# Fourier transforms of homogeneous distribution

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## Abstract

Fourier transforms of homogeneous distribution ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze CARLOS LEMOINE Fourier transforms of homogeneous distribution Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 26, no 1 (1972), p. 117-149. <http://www.numdam.org/item?id=ASNSP_1972_3_26_1_117_0> © Scuola Normale Superiore, Pisa, 1972, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ FOURIER TRANSFORMS OF HOMOGENEOUS DISTRIBUTION CARLOS LEMOINE Introduction. The purpose of the paper * is to study the relations between the regu- larity of a homogeneous distribution and that of its-Fourier transform ; this problem has been treated by Calderon, Zygmund and Hormander, our results are extensions of theirs. In preparing the basis for our study we obtain a characterization of the continuos linear maps, from the space of distributions in the unit sphere into itself, that commute with rotations (Chap 1). A clear presenta- tion of the spaces in a compact manifolds, is given in chap 2. CHAPTER I SPHERICAL HARMONICS Summary. 1.1. Some notations are introduced, and some fundamental facts are recalled. 1.2. The expansion of a distribution on the unit sphere in a convergent series of spherical harmonics is given and also a characterization of the continuous linear maps from D (2:) to D’ (2:) that commute with rotations. As an application we consider the operators Ja defined by ~ Pervenuto alla Redazione

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