Investigation of smooth functions and analytic sets using fractal dimensions

D'Aniello, Emma

Bollettino dell'Unione Matematica Italiana, Tome 7-A (2004), p. 637-646 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We start from the following problem: given a function $f \colon [0, 1] \to [0, 1]$ what can be said about the set of points in the range where level sets are «big» according to an opportune definition. This yields the necessity of an analysis of the structure of level sets of $C^{n}$ functions. We investigate the analogous problem for $C^{n, a}$ functions. These are in a certain way intermediate between $C^{n}$ and $C^{n+1}$ functions. The results involve a mixture of Real Analysis, Geometric Measure Theory and Classical Descriptive Set Theory.

Résumé

Si parte dal seguente problema: data una funzione $f \colon [0, 1] \to [0, 1]$, cosa si può dire riguardo l'insieme dei punti nel codominio in cui gli insiemi di livello sono grandi secondo una opportuna definizione. Ciò porta alla necessità di analizzare la struttura degli insiemi di livello per funzioni di classe $C^{n}$. Analogo problema viene affrontato per le funzioni di classe $C^{n, a}$ che sono in un certo senso intermedie fra quelle di classe $C^{n}$ e quelle di classe $C^{n+1}$. I risultati coinvolgono strumenti di analisi reale, teoria geometrica della misura e teoria descrittiva classica degli insiemi.

Détail

Publié le : 2004-10-01

MR 2101655 | Zbl 1178.26005

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