Singular distributions, dimension of support, and symmetry of Fourier transform

Kozma, Gady; Olevskiĭ, Alexander

Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1205-1226.

On étudie la “symétrie de Fourier” des mesures et des distributions sur le cercle en rapport avec la dimension de leurs supports. Les résultats essentiels du présent travail sont les suivants  :

(i) L’extension unilatérale du théorème de Frostman qui met en rapport la vitesse de décroissance de la transformation de Fourier d’une distribution et la dimension de Hausdorf de son support.

(ii) La construction des compacts d’une taille “critique” qui peut supporter des distributions (voire des pseudo-fonctions) avec une partie anti-analytique appartenant à $l^2$.

On donne également quelques exemples de l’asymétrie qui peut se produire pour des mesures à “petit” support. Plusieurs questions ouvertes sont formulées.

We study the “Fourier symmetry” of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are:

(i) A one-side extension of Frostman’s theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of its support;

(ii) A construction of compacts of “critical” size, which support distributions (even pseudo-functions) with anti-analytic part belonging to $l^2$.

We also give examples of non-symmetry which may occur for measures with “small” support. A number of open questions are stated.

Classification : 42A63, 42A50, 42A20, 28A80
Keywords: Hausorff dimension, Frostman’s theorem, Fourier symmetry
Mot clés : Dimension de Hausdorff, Théorème de Frostman, Symétrie de Fourier.

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