Wavelet techniques for pointwise regularity

Jaffard, Stéphane

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 1, pp. 3-33.

Soit $E$ un espace de Banach (ou un quasi-Banach) invariant par translation et dilatation (typiquement un espace de Besov ou de Sobolev homogène). Nous introduisons une définition générale de régularité ponctuelle associée à $E$, et notée $C_E^{\alpha }\left(x_0\right)$. Nous montrons comment les propriétés de $E$ se traduisent en propriétés de $C_E^{\alpha }\left(x_0\right)$. Nous donnons également des application en analyse multifractale.

Let $E$ be a Banach (or quasi-Banach) space which is shift and scaling invariant (typically a homogeneous Besov or Sobolev space). We introduce a general definition of pointwise regularity associated with $E$, and denoted by $C_E^{\alpha }\left(x_0\right)$. We show how properties of $E$ are transferred into properties of $C_E^{\alpha }\left(x_0\right)$. Applications are given in multifractal analysis.

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