The class of convolution operators on the Marcinkiewicz spaces
Annales de l'Institut Fourier, Volume 31 (1981) no. 3, pp. 225-243.

Let 𝒯 X denote the operator-norm closure of the class of convolution operators Φ μ :XX where X is a suitable function space on R. Let r p be the closed subspace of regular functions in the Marinkiewicz space p , 1p<. We show that the space 𝒯 r p is isometrically isomorphic to 𝒯 L p and that strong operator sequential convergence and norm convergence in 𝒯 r p coincide. We also obtain some results concerning convolution operators under the Wiener transformation. These are to improve a Tauberian theorem of Wiener on 2 .

On considère la classe des opérateurs de convolution Φ μ :XXX est un espace convenable de fonctions sur R. Soit 𝒯 X la fermeture de cette classe dans la norme des opérateurs. Soit r p le sous-espace des fonctions régulières dans l’espace de Marcinkiewicz p , 1p<. Nous montrons que l’espace 𝒯 r p est isométriquement isomorphe à 𝒯 L p et que la convergence d’une suite d’opérateurs dans la topologie forte des opérateurs est équivalente à la convergence en norme. Nous obtenons aussi quelques résultats sur l’action de la transformation de Wiener sur les opérateurs de convolution, et comme application, nous trouvons une extension d’un théorème taubérien de Wiener.

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     title = {The class of convolution operators on the {Marcinkiewicz} spaces},
     journal = {Annales de l'Institut Fourier},
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Lau, Ka-Sing. The class of convolution operators on the Marcinkiewicz spaces. Annales de l'Institut Fourier, Volume 31 (1981) no. 3, pp. 225-243. doi : 10.5802/aif.845. http://www.numdam.org/articles/10.5802/aif.845/

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