An h-space is a compact set with respect to a quasi-metric and endowed with a Borel measure such that the measure of a ball of radius r is equivalent to h(r), for some function h. Applying an approach introduced by Triebel in [28] we define Besov spaces of generalised smoothness on h-spaces. We describe the techniques and tools used in this construction, namely snowflaked transforms and charts. This approach relies on using what is known for function spaces on some fractal sets, which are themselves defined as traces of convenient function spaces on ℝⁿ. It has turned out to be important to obtain new properties and characterisations for the elements of these spaces, for example, to guarantee the independence of the charts used. So we also present results for Besov spaces of generalised smoothness on ℝⁿ and some special fractal sets, namely characterisations by differences and a homogeneity property (on ℝⁿ) and non-smooth atomic decompositions.
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm460-0-1,
author = {Ant\'onio M. Caetano and Sofia Lopes},
title = {Homogeneity, non-smooth atoms and Besov spaces of generalised smoothness on quasi-metric spaces},
series = {GDML\_Books},
year = {2009},
zbl = {1172.46019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm460-0-1}
}
António M. Caetano; Sofia Lopes. Homogeneity, non-smooth atoms and Besov spaces of generalised smoothness on quasi-metric spaces. GDML_Books (2009), http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm460-0-1/