^\ast(t) : \|f|X\| \leq 1\right\}$, $0<t<1$. It turns out that in rearrangement-invariant spaces there is a connection between ${mathcal E}^X_{\mathsf G}$ and the
fundamental function $\varphi_X$; we derive further properties and give some examples. The pair ${\mathfrak E}_{\mathsf G}( X ) = \left({\mathcal E}_{\mathsf G}^X(t), u_X\right)$ is called {\em growth envelope} of $X$, where $u_X \in (0,\infty]$, is some additional index providing
an even finer description of the unboundedness of functions belonging to the space $X$; one verifies, for instance, ${\mathfrak E}_{\mathsf G}( L_{p,q} ) =(t^{-1/p}, q )$, but we also obtain characterisations for spaces of type $B^s_{p,q}$, when
$n (\frac1p-1)_+ \leq s \leq \frac{n}{p}$.\\
Instead of investigating the growth of functions one can also focus on their smoothness, i.e. for $X\hookrightarrow C$ it makes sense to replace $f^\ast(t)$ by $\frac{\omega(f,t)}{t}$, where $\omega(f,t)$ is the modulus of continuity. Now the {\em continuity envelope function}
${\mathcal E}_{\mathsf C}^X$ and the continuity envelope ${\mathfrak E}_{\mathsf C}$ are introduced completely parallel to ${\mathcal E}_{\mathsf G}^X$ and ${\mathfrak E}_{\mathsf G}$, respectively, and similar questions are studied.
Finally we look at envelope functions from various points of view and discover some interesting interplay between ${\mathcal E}_{\mathsf G}^X$ and ${\mathcal E}^X_{\mathsf C}$.