Envelopes in function spaces - a first approach.

by    D.D. Haroske

Preprint series: 01-09, Analysis

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

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}$.

Upload: 2001-08-28

Update: 2005 -11 -22

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