On the continuity of Fourier multipliers on the homogeneous Sobolev spaces ${\dot{W}^1_1(R^d)}$

Kazaniecki, Krystian; Wojciechowski, Michał

doi:10.5802/aif.3036

On the continuity of Fourier multipliers on the homogeneous Sobolev spaces

{\dot{W}}_{1}^{1} (R^{d})

[Sur la continuité de multiplicateurs de Fourier sur les espaces de Sobolev homogènes

{\dot{W}}_{1}^{1} (R^{d})

]

Kazaniecki, Krystian ¹ ; Wojciechowski, Michał ²

¹ University of Warsaw Institute of Mathematics ul. Banacha 2 02-097 Warszawa,(Poland)
² Polish Academy of Sciences Institute of Mathematics ul. Śniadeckich 8 00-956 Warszawa, (Poland)

Annales de l'Institut Fourier, Tome 66 (2016) no. 3, pp. 1247-1260.

Résumé
Abstract

Dans cet article, nous prouvons que chaque multiplicateur de Fourier sur l’espace homogène ${\dot{W}}_{1}^{1} (ℝ^{d})$ de Sobolev est une fonction continue. Notre théorème est une généralisation du résultat de A. Bonami et S. Poornima sur les multiplicateurs de Fourier, qui sont des fonctions homogènes de degré zéro.

In this paper we prove that every Fourier multiplier on the homogeneous Sobolev space ${\dot{W}}_{1}^{1} (ℝ^{d})$ is a continuous function. This theorem is a generalization of the result of A. Bonami and S. Poornima for Fourier multipliers, which are homogeneous functions of degree zero.

Reçu le : 2015-01-02
Révisé le : 2015-06-21
Accepté le : 2015-09-10
Publié le : 2016-12-14

DOI : 10.5802/aif.3036

Classification : 42B15, 43A22
Keywords: Fourier multipliers, Sobolev spaces, Riesz product
Mot clés : multiplicateurs de Fourier, espaces de Sobolev, Produits de Riesz

Affiliations des auteurs :

Kazaniecki, Krystian ¹ ; Wojciechowski, Michał ²

¹ University of Warsaw Institute of Mathematics ul. Banacha 2 02-097 Warszawa,(Poland)
² Polish Academy of Sciences Institute of Mathematics ul. Śniadeckich 8 00-956 Warszawa, (Poland)

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     journal = {Annales de l'Institut Fourier},
     pages = {1247--1260},
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Kazaniecki, Krystian; Wojciechowski, Michał. On the continuity of Fourier multipliers on the homogeneous Sobolev spaces ${\dot{W}^1_1(R^d)}$. Annales de l'Institut Fourier, Tome 66 (2016) no. 3, pp. 1247-1260. doi : 10.5802/aif.3036. http://www.numdam.org/articles/10.5802/aif.3036/

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