Étude dans un cadre hilbertien des algèbres de Beurling munies d'un poids radial à croissance rapide. (Study in a Hilbertian framework of Beurling algebras equipped with a radial weight having rapid growth) (Q921442)

The paper starts with a discussion of weights \(\omega\) for which \({\mathcal F}^{-1}(L^ 2_{\omega}({\mathbb{R}}^ n))\) is a Banach algebra under pointwise multiplication. For \(\omega\) of polynomial type, these are the well-known \(L^ 2\)-Sobolev spaces, but the main interest here is on radial and non-quasianalytic weights (such as the so-called Gevrey weights \(x\mapsto \exp (| x|^{\alpha})\), for \(0\leq \alpha <1)\), the difficulty being the fact that these spaces are not dilation- invariant. As applications of the basic results (e.g., equivalent norms) the author presents results on traces (over lower-dimensional subspaces), and characterizations of multiplier algebras, as well as a characterization of operating functions on these Banach algebras.