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On fractals which are not so terrible

100%
Caetano A.
Fundamenta Mathematicae
|
2002
|
tom 171
|
nr 3
249-266
EN
The notion of NST domain and the closely related notion of ball condition, both topological in nature and quite useful within the theory of function spaces, are compared with each other (and with the older concept of porosity) and also with other notions of interest, like those of d-set and of interior regular domain, which have a measure-theoretical nature. Also, after extending the idea of NST (not so terrible) to a larger class of sets, the property is studied in the context of anisotropic self-affine fractals.
2
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Homogeneity, non-smooth atoms and Besov spaces of generalised smoothness on quasi-metric spaces

68%
Caetano A., Lopes S.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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.
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Traces of Besov spaces on fractal h-sets and dichotomy results

63%
Caetano A., Haroske D.
Studia Mathematica
|
2015
|
tom 231
|
nr 2
117-147
EN
We study the existence of traces of Besov spaces on fractal h-sets Γ with a special focus on assumptions necessary for this existence; in other words, we present criteria for the non-existence of traces. In that sense our paper can be regarded as an extension of Bricchi (2004) and a continuation of Caetano (2013). Closely connected with the problem of existence of traces is the notion of dichotomy in function spaces: We can prove that-depending on the function space and the set Γ-there occurs an alternative: either the trace on Γ exists, or smooth functions compactly supported outside Γ are dense in the space. This notion was introduced by Triebel (2008) for the special case of d-sets.
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