Multipliers on $${{\mathcal {S}}}_{\omega }({{\mathbb {R}}}^N)$$

In this paper we give different estimates between Lebesgue norms of quadratic time-frequency representations. We show that, in some cases, it is not possible to have such bounds in classical $$L^p$$ L p spaces, but the Lebesgue norm needs to be suitably weighted. This leads to consider weights of polynomial type, and, more generally, of ultradifferentiable type, and this, in turn, gives rise to use as functional setting the ultradifferentiable classes. As applications of such estimates we deduce uncertainty principles both of Donoho-Stark type and of local type for representations.

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Convolutors on $${\mathcal S}_{\omega }({\mathbb R}^N)$$

Albanese

Mele

2021

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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In this paper we continue the study of the spaces $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) and $${\mathcal O}_{C,\omega }({\mathbb R}^N)$$ O C , ω ( R N ) undertaken in Albanese and Mele (J Pseudo-Differ Oper Appl, 2021). We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) is the space of convolutors of the space $${\mathcal S}_\omega ({\mathbb R}^N)$$ S ω ( R N ) of the $$\omega $$ ω -ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $${\mathcal S}'_\omega ({\mathbb R}^N)$$ S ω ′ ( R N ) . We also establish that the Fourier transform is an isomorphism from $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) onto $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) . In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $${\mathcal L}_b({\mathcal S}_\omega ({\mathbb R}^N))$$ L b ( S ω ( R N ) ) and the last space is endowed with its natural lc-topology.

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Multipliers on $${{\mathcal {S}}}_{\omega }({{\mathbb {R}}}^N)$$

Abstract: The aim of this paper is to introduce and to study the space $${{\mathcal {O}}}_{M,\omega }({{\mathbb {R}}}^N)$$ O M , ω ( R N … Show more

Cited by 3 publications

References 21 publications

Multiplication and Convolution Topological Algebras in Spaces of ω-Ultradifferentiable Functions of Beurling Type

Multiplication and Convolution Topological Algebras in Spaces of ω-Ultradifferentiable Functions of Beurling Type

Mutual estimates of time-frequency representations and uncertainty principles

Convolutors on $${\mathcal S}_{\omega }({\mathbb R}^N)$$

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