Morneau-Guérin, Frédéric 
ORCID: https://orcid.org/0000-0001-7610-4648
(2026).
Convolution weights on discrete abelian groups. Acta Scientiarum Mathematicarum. 10.1007/s44146-026-00274-1
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| Catégorie de document : | Articles de revues |
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| Évaluation par un comité de lecture : | Oui |
| Étape de publication : | Publié |
| Résumé : | Let G be a discrete abelian group and let w be a weight on G. A classical sufficient condition for the weighted space l^p(G,w) to be stable under convolution is the subconvolutive inequality w^{-q} * w^{-q} is less than or equal to C w^{-q}, where q is the conjugate exponent of p. In work originating with Nikol'skii, this condition was shown to imply that l^p(G,w) forms a convolution algebra. Whether the condition is also necessary for discrete abelian groups appears to have remained open. In this paper, we answer this question in the negative. We construct a discrete abelian group G and a weight w such that l^2(G,w) is stable under convolution, while the above subconvolutive condition fails for every constant C. Consequently, the subconvolutive condition, although sufficient for convolution stability, is not necessary. To the best of the author's knowledge, this yields the first such counterexample on a discrete \emph{abelian} group; the counterexamples previously known on discrete groups rest on a cardinality obstruction that cannot arise in the abelian setting. |
| Adresse de la version officielle : | https://link.springer.com/article/10.1007/s44146-0... |
| Déposant: | Morneau-Guérin, Frédéric |
| Responsable : | Frédéric Morneau-Guérin |
| Dépôt : | 21 sept. 2026 18:30 |
| Dernière modification : | 29 sept. 2026 22:36 |
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